Discrete Framelet Transforms

Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Discrete wavelet/framelet transforms are the backbone of wavelet theory for its applications in a wide scope of areas. In this chapter we study algorithmic aspects and key properties of wavelets and framelets in the discrete setting. First, we introduce a standard (both one-level and multilevel) discrete framelet transform and filter banks. Then we investigate three fundamental properties of a standard discrete framelet transform: perfect reconstruction, sparsity, and stability; these properties are very much desired and crucial in successful applications of wavelets and framelets.

Discrete wavelet/framelet transforms are the backbone of wavelet theory for its applications in a wide scope of areas. In this chapter we study algorithmic aspects and key properties of wavelets and framelets in the discrete setting. First, we introduce a standard (both one-level and multilevel) discrete framelet transform and filter banks. Then we investigate three fundamental properties of a standard discrete framelet transform: perfect reconstruction, sparsity, and stability; these properties are very much desired and crucial in successful applications of wavelets and framelets. Furthermore, we discuss several variants of a standard discrete framelet transform such as nonstationary discrete framelet transforms and undecimated discrete framelet transforms. We fully analyze, in the discrete setting, basic properties related to sparsity such as vanishing moments, sum rules, polynomial reproduction, linear-phase moments, and symmetry. Next, we introduce a general discrete framelet transform that is based on the oblique extension principle, which allows us to increase vanishing moments of high-pass filters in a filter bank. Finally, we describe in detail several algorithms to concretely implement a discrete framelet transform and its variants for processing signals on a bounded interval. We also discuss such algorithms implemented equivalently and completely in the frequency domain using the discrete Fourier transform.

1.1 Perfect Reconstruction of Discrete Framelet Transforms

In this section we introduce one-level (standard) discrete framelet transforms (DFrT). There are three fundamental properties of a discrete framelet transform: perfect reconstruction, sparsity, and stability. In this section we study the perfect reconstruction property.

1.1.1 One-Level Standard Discrete Framelet Transforms

To introduce a discrete framelet transform, we need some definitions and notation. By \(l(\mathbb{Z})\) we denote the linear space of all sequences \(v =\{ v(k)\}_{k\in \mathbb{Z}}: \mathbb{Z} \rightarrow \mathbb{C}\) of complex numbers on \(\mathbb{Z}\). A one-dimensional discrete input signal is often regarded as an element in \(l(\mathbb{Z})\). Similarly, by \(l_{0}(\mathbb{Z})\) we denote the linear space of all sequences \(u =\{ u(k)\}_{k\in \mathbb{Z}}: \mathbb{Z} \rightarrow \mathbb{C}\) on \(\mathbb{Z}\) such that u(k) ≠ 0 only for finitely many \(k \in \mathbb{Z}\). An element in \(l_{0}(\mathbb{Z})\) is often regarded as a finitely supported filter or mask in the literature of wavelet analysis (which is also called a finite-impulse-response filter in engineering). In this book we often use u for a general filter and v for a general signal. It is often convenient to use the formal Fourier series (or symbol) \(\widehat{v}\) of a sequence \(v =\{ v(k)\}_{k\in \mathbb{Z}}\), which is defined to be
$$\displaystyle{ \widehat{v}(\xi ):=\sum _{k\in \mathbb{Z}}v(k)e^{-ik\xi },\qquad \xi \in \mathbb{R}, }$$
(1.1.1)
where i in this book always denotes the imaginary unit. For \(v \in l_{0}(\mathbb{Z})\), the Fourier series \(\widehat{v}\) is a 2π-periodic trigonometric polynomial. See Appendix A for a brief introduction to Fourier series.
A discrete framelet transform can be described using two linear operators—the subdivision operator and the transition operator. For a filter \(u \in l_{0}(\mathbb{Z})\) and a sequence \(v \in l(\mathbb{Z})\), the subdivision operator \(\mathcal{S}_{u}: l(\mathbb{Z}) \rightarrow l(\mathbb{Z})\) is defined to be
$$\displaystyle{ [\mathcal{S}_{u}v](n):= 2\sum _{k\in \mathbb{Z}}v(k)u(n - 2k),\qquad n \in \mathbb{Z} }$$
(1.1.2)
and the transition operator \(\mathcal{T}_{u}: l(\mathbb{Z}) \rightarrow l(\mathbb{Z})\) is defined to be
$$\displaystyle{ [\mathcal{T}_{u}v](n):= 2\sum _{k\in \mathbb{Z}}v(k)\overline{u(k - 2n)},\qquad n \in \mathbb{Z}. }$$
(1.1.3)
The transition operator plays the role of coarsening and frequency-separating a signal to lower resolution levels; while the subdivision operator plays the role of refining and predicting a signal to higher resolution levels.
In terms of Fourier series, the subdivision operator \(\mathcal{S}_{u}\) in (1.1.2) and the transition operator \(\mathcal{T}_{u}\) in (1.1.3) can be equivalently rewritten as
$$\displaystyle{ \widehat{\mathcal{S}_{u}v}(\xi ) = 2\widehat{v}(2\xi )\widehat{u}(\xi ),\qquad \xi \in \mathbb{R} }$$
(1.1.4)
and
$$\displaystyle{ \widehat{\mathcal{T}_{u}v}(\xi ) =\widehat{ v}(\xi /2)\overline{\widehat{u}(\xi /2)} +\widehat{ v}(\xi /2+\pi )\overline{\widehat{u}(\xi /2+\pi )},\qquad \xi \in \mathbb{R} }$$
(1.1.5)
for \(u,v \in l_{0}(\mathbb{Z})\), where \(\overline{c}\) denotes the complex conjugate of a complex number \(c \in \mathbb{C}\). Though most results in this book can be stated and proofs can be carried out equivalently in the space/time domain, to understand wavelets and framelets better, we shall take a frequency/Fourier based approach as the main theme of this book.
A one-level standard discrete framelet transform has two parts: a one-level discrete framelet decomposition and a one-level discrete framelet reconstruction. A set \(\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\) of filters \(\tilde{u}_{0},\ldots,\tilde{u}_{s} \in l_{0}(\mathbb{Z})\) forms a filter bank for decomposition. For a given signal \(v \in l(\mathbb{Z})\), a one-level discrete framelet decomposition employing the filter bank \(\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\) is
$$\displaystyle{ w_{\ell}:= \tfrac{\sqrt{2}} {2} \;\mathcal{T}_{\tilde{u}_{\ell}}v,\qquad \ell = 0,\ldots,s, }$$
(1.1.6)
where w are called sequences of framelet coefficients of the input signal v. We can group all sequences of framelet coefficients in (1.1.6) together and define a discrete framelet analysis (or decomposition) operator \(\widetilde{\mathcal{W}}: l(\mathbb{Z}) \rightarrow (l(\mathbb{Z}))^{1\times (s+1)}\) employing the filter bank \(\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\) as follows:
$$\displaystyle{ \widetilde{\mathcal{W}}v:= \tfrac{\sqrt{2}} {2} (\mathcal{T}_{\tilde{u}_{0}}v,\ldots,\mathcal{T}_{\tilde{u}_{s}}v),\qquad v \in l(\mathbb{Z}). }$$
(1.1.7)
Let {u 0, , u s } with \(u_{0},\ldots,u_{s} \in l_{0}(\mathbb{Z})\) be a filter bank for reconstruction. A one-level discrete framelet reconstruction employing the filter bank {u 0, , u s } can be described by a discrete framelet synthesis (or reconstruction) operator \(\mathcal{V}: (l(\mathbb{Z}))^{1\times (s+1)} \rightarrow l(\mathbb{Z})\) which is defined to be
$$\displaystyle{ \mathcal{V}(w_{0},\ldots,w_{s}):= \frac{\sqrt{2}} {2} \sum _{\ell=0}^{s}\mathcal{S}_{ u_{\ell}}w_{\ell},\qquad w_{0},\ldots,w_{s} \in l(\mathbb{Z}). }$$
(1.1.8)
Fig. 1.1

Diagram of a one-level discrete framelet transform using a filter bank \(\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\) for decomposition and a filter bank {u 0, , u s } for reconstruction

Throughout the book we denote a discrete framelet analysis operator employing the filter bank {u 0, , u s } by \(\mathcal{W}\) and similarly, a discrete framelet synthesis operator employing the filter bank \(\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\) by \(\widetilde{\mathcal{V}}\). See Fig. 1.1 for a diagram of a one-level discrete framelet transform using a filter bank \(\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\) for decomposition and a filter bank {u 0, , u s } for reconstruction.

1.1.2 Perfect Reconstruction of Discrete Framelet Transforms

We say that a filter bank \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\) (or more precisely, its associated discrete framelet transform) has the perfect reconstruction property if \(\mathcal{V}\widetilde{\mathcal{W}}v = v\) for all input signals \(v \in l(\mathbb{Z})\). A necessary and sufficient condition for the perfect reconstruction property of a general one-level discrete framelet transform is as follows:

Theorem 1.1.1

Let \(\tilde{u}_{0},\ldots,\tilde{u}_{s},u_{0},\ldots,u_{s} \in l_{0}(\mathbb{Z})\) . Then the following are equivalent:
  1. (i)
    The filter bank\((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\)has the following perfect reconstruction property: for all\(v \in l(\mathbb{Z})\),
    $$\displaystyle{ v = \mathcal{V}\widetilde{\mathcal{W}}v = \frac{1} {2}\sum _{\ell=0}^{s}\mathcal{S}_{ u_{\ell}}\mathcal{T}_{\tilde{u}_{\ell}}v, }$$
    (1.1.9)
    where\(\widetilde{\mathcal{W}}\)and\(\mathcal{V}\)are defined in (1.1.7) and (1.1.8), respectively.
     
  2. (ii)

    The identity in (1.1.9) holds for all\(v \in l_{0}(\mathbb{Z})\).

     
  3. (iii)
    (1.1.9) holds for the two sequences\(v =\boldsymbol{\delta }\)and\(\boldsymbol{\delta }(\cdot - 1)\), more explicitly,
    $$\displaystyle{ \sum _{\ell=0}^{s}\sum _{ k\in \mathbb{Z}}\overline{\tilde{u}_{\ell}(\gamma +2k)}u_{\ell}(n +\gamma +2k) = \frac{1} {2}\boldsymbol{\delta }(n),\qquad \forall \;\gamma \in \{ 0,1\},\;n \in \mathbb{Z}, }$$
    (1.1.10)
    where\(\boldsymbol{\delta }\)is the Dirac (or Kronecker) sequence such that
    $$\displaystyle{ \boldsymbol{\delta }(0) = 1\qquad \mathit{\mbox{ and}}\qquad \boldsymbol{\delta }(k) = 0,\qquad \forall \;k\neq 0. }$$
    (1.1.11)
     
  4. (iv)
    The following perfect reconstruction condition holds: for all \(\xi \in \mathbb{R}\) ,
    $$\displaystyle{ \widehat{\tilde{u}_{0}}(\xi )\overline{\widehat{u_{0}}(\xi )} + \widehat{\tilde{u}_{1}}(\xi )\overline{\widehat{u_{1}}(\xi )} + \cdots + \widehat{\tilde{u}_{s}}(\xi )\overline{\widehat{u_{s}}(\xi )} = 1, }$$
    (1.1.12)
    $$\displaystyle{ \widehat{\tilde{u}_{0}}(\xi )\overline{\widehat{u_{0}}(\xi +\pi )} +\widehat{\tilde{ u}_{1}}(\xi )\overline{\widehat{u_{1}}(\xi +\pi )} + \cdots +\widehat{\tilde{ u}_{s}}(\xi )\overline{\widehat{u_{s}}(\xi +\pi )} = 0. }$$
    (1.1.13)
     

Proof

Since \(\{\boldsymbol{\delta },\boldsymbol{\delta }(\cdot - 1)\} \subseteq l_{0}(\mathbb{Z}) \subseteq l(\mathbb{Z})\), we trivially have (i)⇒(ii)⇒(iii). By (1.1.2) and (1.1.3), it is straightforward to see that (1.1.9) holds for \(v =\boldsymbol{\delta },\boldsymbol{\delta }(\cdot - 1)\) if and only if (1.1.10) holds. Using (1.1.4) and (1.1.5) for \(v \in l_{0}(\mathbb{Z})\), we have
$$\displaystyle{\frac{1} {2}\widehat{[\mathcal{S}_{u_{\ell}}\mathcal{T}_{\tilde{u}_{\ell}}v]}(\xi ) =\widehat{ \mathcal{T}_{\tilde{u}_{\ell}}v}(2\xi )\widehat{u_{\ell}}(\xi ) =\widehat{ v}(\xi )\overline{\widehat{\tilde{u}_{\ell}}(\xi )}\widehat{u_{\ell}}(\xi ) +\widehat{ v}(\xi +\pi )\overline{\widehat{\tilde{u}_{\ell}}(\xi +\pi )}\widehat{u_{\ell}}(\xi ).}$$
Therefore, for \(v \in l_{0}(\mathbb{Z})\), (1.1.9) holds if and only if
$$\displaystyle{ \overline{\widehat{v}(\xi )} = \overline{\widehat{v}(\xi )}\sum _{\ell=0}^{s}\widehat{\tilde{u}_{\ell}}(\xi )\overline{\widehat{u_{\ell}}(\xi )} + \overline{\widehat{v}(\xi +\pi )}\sum _{\ell =0}^{s}\widehat{\tilde{u}_{\ell}}(\xi +\pi )\overline{\widehat{u_{\ell}}(\xi )}. }$$
(1.1.14)
To prove (iii)⇒(iv), plugging \(v =\boldsymbol{\delta }\) into (1.1.14) and noting \(\widehat{\boldsymbol{\delta }}(\xi ) = 1\), we see that (1.1.14) becomes
$$\displaystyle{1 =\sum _{ \ell=0}^{s}\widehat{\tilde{u}_{\ell}}(\xi )\overline{\widehat{u_{\ell}}(\xi )} +\sum _{ \ell =0}^{s}\widehat{\tilde{u}_{\ell}}(\xi +\pi )\overline{\widehat{u_{\ell}}(\xi )}.}$$
Plugging \(v =\boldsymbol{\delta } (\cdot - 1)\) into (1.1.14) and noting \(\widehat{\boldsymbol{\delta }(\cdot - 1)}(\xi ) = e^{-i\xi }\), we conclude from (1.1.14) that
$$\displaystyle{1 =\sum _{ \ell=0}^{s}\widehat{\tilde{u}_{\ell}}(\xi )\overline{\widehat{u_{\ell}}(\xi )} -\sum _{\ell =0}^{s}\widehat{\tilde{u}_{\ell}}(\xi +\pi )\overline{\widehat{u_{\ell}}(\xi )}.}$$
From these two identities, by adding or subtracting one from the other, we conclude that (1.1.12) and (1.1.13) must hold. Therefore, (iii)⇒(iv).

If (1.1.12) and (1.1.13) are satisfied, then it is straightforward to see that (1.1.14) holds for all \(v \in l_{0}(\mathbb{Z})\). That is, we proved (iv)⇒(ii).

To complete the proof, we prove (ii)⇒(i) by using the locality of the subdivision and transition operators. Let \(v \in l(\mathbb{Z})\). Since all the filters are finitely supported, there exists a positive integer N such that all filters \(u_{0},\ldots,u_{s},\tilde{u}_{0},\ldots,\tilde{u}_{s}\) are supported inside [−N, N]. Let \(n \in \mathbb{Z}\) be fixed. Define a finitely supported sequence \(v_{n} \in l_{0}(\mathbb{Z})\) by v n (k): = v(k) for all \(k \in \mathbb{Z} \cap [n - 2N,n + 2N]\), and v n (k) = 0 otherwise. For all \(k \in \mathbb{Z} \cap [\frac{n-N} {2}, \frac{n+N} {2} ]\), since all involved filters are supported inside [−N, N], we have
$$\displaystyle\begin{array}{rcl} [\mathcal{T}_{\tilde{u}_{\ell}}v](k)& =& 2\sum _{j\in \mathbb{Z}}v(\,j)\overline{\tilde{u}_{\ell}(\,j - 2k)} = 2\sum _{j=n-2N}^{n+2N}v(\,j)\overline{\tilde{u}_{\ell}(\,j - 2k)} {}\\ & =& 2\sum _{j=n-2N}^{n+2N}v_{ n}(\,j)\overline{\tilde{u}_{\ell}(\,j - 2k)} = [\mathcal{T}_{\tilde{u}_{\ell}}v_{n}](k). {}\\ \end{array}$$
Therefore, we deduce that
$$\displaystyle\begin{array}{rcl} \frac{1} {2}\sum _{\ell=0}^{s}[\mathcal{S}_{ u_{\ell}}\mathcal{T}_{\tilde{u}_{\ell}}v](n)& =& \sum _{\ell=0}^{s}\sum _{ k\in \mathbb{Z}\cap [\frac{n-N} {2},\frac{n+N} {2} ]}[\mathcal{T}_{\tilde{u}_{\ell}}v](k)u_{\ell}(n - 2k) {}\\ & =& \sum _{\ell=0}^{s}\sum _{ k\in \mathbb{Z}\cap [\frac{n-N} {2},\frac{n+N} {2} ]}[\mathcal{T}_{\tilde{u}_{\ell}}v_{n}](k)u_{\ell}(n - 2k) {}\\ & =& \frac{1} {2}\sum _{\ell=0}^{s}[\mathcal{S}_{ u_{\ell}}\mathcal{T}_{\tilde{u}_{\ell}}v_{n}](n) = v_{n}(n) = v(n), {}\\ \end{array}$$
where we used (ii) in the second-to-last identity. Hence, (ii)⇒(i). □
The equivalence between items (ii) and (iii) of Theorem 1.1.1 can be easily understood through the following simple relation: for \(m \in \mathbb{Z}\),
$$\displaystyle{\frac{1} {2}\sum _{\ell=0}^{s}\mathcal{S}_{ u_{\ell}}\mathcal{T}_{\tilde{u}_{\ell}}(v(\cdot - 2m)) = \frac{1} {2}\sum _{\ell=0}^{s}\mathcal{S}_{ u_{\ell}}([\mathcal{T}_{\tilde{u}_{\ell}}v](\cdot - m)) = \left [\frac{1} {2}\sum _{\ell=0}^{s}(\mathcal{S}_{ u_{\ell}}\mathcal{T}_{\tilde{u}_{\ell}}v)\right ](\cdot - 2m).}$$
Therefore, if the identity in (1.1.9) holds for a particular sequence v, then it also holds for all \(v(\cdot - 2m),m \in \mathbb{Z}\). Note that the space \(l_{0}(\mathbb{Z})\) is generated by the finite linear combinations of \(\boldsymbol{\delta }(\cdot - k),k \in \mathbb{Z}\). Now it is not surprising to see the equivalence between items (ii) and (iii) of Theorem 1.1.1.
The perfect reconstruction condition in (1.1.12) and (1.1.13) can be equivalently rewritten into the following matrix form:
$$\displaystyle{ \left [\begin{array}{*{10}c} \widehat{\tilde{u}_{0}}(\xi ) &\cdots & \widehat{\tilde{u}_{s}}(\xi ) \\ \widehat{\tilde{u}_{0}}(\xi +\pi )&\cdots &\widehat{\tilde{u}_{s}}(\xi +\pi ) \end{array} \right ]\left [\begin{array}{*{10}c} \widehat{u_{0}}(\xi ) &\cdots & \widehat{u_{s}}(\xi ) \\ \widehat{u_{0}}(\xi +\pi )&\cdots &\widehat{u_{s}}(\xi +\pi ) \end{array} \right ]^{\star } = I_{ 2}, }$$
(1.1.15)
where I 2 denotes the 2 × 2 identity matrix and A denotes the transpose of the complex conjugate of a matrix A, that is, \(A^{\star }:= \overline{A}^{\mathsf{T}}\).

A filter bank satisfying the perfect reconstruction condition in (1.1.15) is called a dual framelet filter bank. It is trivial from (1.1.15) that \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\) is a dual framelet filter bank if and only if \((\{u_{0},\ldots,u_{s}\},\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\})\) is a dual framelet filter bank. In other words, \(\mathcal{V}\widetilde{\mathcal{W}} =\mathrm{ Id}_{\,l(\mathbb{Z})}\) if and only if \(\widetilde{\mathcal{V}}\mathcal{W} =\mathrm{ Id}_{\,l(\mathbb{Z})}\).

In particular, a dual framelet filter bank with s = 1 is called a biorthogonal wavelet filter bank which, by the following result, is a nonredundant filter bank.

Proposition 1.1.2

Let\((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\)be a dual framelet filter bank. Let the discrete framelet analysis operator\(\widetilde{\mathcal{W}}: l(\mathbb{Z}) \rightarrow (l(\mathbb{Z}))^{1\times (s+1)}\)and the discrete framelet synthesis operator\(\mathcal{V}: (l(\mathbb{Z}))^{1\times (s+1)} \rightarrow l(\mathbb{Z})\)be defined in (1.1.7) and (1.1.8), respectively. Then the following statements are equivalent:
  1. (i)

    \(\widetilde{\mathcal{W}}\) is onto.

     
  2. (ii)

    \(\mathcal{V}\) is one-to-one.

     
  3. (iii)

    \(\mathcal{V}\widetilde{\mathcal{W}} =\mathrm{ Id}_{\,l(\mathbb{Z})}\) and \(\widetilde{\mathcal{W}}\mathcal{V} =\mathrm{ Id}_{(l(\mathbb{Z}))^{1\times (s+1)}}\) , that is, \(\mathcal{V}^{-1} =\widetilde{ \mathcal{W}}\) and \(\widetilde{\mathcal{W}}^{-1} = \mathcal{V}\) .

     
  4. (iv)

    s = 1.

     

Proof

It is trivial that (iii) implies both (i) and (ii). Note that \(\mathcal{V}\widetilde{\mathcal{W}} =\mathrm{ Id}_{\,l(\mathbb{Z})}\) follows directly from the perfect reconstruction property. If (i) holds, by \(\widetilde{\mathcal{W}}\mathcal{V}\widetilde{\mathcal{W}}v =\widetilde{ \mathcal{W}}v\) and \(\widetilde{\mathcal{W}}(l(\mathbb{Z}))^{1\times (s+1)} = l(\mathbb{Z})\), we must have \(\widetilde{\mathcal{W}}\mathcal{V} =\mathrm{ Id}_{(l(\mathbb{Z}))^{1\times (s+1)}}\). Thus, (i)⇒(iii). By \(\mathcal{V}\widetilde{\mathcal{W}} =\mathrm{ Id}_{\,l(\mathbb{Z})}\), we have \(\mathcal{V}\widetilde{\mathcal{W}}\mathcal{V} = \mathcal{V}\) and therefore, \(\mathcal{V}(\widetilde{\mathcal{W}}\mathcal{V}-\mathrm{ Id}_{(l(\mathbb{Z}))^{1\times (s+1)}}) = 0\). Since \(\mathcal{V}\) is one-to-one by item (ii), we must have \(\widetilde{\mathcal{W}}\mathcal{V} =\mathrm{ Id}_{(l(\mathbb{Z}))^{1\times (s+1)}}\) and hence, (ii)⇒(iii).

We now prove (iii) ⇔ (iv). If (iii) holds, then \(\widetilde{\mathcal{W}}\mathcal{V} =\mathrm{ Id}_{(l(\mathbb{Z}))^{1\times (s+1)}}\) implies
$$\displaystyle{ \frac{1} {2}\mathcal{T}_{\tilde{u}_{\ell}}\Big(\mathcal{S}_{u_{0}}w_{0} + \cdots + \mathcal{S}_{u_{s}}w_{s}\Big) = w_{\ell},\;\forall \;w_{0},\ldots,w_{s} \in l_{0}(\mathbb{Z}),\,\ell= 0,\ldots,s. }$$
(1.1.16)
Taking Fourier series on both sides of (1.1.16), we see that (1.1.16) is equivalent to
$$\displaystyle\begin{array}{rcl} & & \big[\widehat{u_{0}}(\xi /2)\overline{\widehat{\tilde{u}_{\ell}}(\xi /2)} +\widehat{ u_{0}}(\xi /2+\pi )\overline{\widehat{\tilde{u}_{\ell}}(\xi /2+\pi )}\big]\widehat{w_{0}}(\xi ) + \cdots {}\\ & & \qquad +\big [\widehat{u_{s}}(\xi /2)\overline{\widehat{\tilde{u}_{\ell}}(\xi /2)} +\widehat{ u_{s}}(\xi /2+\pi )\overline{\widehat{\tilde{u}_{\ell}}(\xi /2+\pi )}\big]\widehat{w_{s}}(\xi ) =\widehat{ w_{\ell}}(\xi ) {}\\ \end{array}$$
for all = 0, , s. It is trivial to see that the above identities hold if and only if
$$\displaystyle{\overline{\widehat{u_{m}}(\xi /2)}\widehat{\tilde{u}_{\ell}}(\xi /2) + \overline{\widehat{u_{m}}(\xi /2+\pi )}\widehat{\tilde{u}_{\ell}}(\xi /2+\pi ) =\boldsymbol{\delta } (\ell-m),\qquad \ell,m = 0,\ldots,s,}$$
where \(\boldsymbol{\delta }\) is the Dirac sequence defined in (1.1.11). Consequently, we can rewrite the above identities into the following matrix form:
$$\displaystyle{ \left [\begin{array}{*{10}c} \widehat{u_{0}}(\xi ) &\cdots & \widehat{u_{s}}(\xi ) \\ \widehat{u_{0}}(\xi +\pi )&\cdots &\widehat{u_{s}}(\xi +\pi ) \end{array} \right ]^{\star }\left [\begin{array}{*{10}c} \widehat{\tilde{u}_{0}}(\xi ) &\cdots & \widehat{\tilde{u}_{s}}(\xi ) \\ \widehat{\tilde{u}_{0}}(\xi +\pi )&\cdots &\widehat{\tilde{u}_{s}}(\xi +\pi ) \end{array} \right ] = I_{s+1}, }$$
(1.1.17)
where I s+1 denotes the (s + 1) × (s + 1) identity matrix. Noting that the traces of the matrices on the left-hand sides of (1.1.15) and (1.1.17) must be the same, we conclude from (1.1.15) and (1.1.17) that s = 1. Therefore, (iii)⇒(iv).

Conversely, if s = 1, then the two matrices in (1.1.15) are square matrices and hence (1.1.15) directly implies (1.1.17). By the above argument, (1.1.16) must hold. Using the locality of the subdivision and transition operators as in the proof of Theorem 1.1.1, we see that (1.1.16) holds for all \(w_{0},\ldots,w_{s} \in l(\mathbb{Z})\). Thus, (iii) must hold. □

Consequently, under a biorthogonal wavelet filter bank, any input signal \(v \in l(\mathbb{Z})\) has a nonredundant representation \(v = \mathcal{V}w\) with the unique choice \(w =\widetilde{ \mathcal{W}}v\); while under a dual framelet filter bank with s > 1, an input signal v can be represented as \(v = \mathcal{V}w\) from infinitely many \(w \in (l(\mathbb{Z}))^{1\times (s+1)}\) of framelet coefficients.

Quite often, one only needs to deal with v in the space \(l_{2}(\mathbb{Z})\), which is equipped with the following inner product:
$$\displaystyle{\langle v,w\rangle:=\sum _{k\in \mathbb{Z}}v(k)\overline{w(k)},\qquad v,w \in l_{2}(\mathbb{Z})}$$
and \(\|v\|_{l_{2}(\mathbb{Z})}^{2}:=\langle v,v\rangle <\infty\). For \(v \in l_{2}(\mathbb{Z})\), its Fourier series \(\widehat{v}\) is a 2π-periodic square integrable function on \(\mathbb{R}\) satisfying \(\widehat{v}(\xi +2\pi ) =\widehat{ v}(\xi )\) and \(\frac{1} {2\pi }\int _{-\pi }^{\pi }\vert \widehat{v}(\xi )\vert ^{2}d\xi =\| v\|_{ l_{2}(\mathbb{Z})}^{2} =\sum _{ k\in \mathbb{Z}}\vert v(k)\vert ^{2}\). See Appendix A for a brief introduction to Fourier series.

In the following we explain the role played by the factor \(\frac{\sqrt{ 2}} {2}\) in (1.1.7) and (1.1.8). To do so, we need the following duality relation between the subdivision operator \(\mathcal{S}_{u}\) and the transition operator \(\mathcal{T}_{u}\) acting on the space \(l_{2}(\mathbb{Z})\).

Lemma 1.1.3

Let \(u \in l_{0}(\mathbb{Z})\) be a finitely supported filter on \(\mathbb{Z}\) . Then \(\mathcal{S}_{u}: l_{2}(\mathbb{Z}) \rightarrow l_{2}(\mathbb{Z})\) is the adjoint operator of \(\mathcal{T}_{u}: l_{2}(\mathbb{Z}) \rightarrow l_{2}(\mathbb{Z})\) , that is, \(\mathcal{T}_{u}^{\star } = \mathcal{S}_{u}\) :
$$\displaystyle{ \langle \mathcal{S}_{u}v,w\rangle =\langle \mathcal{T}_{u}^{\star }v,w\rangle:=\langle v,\mathcal{T}_{ u}w\rangle,\qquad \forall \;v,w \in l_{2}(\mathbb{Z}). }$$
(1.1.18)

Proof

Applying (1.1.4) and (1.1.18), we have
$$\displaystyle\begin{array}{rcl} \langle \mathcal{S}_{u}v,w\rangle & =& \frac{1} {\pi } \int _{-\pi }^{\pi }\widehat{v}(2\xi )\widehat{u}(\xi )\overline{\widehat{w}(\xi )}d\xi = \frac{1} {2\pi }\int _{-2\pi }^{2\pi }\widehat{v}(\xi )\widehat{u}(\xi /2)\overline{\widehat{w}(\xi /2)}d\xi {}\\ & =& \frac{1} {2\pi }\int _{-\pi }^{\pi }\widehat{v}(\xi )\overline{\widehat{\mathcal{T}_{ u}w}(\xi )}d\xi =\langle v,\mathcal{T}_{u}w\rangle =\langle \mathcal{T}_{u}^{\star }v,w\rangle. {}\\ \end{array}$$
Hence, (1.1.18) holds. □
Note that the space \((l_{2}(\mathbb{Z}))^{1\times (s+1)}\) is equipped with the following inner product:
$$\displaystyle\begin{array}{rcl} \langle (w_{0},\ldots,w_{s}),(\tilde{w}_{0},\ldots,\tilde{w}_{s})\rangle &:=& \langle w_{0},\tilde{w}_{0}\rangle + \cdots +\langle w_{s},\tilde{w}_{s}\rangle, {}\\ & & w_{0},\ldots,w_{s},\tilde{w}_{0},\ldots,\tilde{w}_{s} \in l_{2}(\mathbb{Z}) {}\\ \end{array}$$
and
$$\displaystyle{\|(w_{0},\ldots,w_{s})\|_{(l_{2}(\mathbb{Z}))^{1\times (s+1)}}^{2}:=\| w_{ 0}\|_{l_{2}(\mathbb{Z})}^{2} + \cdots +\| w_{ s}\|_{l_{2}(\mathbb{Z})}^{2}.}$$
For a filter bank {u 0, , u s }, recall that
$$\displaystyle{ \mathcal{W}: l_{2}(\mathbb{Z}) \rightarrow (l_{2}(\mathbb{Z}))^{1\times (s+1)},\quad \mathcal{W}v:= \tfrac{\sqrt{2}} {2} (\mathcal{T}_{u_{0}}v,\ldots,\mathcal{T}_{u_{s}}v),\quad v \in l_{2}(\mathbb{Z}) }$$
(1.1.19)
and
$$\displaystyle{ \begin{array}{rl} \mathcal{V}:&(l_{2}(\mathbb{Z}))^{1\times (s+1)} \rightarrow l_{2}(\mathbb{Z})\quad \mbox{ with}\quad \\ &\mathcal{V}(w_{0},\ldots,w_{s}):= \tfrac{\sqrt{2}} {2} \sum _{\ell=0}^{s}\mathcal{S}_{ u_{\ell}}w_{\ell},\quad w_{0},\ldots,w_{s} \in l_{2}(\mathbb{Z}). \end{array} }$$
(1.1.20)
The adjoint operators of \(\mathcal{W}\) and \(\mathcal{V}\) are defined to be
$$\displaystyle{ \mathcal{W}^{\star }: (l_{ 2}(\mathbb{Z}))^{1\times (s+1)} \rightarrow l_{ 2}(\mathbb{Z})\qquad \mbox{ through}\qquad \langle v,\mathcal{W}^{\star }w\rangle:=\langle \mathcal{W}v,w\rangle }$$
(1.1.21)
and
$$\displaystyle{ \mathcal{V}^{\star }: l_{ 2}(\mathbb{Z}) \rightarrow (l_{2}(\mathbb{Z}))^{1\times (s+1)}\qquad \mbox{ through}\qquad \langle \mathcal{V}^{\star }v,w\rangle:=\langle v,\mathcal{V}w\rangle }$$
(1.1.22)
for all \(v \in l_{2}(\mathbb{Z})\) and \(w \in (l_{2}(\mathbb{Z}))^{1\times (s+1)}\). By Lemma 1.1.3, it is easy to directly check that \(\mathcal{W}^{\star } = \mathcal{V}\) and \(\mathcal{V}^{\star } = \mathcal{W}\).

The role played by \(\frac{\sqrt{2}} {2}\) in (1.1.7) and (1.1.8) is explained by the following result:

Theorem 1.1.4

Let\(u_{0},\ldots,u_{s} \in l_{0}(\mathbb{Z})\)be finitely supported sequences on\(\mathbb{Z}\). Let\(\mathcal{W}: l_{2}(\mathbb{Z}) \rightarrow (l_{2}(\mathbb{Z}))^{1\times (s+1)}\)be defined in (1.1.19). Then the following are equivalent:
  1. (i)
    \(\|\mathcal{W}v\|_{(l_{2}(\mathbb{Z}))^{1\times (s+1)}}^{2} =\| v\|_{l_{2}(\mathbb{Z})}^{2}\) for all \(v \in l_{2}(\mathbb{Z})\) , that is,
    $$\displaystyle{\|\mathcal{T}_{u_{0}}v\|_{l_{2}(\mathbb{Z})}^{2} + \cdots +\| \mathcal{T}_{ u_{s}}v\|_{l_{2}(\mathbb{Z})}^{2} = 2\|v\|_{ l_{2}(\mathbb{Z})}^{2},\qquad \forall \;v \in l_{ 2}(\mathbb{Z}).}$$
     
  2. (ii)

    \(\langle \mathcal{W}v,\mathcal{W}\tilde{v}\rangle =\langle v,\tilde{v}\rangle\) for all \(v,\tilde{v} \in l_{2}(\mathbb{Z})\) .

     
  3. (iii)

    \(\mathcal{W}^{\star }\mathcal{W} =\mathrm{ Id}_{\,l_{2}(\mathbb{Z})}\) , that is, \(\mathcal{W}^{\star }\mathcal{W}v = v\) for all \(v \in l_{2}(\mathbb{Z})\) .

     
  4. (iv)
    The filter bank {u 0, , u s } satisfies the perfect reconstruction condition:
    $$\displaystyle{ \left [\begin{array}{*{10}c} \widehat{u_{0}}(\xi ) &\cdots & \widehat{u_{s}}(\xi ) \\ \widehat{u_{0}}(\xi +\pi )&\cdots &\widehat{u_{s}}(\xi +\pi )\end{array} \right ]\left [\begin{array}{*{10}c} \widehat{u_{0}}(\xi ) &\cdots & \widehat{u_{s}}(\xi ) \\ \widehat{u_{0}}(\xi +\pi )&\cdots &\widehat{u_{s}}(\xi +\pi )\end{array} \right ]^{\star } = I_{ 2},\quad \xi \in \mathbb{R}. }$$
    (1.1.23)
     

Proof

Obviously, (ii)⇒(i). Note that (i) implies \(\langle \mathcal{W}^{\star }\mathcal{W}v,v\rangle =\langle \mathcal{W}v,\mathcal{W}v\rangle =\langle v,v\rangle\). Using the well-known polarization identity in Exercise 1.2, it is straightforward to see that \(\langle \mathcal{W}v,\mathcal{W}\tilde{v}\rangle =\langle \mathcal{W}^{\star }\mathcal{W}v,\tilde{v}\rangle =\langle v,\tilde{v}\rangle\). Hence, (i)⇒(ii). The equivalence between (ii) and (iii) is trivial. Note that \(\mathcal{W}^{\star } = \mathcal{V}\). The equivalence between (iii) and (iv) follows directly from Theorem 1.1.1. □

A filter bank {u 0, , u s } satisfying the perfect reconstruction condition in (1.1.23) is called a tight framelet filter bank. In particular, a tight framelet filter bank with s = 1 is called an orthogonal wavelet filter bank. By Theorem 1.1.4, if (1.1.23) is satisfied, then the energy is preserved after a framelet decomposition: \(\sum _{\ell=0}^{s}\|w_{\ell}\|_{l_{2}(\mathbb{Z})}^{2} =\| \mathcal{W}v\|_{l_{2}(\mathbb{Z})}^{2} =\| v\|_{l_{2}(\mathbb{Z})}^{2}\) for all \(v \in l_{2}(\mathbb{Z})\), where \((w_{0},\ldots,w_{s}):= \mathcal{W}v\) is the sequence of framelet coefficients.

By Proposition 1.1.2 and Theorem 1.1.4, we have the following result on orthogonal wavelet filter banks.

Proposition 1.1.5

Let {u 0, , u s } be a tight framelet filter bank. Define\(\mathcal{W}\)as in (1.1.19) and\(\mathcal{V}\)as in (1.1.20). Then the following are equivalent:
  1. (1)

    \(\mathcal{W}\) is an invertible orthogonal mapping satisfying \(\langle \mathcal{W}v,\mathcal{W}\tilde{v}\rangle =\langle v,\tilde{v}\rangle\) , \(v,\tilde{v} \in l_{2}(\mathbb{Z})\) .

     
  2. (2)

    \(\mathcal{V}\) is an invertible orthogonal mapping such that for all \(w_{0},\ldots,w_{s},\tilde{w}_{0},\ldots,\tilde{w}_{s} \in l_{2}(\mathbb{Z})\) , \(\langle \mathcal{V}(w_{0},\ldots,w_{s}),\mathcal{V}(\tilde{w}_{0},\ldots,\tilde{w}_{s})\rangle =\langle (w_{0},\ldots,w_{s}),(\tilde{w}_{0},\ldots,\tilde{w}_{s})\rangle.\)

     
  3. (3)

    \(\mathcal{W}^{\star }\mathcal{W} =\mathrm{ Id}_{\,l_{2}(\mathbb{Z})}\) and \(\mathcal{W}\mathcal{W}^{\star } =\mathrm{ Id}_{\,(l_{2}(\mathbb{Z}))^{1\times (s+1)}}\) .

     
  4. (4)

    s = 1.

     

We shall discuss how to design orthogonal or biorthogonal wavelet filter banks in Chap.  2, and tight or dual framelet filter banks in Chap.  3

1.1.3 Some Examples of Wavelet or Framelet Filter Banks

In the following, let us provide a few examples to illustrate various types of filter banks. For a filter \(u =\{ u(k)\}_{k\in \mathbb{Z}}\) such that u(k) = 0 for all \(k \in \mathbb{Z}\setminus [m,n]\) and u(m)u(n) ≠ 0, we denote by \(\mathop{\mathrm{fsupp}}\nolimits (u):= [m,n]\) as its filter support. To list the filter u, we shall adopt the following notation throughout the book:
$$\displaystyle{ u =\{ u(m),u(m + 1),\ldots,u(-1),\underline{\mathbf{u(0)}},u(1),\ldots,u(n - 1),u(n)\}_{[m,n]}, }$$
(1.1.24)
where we underlined and boldfaced the number u(0) to indicate its position at 0.

Example 1.1.1

{u 0, u 1} is an orthogonal wavelet filter bank (called the Haar orthogonal wavelet filter bank), where
$$\displaystyle{ u_{0} =\{\underline{ \tfrac{\mathbf{1}} {\mathbf{2}}}, \tfrac{1} {2}\}_{[0,1]},\quad u_{1} =\{\underline{ -\tfrac{\mathbf{1}} {\mathbf{2}}}, \tfrac{1} {2}\}_{[0,1]}. }$$
(1.1.25)

Example 1.1.2

\((\{\tilde{u}_{0},\tilde{u}_{1}\},\{u_{0},u_{1}\})\) is a biorthogonal wavelet filter bank, where
$$\displaystyle{ \begin{array}{rlll} &\tilde{u}_{0} =\{ -\tfrac{1} {8}, \tfrac{1} {4},\underline{ \tfrac{\mathbf{3}} {\mathbf{4}}}, \tfrac{1} {4},-\tfrac{1} {8}\}_{[-2,2]},\quad &&\tilde{u}_{1} =\{\underline{ \mathbf{-\tfrac{1} {4}}}, \tfrac{1} {2},-\tfrac{1} {4}\}_{[0,2]},\quad \\ &u_{0} =\{ \tfrac{1} {4},\underline{ \tfrac{\mathbf{1}} {\mathbf{2}}}, \tfrac{1} {4}\}_{[-1,1]},\quad \qquad \quad &&u_{1} =\{ -\tfrac{1} {8},\underline{\mathbf{-\tfrac{1} {4}}}, \tfrac{3} {4},-\tfrac{1} {4},-\tfrac{1} {8}\}_{[-1,3]}. \end{array} }$$

Example 1.1.3

{u 0, u 1, u 2} is a tight framelet filter bank, where
$$\displaystyle{u_{0} =\{ \tfrac{1} {4},\underline{ \tfrac{\mathbf{1}} {\mathbf{2}}}, \tfrac{1} {4}\}_{[-1,1]},\quad u_{1} =\{ -\tfrac{\sqrt{2}} {4},\underline{\mathbf{0}}, \tfrac{\sqrt{2}} {4} \}_{[-1,1]},\quad u_{2} =\{ -\tfrac{1} {4},\underline{ \tfrac{\mathbf{1}} {\mathbf{2}}},-\tfrac{1} {4}\}_{[-1,1]}.}$$

Example 1.1.4

\((\{\tilde{u}_{0},\tilde{u}_{1},\tilde{u}_{2}\},\{u_{0},u_{1},u_{2}\})\) is a dual framelet filter bank, where
$$\displaystyle{ \begin{array}{rlllll} &\tilde{u}_{0} =\{\underline{ \tfrac{\mathbf{1}} {\mathbf{2}}}, \tfrac{1} {2}\}_{[0,1]},\qquad \qquad &&\tilde{u}_{1} =\{ -\tfrac{1} {2},\underline{ \tfrac{\mathbf{1}} {\mathbf{2}}}\}_{[-1,0]},\qquad &&\tilde{u}_{2} =\{\underline{ \mathbf{-\tfrac{1} {2}}}, \tfrac{1} {2}\}_{[0,1]}, \\ &u_{0} =\{ \tfrac{1} {8},\underline{ \tfrac{\mathbf{3}} {\mathbf{8}}}, \tfrac{3} {8}, \tfrac{1} {8}\}_{[-1,2]},\quad &&u_{1} =\{ -\tfrac{1} {4},\underline{ \tfrac{\mathbf{1}} {\mathbf{4}}}\}_{[-1,0]},\qquad &&u_{2} =\{ -\tfrac{1} {8},\underline{\mathbf{-\tfrac{3} {8}}}, \tfrac{3} {8}, \tfrac{1} {8}\}_{[-1,2]}. \end{array} }$$
At the end of this section, we illustrate a one-level discrete framelet transform using the Haar orthogonal wavelet filter bank in (1.1.25). Let
$$\displaystyle{ v =\{ -21,-22,-23,-23,-25,38,36,34\}_{[0,7]} }$$
(1.1.26)
be a test input signal. Note that
$$\displaystyle{[\mathcal{T}_{u_{0}}v](n) = v(2n + 1) + v(2n),\qquad [\mathcal{T}_{u_{1}}v](n) = v(2n + 1) - v(2n),\qquad n \in \mathbb{Z}.}$$
Therefore, we have the wavelet coefficients:
$$\displaystyle{w_{0} = \tfrac{\sqrt{2}} {2} \{ - 43,-46,13,70\}_{[0,3]},\qquad w_{1} = \tfrac{\sqrt{2}} {2} \{ - 1,0,63,-2\}_{[0,3]}.}$$
On the other hand, we have
$$\displaystyle{ \begin{array}{rlll} &[\mathcal{S}_{u_{0}}w_{0}](2n) = w_{0}(n),\qquad &&[\mathcal{S}_{u_{0}}w_{0}](2n + 1) = w_{0}(n),\quad n \in \mathbb{Z}, \\ &[\mathcal{S}_{u_{1}}w_{1}](2n) = -w_{1}(n),\qquad &&[\mathcal{S}_{u_{1}}w_{1}](2n + 1) = w_{1}(n),\quad n \in \mathbb{Z}. \end{array} }$$
Hence, we have
$$\displaystyle\begin{array}{rcl} & & \tfrac{\sqrt{2}} {2} \mathcal{S}_{u_{0}}w_{0} = \tfrac{1} {2}\{ - 43,-43,-46,-46,13,13,70,70\}_{[0,7]},\quad {}\\ & & \tfrac{\sqrt{2}} {2} \mathcal{S}_{u_{1}}w_{1} = \tfrac{1} {2}\{1,-1,0,0,-63,63,2,-2\}_{[0,7]}. {}\\ \end{array}$$
Clearly, we have the perfect reconstruction of the original input signal v:
$$\displaystyle{\tfrac{\sqrt{2}} {2} \mathcal{S}_{u_{0}}w_{0} + \tfrac{\sqrt{2}} {2} \mathcal{S}_{u_{1}}w_{1} =\{ -21,-22,-23,-23,-25,38,36,34\}_{[0,7]} = v}$$
and the following energy-preserving identity
$$\displaystyle{\|w_{0}\|_{l_{2}(\mathbb{Z})}^{2} +\| w_{ 1}\|_{l_{2}(\mathbb{Z})}^{2} = 4517 + 1987 = 6504 =\| v\|_{ l_{2}(\mathbb{Z})}^{2}.}$$
The subdivision operator and the transition operator in applications are often implemented through the widely used convolution operation in mathematics and engineering. For \(u \in l_{0}(\mathbb{Z})\) and \(v \in l(\mathbb{Z})\), the convolution uv is defined to be
$$\displaystyle{ [u {\ast} v](n):=\sum _{k\in \mathbb{Z}}u(k)v(n - k),\qquad n \in \mathbb{Z}. }$$
(1.1.27)
By the definition of the convolution in (1.1.27), we note that \(\widehat{u {\ast} v}(\xi ) =\widehat{ u}(\xi )\widehat{v}(\xi )\). To implement the subdivision and transition operators using the convolution operation, we also need the upsampling and downsampling operators on sequences in \(l(\mathbb{Z})\). The downsampling (or decimation) operator \(\downarrow \!\mathsf{d}: l(\mathbb{Z}) \rightarrow l(\mathbb{Z})\) and the upsampling operator \(\uparrow \!\mathsf{d}: l(\mathbb{Z}) \rightarrow l(\mathbb{Z})\) with a sampling factor \(\mathsf{d} \in \mathbb{Z}\setminus \{0\}\) are given by
$$\displaystyle{ [v\! \downarrow \!\mathsf{d}](n):= v(\mathsf{d}n)\quad \mbox{ and}\quad [v\! \uparrow \!\mathsf{d}](n):= \left \{\begin{array}{@{}l@{\quad }l@{}} v(n/\mathsf{d}),\quad &\text{if }n/\mathsf{d}\text{ is an integer},\\ 0, \quad &\text{otherwise}, \end{array} \right. }$$
(1.1.28)
for \(n \in \mathbb{Z}\). For a sequence \(v =\{ v(k)\}_{k\in \mathbb{Z}}\), we denote its complex conjugate sequence reflected about the origin by v, which is defined to be
$$\displaystyle{v^{\star }(k):= \overline{v(-k)},\qquad k \in \mathbb{Z}.}$$
Note that \(\widehat{v^{\star }}(\xi ) = \overline{\widehat{v}(\xi )}\). Now the subdivision operator \(\mathcal{S}_{u}\) in (1.1.2) and the transition operator \(\mathcal{T}_{u}\) in (1.1.3) can be equivalently expressed as follows:
$$\displaystyle{ \mathcal{S}_{u}v = 2(v\! \uparrow \! 2) {\ast} u\quad \mbox{ and}\quad \mathcal{T}_{u}v = 2(v {\ast} u^{\star })\! \downarrow \! 2. }$$
(1.1.29)
For \(u =\{ u(k)\}_{k\in \mathbb{Z}}\) and \(\gamma \in \mathbb{Z}\), we define the associated coset sequence u[γ] of u at the coset \(\gamma +2\mathbb{Z}\) by
$$\displaystyle{ \widehat{u^{[\gamma ]}}(\xi ):=\sum _{ k\in \mathbb{Z}}u(\gamma +2k)e^{-ik\xi },\;\mbox{ i.e.},\;u^{[\gamma ]} = u(\gamma +\cdot )\! \downarrow \! 2 =\{ u(\gamma +2k)\}_{ k\in \mathbb{Z}}. }$$
(1.1.30)
Using the coset sequences of u, we can rewrite (1.1.29) as
$$\displaystyle{[\mathcal{S}_{u}v]^{[0]} = 2v {\ast} u^{[0]},\quad [\mathcal{S}_{ u}v]^{[1]} = 2v {\ast} u^{[1]},\quad \mathcal{T}_{ u}v = 2\big(v^{[0]} {\ast} (u^{[0]})^{\star } + v^{[1]} {\ast} (u^{[1]})^{\star }\big).}$$

1.2 Sparsity of Discrete Framelet Transforms

Sparse representation for smooth or piecewise smooth signals is a highly desired property of a discrete transform in applications. To achieve sparsity, it is desirable to have as many as possible negligible framelet coefficients for smooth signals. In this section, we study several basic mathematical properties that are closely related to sparsity of a discrete framelet transform in the discrete setting, in particular, properties such as vanishing moments, sum rules, polynomial reproduction, linear-phase moments, and symmetry. For the convenience of the reader, basic definitions such as vanishing moments, sum rules, linear-phase moments, and symmetry will be repeated at the beginning of Chap.  2.

1.2.1 Convolution and Transition Operators on Polynomial Spaces

Smooth signals are theoretically modeled by polynomials of various degrees. Let \(\mathsf{p}: \mathbb{R} \rightarrow \mathbb{C}\) be a polynomial, that is, p(x) = j = 0 m p j x j with \(p_{0},\ldots,p_{m} \in \mathbb{C}\) and a nonnegative integer m; if the leading coefficient p m ≠ 0, then we define \(\deg (\mathsf{p}) = m\), which is the degree of the polynomial \(\mathsf{p}\). For the zero polynomial, we use the convention \(\deg (0) = -\infty\). Sampling a polynomial \(\mathsf{p}\) on the integer lattice \(\mathbb{Z}\), we have a polynomial sequence \(\mathsf{p}\vert _{\mathbb{Z}}: \mathbb{Z} \rightarrow \mathbb{C}\) which is given by \([\mathsf{p}\vert _{\mathbb{Z}}](k) = \mathsf{p}(k),k \in \mathbb{Z}\). If a sequence \(v =\{ v(k)\}_{k\in \mathbb{Z}}\) is a polynomial sequence, then a polynomial \(\mathsf{p}\), satisfying v(k) = p(k) for all \(k \in \mathbb{Z}\), is uniquely determined. Therefore, for simplicity of presentation, we shall use \(\mathsf{p}\) to denote both a polynomial function \(\mathsf{p}\) on \(\mathbb{R}\) and its induced polynomial sequence \(\mathsf{p}\vert _{\mathbb{Z}}\) on \(\mathbb{Z}\). One can easily tell them apart from the context. In case of confusion, we explicitly use \(\mathsf{p}\vert _{\mathbb{Z}}\) instead of \(\mathsf{p}\).

Define \(\mathbb{N}_{0}:= \mathbb{N} \cup \{ 0\}\), the set of all nonnegative integers. For \(m \in \mathbb{N}_{0}\), \(\mathbb{P}_{m}\) denotes the space of all polynomials of degree no more than m. In particular, \(\mathbb{P}:= \cup _{m=0}^{\infty }\mathbb{P}_{m}\) denotes the space of all polynomials on \(\mathbb{R}\). For a polynomial \(\mathsf{p}(x) =\sum _{ j=0}^{\infty }p_{j}x^{j} \in \mathbb{P}\) and a smooth function f(ξ), p(n) is the nth derivative of \(\mathsf{p}\) and we use the following polynomial differentiation operator:
$$\displaystyle{ \mathsf{p}\big(x - i\tfrac{d} {d\xi }\big)\mathbf{f}(\xi ):=\sum _{ j=0}^{\infty }p_{ j}\big(x - i\tfrac{d} {d\xi }\big)^{j}\mathbf{f}(\xi ). }$$
(1.2.1)
By the definition of \(\mathsf{p}\big(x - i\tfrac{d} {d\xi }\big)\) in (1.2.1) and the Taylor expansion of p(y + z) at the point y, we deduce \(\mathsf{p}(y + z) =\sum _{ j=0}^{\infty }\mathsf{p}^{(\,j)}(y)\frac{z^{j}} {j!}\) and hence,
$$\displaystyle{ \mathsf{p}\big(x - i\tfrac{d} {d\xi }\big)\mathbf{f}(\xi ) =\sum _{ j=0}^{\infty }\frac{(-i)^{\,j}} {j!} \mathsf{p}^{(\,j)}(x)\mathbf{f}^{(\,j)}(\xi ) =\sum _{ j=0}^{\infty }\frac{x^{j}} {j!} \mathsf{p}^{(\,j)}\big(-i\tfrac{d} {d\xi }\big)\mathbf{f}(\xi ). }$$
(1.2.2)
Using the Leibniz differentiation formula and (1.2.2), we have the following generalized product rule for differentiation:
$$\displaystyle{ \mathsf{p}\big(x - i\tfrac{d} {d\xi }\big)\big(\mathbf{g}(\xi )\mathbf{f}(\xi )\big) =\sum _{ j=0}^{\infty }\frac{(-i)^{\,j}} {j!} \mathbf{g}^{(\,j)}(\xi )\mathsf{p}^{(\,j)}\big(x - i\tfrac{d} {d\xi }\big)\mathbf{f}(\xi ). }$$
(1.2.3)
It follows directly from (1.2.2) and (1.2.3) that
$$\displaystyle{ \Big[\mathsf{p}\big(-i\tfrac{d} {d\xi }\big)(e^{ix\xi }\mathbf{f}(\xi ))\Big]\Big\vert _{\xi =0} =\Big [\mathsf{p}\big(x - i\tfrac{d} {d\xi }\big)\mathbf{f}(\xi )\Big]\Big\vert _{\xi =0}. }$$
(1.2.4)

To study sparsity of a discrete framelet transform, we have to understand how the subdivision operator and the transition operator act on polynomial spaces. Because the subdivision and transition operators can be expressed via the convolution operation, in the following we first study the convolution operation acting on polynomial spaces.

Lemma 1.2.1

Let \(u =\{ u(k)\}_{k\in \mathbb{Z}} \in l_{0}(\mathbb{Z})\) be a finitely supported sequence on \(\mathbb{Z}\) and \(\mathsf{p} \in \mathbb{P}\) . Then \(\mathsf{p} {\ast} u\) is a polynomial sequence satisfying \(\deg (\mathsf{p} {\ast} u)\leqslant \deg (\mathsf{p})\) and
$$\displaystyle{ \begin{array}{rl} [\mathsf{p} {\ast} u](x)&:=\sum _{k\in \mathbb{Z}}\mathsf{p}(x - k)u(k) =\Big [\mathsf{p}\big(x - i\tfrac{d} {d\xi }\big)\widehat{u}(\xi )\Big]\Big\vert _{\xi =0} \\ & =\sum _{ j=0}^{\infty }\frac{(-i)^{\,j}} {j!} \mathsf{p}^{(\,j)}(x)\widehat{u}^{\,(\,j)}(0) =\sum _{ j=0}^{\infty }\frac{x^{j}} {j!} \Big[\mathsf{p}^{(\,j)}\big(-i\tfrac{d} {d\xi }\big)\widehat{u}(\xi )\Big]\Big\vert _{\xi =0}. \end{array} }$$
(1.2.5)
Moreover, \(\mathsf{p} {\ast} (u\! \uparrow \! 2) = [\mathsf{p}(2\cdot ) {\ast} u](2^{-1}\cdot )\) ,
$$\displaystyle{ \mathsf{p}^{(\,j)} {\ast} u = [\mathsf{p} {\ast} u]^{(\,j)},\quad \forall \;j \in \mathbb{N}_{ 0}\quad \mathit{\mbox{ and}}\quad \mathsf{p}(\cdot - y) {\ast} u = [\mathsf{p} {\ast} u](\cdot - y),\quad \forall \;y \in \mathbb{R}. }$$
(1.2.6)

Proof

By the Taylor expansion \(\mathsf{p}(x - k) =\sum _{ j=0}^{\infty }\mathsf{p}^{(\,j)}(x)\frac{(-k)^{\,j}} {j!}\), we have
$$\displaystyle{[\mathsf{p}{\ast}u](x) =\sum _{k\in \mathbb{Z}}\mathsf{p}(x-k)u(k) =\sum _{k\in \mathbb{Z}}\sum _{j=0}^{\infty }\mathsf{p}^{(\,j)}(x)u(k)\frac{(-k)^{\,j}} {j!} =\sum _{ j=0}^{\infty }\mathsf{p}^{(\,j)}(x)\sum _{ k\in \mathbb{Z}}u(k)\frac{(-k)^{\,j}} {j!}.}$$
By \(\widehat{u}(\xi ) =\sum _{k\in \mathbb{Z}}u(k)e^{-ik\xi }\), we have \(\widehat{u}^{\,(\,j)}(0) =\sum _{k\in \mathbb{Z}}u(k)(-ik)^{\,j} = i^{j}\sum _{k\in \mathbb{Z}}u(k)(-k)^{\,j}\). Now we conclude that
$$\displaystyle{ [\mathsf{p} {\ast} u](x) =\sum _{ j=0}^{\infty }\frac{(-i)^{\,j}} {j!} \mathsf{p}^{(\,j)}(x)\widehat{u}^{\,(\,j)}(0). }$$
(1.2.7)
Therefore, (1.2.5) and (1.2.6) follow directly from (1.2.2) and (1.2.7). □
For smooth functions f and g, we shall use the following big \(\mathbb{O}\) notation:
$$\displaystyle{ \mathbf{f}(\xi ) = \mathbf{g}(\xi ) +\mathbb{ O}(\vert \xi -\xi _{0}\vert ^{m}),\qquad \xi \rightarrow \xi _{ 0} }$$
(1.2.8)
to mean that the derivatives of f and g at ξ = ξ 0 agree to the orders up to m − 1:
$$\displaystyle{\mathbf{f}^{(\,j)}(\xi _{ 0}) = \mathbf{g}^{(\,j)}(\xi _{ 0}),\qquad \forall \;j = 0,\ldots,m - 1.}$$

For a polynomial \(\mathsf{p} \in \mathbb{P}_{m-1}\) of degree less than m, by (1.2.5), it is evident that the polynomial pu depends only on the values \(\widehat{u}(0),\widehat{u}'(0),\ldots,\widehat{u}^{(m-1)}(0)\) of \(\widehat{u}\) at the origin. Consequently, if two sequences \(u,v \in l_{0}(\mathbb{Z})\) satisfy \(\widehat{u}(\xi ) =\widehat{ v}(\xi ) +\mathbb{ O}(\vert \xi \vert ^{m})\) as ξ → 0, then pu = pv for all \(\mathsf{p} \in \mathbb{P}_{m-1}\). For simplicity, in this book we shall frequently use the big \(\mathbb{O}\) notation in (1.2.8).

The action of the transition operator on polynomial spaces is as follows.

Theorem 1.2.2

Let \(u \in l_{0}(\mathbb{Z})\) be a finitely supported sequence on \(\mathbb{Z}\) . For \(\mathsf{p} \in \mathbb{P}\) ,
$$\displaystyle{ \mathcal{T}_{u}\mathsf{p} = 2[\mathsf{p} {\ast} u^{\star }](2\cdot ) = 2\mathsf{p}(2\cdot ) {\ast} u_{\mathsf{ p}} =\sum _{ j=0}^{\infty }\frac{2(-i)^{\,j}} {j!} \mathsf{p}^{(\,j)}(2\cdot )\overline{\widehat{u}^{\,(\,j)}(0)}, }$$
(1.2.9)
where \(u_{\mathsf{p}} \in l_{0}(\mathbb{Z})\) is any finitely supported sequence on \(\mathbb{Z}\) such that
$$\displaystyle{\widehat{u_{\mathsf{p}}}(\xi ) = \overline{\widehat{u}(\xi /2)} +\mathbb{ O}(\vert \xi \vert ^{\deg (\mathsf{p})+1}),\qquad \xi \rightarrow 0.}$$
In particular, for any positive integer \(m \in \mathbb{N}\) , the following are equivalent:
  1. (1)

    \(\mathcal{T}_{u}\mathsf{p} = 0\) for all polynomial sequences \(\mathsf{p} \in \mathbb{P}_{m-1}\) .

     
  2. (2)

    \(\mathcal{T}_{u}\mathsf{q} = 0\) for some polynomial sequence \(\mathsf{q}\) with \(\deg (\mathsf{q}) = m - 1\) .

     
  3. (3)

    \(\widehat{u}(\xi ) =\mathbb{ O}(\vert \xi \vert ^{m})\)as ξ → 0, that is,\(\widehat{u}^{\,(\,j)}(0) = 0\)for all j = 0, , m − 1.

     
  4. (4)

    \(\widehat{u}(\xi ) = (1 - e^{-i\xi })^{m}\mathbf{Q}(\xi )\)for some 2π-periodic trigonometric polynomialQ.

     

Proof

Since \(\mathcal{T}_{u}\mathsf{p} = 2(\mathsf{p} {\ast} u^{\star })\! \downarrow \! 2 = 2[\mathsf{p} {\ast} u^{\star }](2\cdot )\), by Lemma 1.2.1, we see that \(\mathcal{T}_{u}\mathsf{p}\) is a polynomial sequence and
$$\displaystyle{2[\mathsf{p} {\ast} u^{\star }](2\cdot ) =\sum _{ j=0}^{\infty }\frac{2(-i)^{\,j}} {j!} \mathsf{p}^{(\,j)}(2\cdot )\overline{\widehat{u}^{\,(\,j)}(0)} =\sum _{ j=0}^{\infty }\frac{(-i)^{\,j}} {j!} [\mathsf{p}(2\cdot )]^{(\,j)}2^{1-j}\overline{\widehat{u}^{\,(\,j)}(0)}.}$$
By \(\widehat{u_{\mathsf{p}}}^{(\,j)}(0) = 2^{-j}\overline{\widehat{u}^{\,(\,j)}(0)}\) for all \(j = 0,\ldots,\deg (\mathsf{p})\), the identities in (1.2.9) follow directly from (1.2.5). From (1.2.5), we see that
$$\displaystyle{ \mathcal{T}_{u}\mathsf{p}^{(\,j)} = 2^{j+1}\mathsf{p}^{(\,j)}(2\cdot ){\ast}\mathring{u},\qquad \forall \;\mathsf{p} \in \mathbb{P}_{ m-1},\;j \in \mathbb{N}_{0}, }$$
(1.2.10)
where \(\mathring{u}\) is any finitely supported sequence satisfying \(\widehat{\mathring{u}}(\xi ) = \overline{\widehat{u}(\xi /2)} +\mathbb{ O}(\vert \xi \vert ^{m})\) as ξ → 0. For a polynomial q with \(\deg (\mathsf{q}) = m - 1\), the set {q, q′, , q(m−1)} is a basis for \(\mathbb{P}_{m-1}\). Now the equivalence among items (1)–(4) is a direct consequence of (1.2.9) and (1.2.10). □

We say that a filter u (or its Fourier series \(\widehat{u}\) ) has m vanishing moments if any of items (1)–(4) in Theorem 1.2.2 holds. The notion of vanishing moments is important for sparse framelet expansions, since most framelet coefficients are identically zero for any input signal which is a polynomial to certain degree. More precisely, suppose that u has m vanishing moments. For a signal v, if v agrees with some polynomial of degree less than m on the support of u(⋅ − 2n), then by the definition of the transition operator and the definition of vanishing moments, we have \([\mathcal{T}_{u}v](n) = 0\).

Note that \(\mathcal{T}_{u}\mathbb{P}_{m-1} \subseteq \mathbb{P}_{m-1}\) for all \(m \in \mathbb{N}\). In particular, \(\mathcal{T}_{u}\mathbb{P}_{m-1} = \mathbb{P}_{m-1}\) if \(\widehat{u}(0)\neq 0\). Moreover, all the eigenvalues of \(\mathcal{T}_{u}\vert _{\mathbb{P}_{m-1}}\) are \(2\overline{\widehat{u}(0)},\ldots,2^{m}\overline{\widehat{u}(0)}\) (see Exercise 1.15).

1.2.2 Subdivision Operator on Polynomial Spaces

We now investigate the subdivision operator acting on polynomial spaces. In contrast to the case of the transition operator, \(\mathcal{S}_{u}\mathsf{p}\) is not always a polynomial sequence for an input polynomial sequence \(\mathsf{p}\). A simple example is p = 1 and u = {1}[0,0] (that is, \(u =\boldsymbol{\delta }\)). Then \([\mathcal{S}_{u}\mathsf{p}]^{[0]}:= [\mathcal{S}_{u}\mathsf{p}](2\cdot ) = 2\) and \([\mathcal{S}_{u}\mathsf{p}]^{[1]}:= [\mathcal{S}_{u}\mathsf{p}](2 \cdot +1) = 0\).

Lemma 1.2.3

Let \(u =\{ u(k)\}_{k\in \mathbb{Z}} \in l_{0}(\mathbb{Z})\) and \(\mathsf{q}\) be a polynomial. Then the following are equivalent:
  1. (i)

    \(\sum _{k\in \mathbb{Z}}\mathsf{q}(-\tfrac{1} {2} - k)u(1 + 2k) =\sum _{k\in \mathbb{Z}}\mathsf{q}(-k)u(2k)\) , i.e., \((\mathsf{q} {\ast} u^{[1]})(-\tfrac{1} {2}) = (\mathsf{q} {\ast} u^{[0]})(0)\) .

     
  2. (ii)

    \([\mathsf{q}(-i\frac{d} {d\xi })(e^{-i\xi /2}\widehat{u^{[1]}}(\xi ))]\vert _{\xi =0} = [\mathsf{q}(-i\frac{d} {d\xi })\widehat{u^{[0]}}(\xi )]\vert _{\xi =0}\) .

     
  3. (iii)

    \([\mathsf{q}(-\frac{i} {2} \frac{d} {d\xi })\widehat{u}(\xi )]\vert _{\xi =\pi } = 0\) .

     

Proof

(i) ⇔ (ii) follows directly from
$$\displaystyle\begin{array}{rcl} \big[\mathsf{q}(-i\tfrac{d} {d\xi })(e^{-i\xi \gamma /2}\widehat{u^{[\gamma ]}}(\xi ))\big]\big\vert _{\xi =0}& =& \big[\mathsf{q}(-\tfrac{\gamma } {2} - i\tfrac{d} {d\xi })\widehat{u^{[\gamma ]}}(\xi )\big]\big\vert _{\xi =0} = [\mathsf{q} {\ast} u^{[\gamma ]}](-\tfrac{\gamma } {2}) {}\\ & =& \sum _{k\in \mathbb{Z}}\mathsf{q}(-\tfrac{\gamma }{2} - k)u(\gamma +2k) {}\\ \end{array}$$
for \(\gamma \in \mathbb{Z}\), where we used (1.2.4) and (1.2.5). By \(\widehat{u}(\xi ) =\widehat{ u^{[0]}}(2\xi ) + e^{-i\xi }\widehat{u^{[1]}}(2\xi )\), we have \(\widehat{u}(2^{-1}\xi +\pi ) =\widehat{ u^{[0]}}(\xi ) - e^{-i\xi /2}\widehat{u^{[1]}}(\xi )\). Now item (ii) is equivalent to \([\mathsf{q}(-i\frac{d} {d\xi })\widehat{u}(2^{-1}\xi +\pi )]\vert _{\xi =0} = 0\), which is simply item (iii). □

A necessary and sufficient condition for \(\mathcal{S}_{u}\mathsf{p} \in \mathbb{P}\) is as follows.

Theorem 1.2.4

Let \(u =\{ u(k)\}_{k\in \mathbb{Z}} \in l_{0}(\mathbb{Z})\) be a finitely supported sequence on \(\mathbb{Z}\) and \(\mathsf{p} \in \mathbb{P}\) be a polynomial. Then the following are equivalent:
  1. (1)

    \(\mathcal{S}_{u}\mathsf{p}\) is a polynomial sequence, i.e., \(\mathcal{S}_{u}\mathsf{p} \in \mathbb{P}\) .

     
  2. (2)

    \(\sum _{k\in \mathbb{Z}}\mathsf{p}^{(\,j)}(-\tfrac{1} {2} - k)u(1 + 2k) =\sum _{k\in \mathbb{Z}}\mathsf{p}^{(\,j)}(-k)u(2k)\) for all \(j \in \mathbb{N}_{0}\) .

     
  3. (3)

    \([\mathsf{p}^{(\,j)}(-i\frac{d} {d\xi })(e^{-i\xi /2}\widehat{u^{[1]}}(\xi ))]\vert _{\xi =0} = [\mathsf{p}^{(\,j)}(-i\frac{d} {d\xi })\widehat{u^{[0]}}(\xi )]\vert _{\xi =0}\) for all \(j \in \mathbb{N}_{0}\) .

     
  4. (4)

    \([\mathsf{p}^{(\,j)}(-\frac{1} {2} - i\frac{d} {d\xi })\widehat{u^{[1]}}(\xi )]\vert _{\xi =0} = [\mathsf{p}^{(\,j)}(-i\frac{d} {d\xi })\widehat{u^{[0]}}(\xi )]\vert _{\xi =0}\) for all \(j \in \mathbb{N}_{0}\) .

     
  5. (5)

    \([\mathsf{p}^{(\,j)}(-\frac{i} {2} \frac{d} {d\xi })\widehat{u}(\xi )]\vert _{\xi =\pi } = 0\) for all nonnegative integers \(j \in \mathbb{N}_{0}\) .

     
Moreover, if any of the above items (1)–(5) holds, then \(\deg (\mathcal{S}_{u}\mathsf{p})\leqslant \deg (\mathsf{p})\) ,
$$\displaystyle{ \mathcal{S}_{u}\mathsf{p} =\big (\mathsf{p}(2^{-1}\cdot )\big) {\ast} u =\sum _{ j=0}^{\infty }\frac{(-i)^{\,j}} {2^{j}j!} \mathsf{p}^{(\,j)}(2^{-1}\cdot )\widehat{u}^{\,(\,j)}(0), }$$
(1.2.11)
and
$$\displaystyle\begin{array}{rcl} & & \mathcal{S}_{u}(\mathsf{p}^{(\,j)}) = \mathsf{p}^{(\,j)}(2^{-1}\cdot ) {\ast} u = 2^{j}[\mathcal{S}_{ u}\mathsf{p}]^{(\,j)},\qquad j \in \mathbb{N}_{ 0}, {}\\ & & \mathcal{S}_{u}(\mathsf{p}(\cdot - y)) = \mathsf{p}(2^{-1} \cdot -y) {\ast} u = [\mathcal{S}_{ u}\mathsf{p}](\cdot - 2y),\qquad y \in \mathbb{R}. {}\\ \end{array}$$

Proof

By the definition of the subdivision operator \(\mathcal{S}_{u}\) in (1.1.2), for \(n,\gamma \in \mathbb{Z}\),
$$\displaystyle{[\mathcal{S}_{u}\mathsf{p}](\gamma +2n) = 2\sum _{m\in \mathbb{Z}}\mathsf{p}(m)u(\gamma +2n - 2m) = 2\sum _{k\in \mathbb{Z}}\mathsf{p}(2^{-1}(\gamma +2n) - \tfrac{\gamma } {2} - k)u(\gamma +2k).}$$
Hence, \([\mathcal{S}_{u}\mathsf{p}](\gamma +2\cdot )\) is a polynomial sequence on each coset for every \(\gamma \in \mathbb{Z}\). Now it is easy to see that \(\mathcal{S}_{u}\mathsf{p}\) is a polynomial sequence if and only if \(\sum _{k\in \mathbb{Z}}\mathsf{p}(\cdot - \tfrac{\gamma } {2} - k)u(\gamma +2k)\) is independent of γ. Using the Taylor expansion of \(\mathsf{p}\), we have
$$\displaystyle\begin{array}{rcl} \sum _{k\in \mathbb{Z}}\mathsf{p}(x - \tfrac{\gamma } {2} - k)u(\gamma +2k)& =& \sum _{k\in \mathbb{Z}}\sum _{j=0}^{\infty }\frac{x^{j}} {j!} \mathsf{p}^{(\,j)}(-\tfrac{\gamma } {2} - k)u(\gamma +2k) {}\\ & =& \sum _{j=0}^{\infty }\frac{x^{j}} {j!} \sum _{k\in \mathbb{Z}}\mathsf{p}^{(\,j)}(-\tfrac{\gamma } {2} - k)u(\gamma +2k). {}\\ \end{array}$$
Hence, the sequence \(\sum _{k\in \mathbb{Z}}\mathsf{p}(\cdot - \tfrac{\gamma } {2} - k)u(\gamma +2k)\) is independent of γ if and only if all \(\sum _{k\in \mathbb{Z}}\mathsf{p}^{(\,j)}(-\tfrac{\gamma }{2} - k)u(\gamma +2k)\), \(j \in \mathbb{N}_{0}\) are independent of γ, which are obviously equivalent to the conditions in item (2). Thus, we proved (1) ⇔ (2). Moreover, when \(\mathcal{S}_{u}\mathsf{p} \in \mathbb{P}\), the above argument also yields
$$\displaystyle{\mathcal{S}_{u}\mathsf{p} = 2\sum _{k\in \mathbb{Z}}\mathsf{p}(2^{-1} \cdot -\tfrac{\gamma } {2} - k)u(\gamma +2k) =\sum _{k\in \mathbb{Z}}\mathsf{p}(2^{-1}(\cdot - k))u(k),\qquad \forall \;\gamma \in \mathbb{Z},}$$
from which we see that (1.2.11) holds. The equivalence among (2)–(5) follows directly from Lemma 1.2.3. □

For the subdivision operator acting on polynomial spaces, we have

Theorem 1.2.5

Let \(u =\{ u(k)\}_{k\in \mathbb{Z}}\) . For \(m \in \mathbb{N}\) , the following are equivalent:
  1. (1)

    \(\mathcal{S}_{u}\mathbb{P}_{m-1} \subseteq \mathbb{P}\) .

     
  2. (2)

    \(\mathcal{S}_{u}\mathsf{q} \in \mathbb{P}\) for some polynomial \(\mathsf{q} \in \mathbb{P}\) with \(\deg (\mathsf{q}) = m - 1\) .

     
  3. (3)

    \(\mathcal{S}_{u}\mathbb{P}_{m-1} \subseteq \mathbb{P}_{m-1}\) .

     
  4. (4)
    \(\widehat{u}^{\,(\,j)}(\pi ) = 0\)for all j = 0, , m − 1, in other words,
    $$\displaystyle{ \widehat{u}(\xi +\pi ) =\mathbb{ O}(\vert \xi \vert ^{m}),\qquad \xi \rightarrow 0. }$$
    (1.2.12)
     
  5. (5)

    \(\widehat{u}(\xi ) = (1 + e^{-i\xi })^{m}\mathbf{Q}(\xi )\)for some 2π-periodic trigonometric polynomialQ.

     
  6. (6)
    \([e^{-i\xi /2}\widehat{u^{[1]}}(\xi )]^{(\,j)}(0) = [\widehat{u^{[0]}}(\xi )]^{(\,j)}(0)\)for all j = 0, , m − 1, that is,
    $$\displaystyle{ e^{-i\xi /2}\widehat{u^{[1]}}(\xi ) =\widehat{ u^{[0]}}(\xi ) +\mathbb{ O}(\vert \xi \vert ^{m}),\qquad \xi \rightarrow 0, }$$
    (1.2.13)
    or its equivalent form in the space/time domain:
    $$\displaystyle{\sum _{k\in \mathbb{Z}}u(1 + 2k)(1 + 2k)^{j} =\sum _{ k\in \mathbb{Z}}u(2k)(2k)^{j},\qquad \forall \;j = 0,\ldots,m - 1.}$$
     
In particular, if (1.2.12) holds, then for all\(\mathsf{p} \in \mathbb{P}_{m-1}\)and\(v \in l_{0}(\mathbb{Z})\),
$$\displaystyle{ \mathcal{S}_{u}(\mathsf{p} {\ast} v) = 2^{-1}\mathsf{p}(2^{-1}\cdot ) {\ast} [\mathcal{S}_{ u}v] =\sum _{ j=0}^{\infty }\frac{(-i)^{\,j}} {2^{j}j!} \mathsf{p}^{(\,j)}(2^{-1}\cdot )[\widehat{u}\,\widehat{v}(2\cdot )]^{(\,j)}(0), }$$
(1.2.14)
and furthermore,\(\mathcal{S}_{u}\mathbb{P}_{m-1} = \mathbb{P}_{m-1}\)if\(\widehat{u}(0)\neq 0\).

Proof

(1)⇒(2) is obvious. By Theorem  1.2.4, if \(\mathcal{S}_{u}\mathsf{p} \in \mathbb{P}\), then \(\mathcal{S}_{u}\mathsf{p}^{(\,j)} \in \mathbb{P}\) for all \(j \in \mathbb{N}_{0}\). Since {q, q′, , q(m−1)} is a basis for \(\mathbb{P}_{m-1}\), we now see that (2)⇒(1). The equivalence between (1) and (3) follows from Theorem 1.2.4.

Applying Theorem 1.2.4 with p ∈ {1, x, , x m−1} which is a basis for \(\mathbb{P}_{m-1}\), we see that (3) ⇔ (4). (4) ⇔ (5) is trivial. By Theorem 1.2.4 or a direct proof, we have (5) ⇔ (6).

By (1.2.11), it is straightforward to see that (1.2.14) holds. □

We say that a filter u (or its Fourier series \(\widehat{u}\) ) has m sum rules if any of items (1)–(6) in Theorem 1.2.5 is satisfied. If u has m sum rules, then \(\mathcal{S}_{u}\mathbb{P}_{m-1} \subseteq \mathbb{P}_{m-1}\) and all the eigenvalues of \(\mathcal{S}_{u}\vert _{\mathbb{P}_{m-1}}\) are \(\widehat{u}(0),2^{-1}\widehat{u}(0),\ldots,2^{1-m}\widehat{u}(0)\) (see Exercise 1.16).

1.2.3 Linear-Phase Moments and Symmetry Property of Filters

For certain applications, the image of a polynomial under a convolution operation is required to be exactly itself or its translated version. For this purpose, we have the following result:

Lemma 1.2.6

Let\(u \in l_{0}(\mathbb{Z})\)be a finitely supported sequence on\(\mathbb{Z}\). Let\(\mathsf{p}\)be a polynomial and define\(m:=\deg (\mathsf{p}) + 1\). For a real number\(c \in \mathbb{R}\), the identity\(\mathsf{p} {\ast} u = \mathsf{p}(\cdot - c)\)holds if and only if u has m linear-phase moments with phase c:
$$\displaystyle{ \widehat{u}(\xi ) = e^{-ic\xi } +\mathbb{ O}(\vert \xi \vert ^{m}),\qquad \xi \rightarrow 0. }$$
(1.2.15)

Proof

By Lemma 1.2.1, we have (1.2.7). On the other hand, using the Taylor expansion of \(\mathsf{p}\), we have
$$\displaystyle{ \mathsf{p}(x - c) =\sum _{ j=0}^{m-1}\mathsf{p}^{(\,j)}(x)\frac{(-c)^{\,j}} {j!} =\sum _{ j=0}^{m-1}\frac{(-i)^{\,j}} {j!} \mathsf{p}^{(\,j)}(x)(-ic)^{\,j}. }$$
(1.2.16)
Comparing the coefficients of p( j), j = 0, , m − 1 in both (1.2.7) and (1.2.16), we see that pu = p(⋅ − c) if and only if \(\widehat{u}^{\,(\,j)}(0) = (-ic)^{\,j}\) for all j = 0, , m − 1, which can be equivalently rewritten as (1.2.15). □

If a filter has linear-phase moments, then the action of the subdivision operator and the transition operator on polynomial spaces has some particular structure.

Proposition 1.2.7

Let \(u \in l_{0}(\mathbb{Z})\) and \(c \in \mathbb{R}\) . Then u has m linear-phase moments with phase c if and only if \(\mathcal{T}_{u}\mathsf{p} = 2\mathsf{p}(2 \cdot +c)\) for all \(\mathsf{p} \in \mathbb{P}_{m-1}\) (or for some polynomial \(\mathsf{p}\) with \(\deg (\mathsf{p}) = m - 1\) ). Similarly, u has m sum rules and m linear-phase moments with phase c if and only if \(\mathcal{S}_{u}\mathsf{p} = \mathsf{p}(2^{-1}(\cdot - c))\) for all \(\mathsf{p} \in \mathbb{P}_{m-1}\) (or for some polynomial \(\mathsf{p}\) with \(\deg (\mathsf{p}) = m - 1\) ).

Proof

The first part is a direct consequence of (1.2.9) in Theorem 1.2.2 and Lemma 1.2.6. The second part is a direct consequence of (1.2.11) and Theorem 1.2.5. □

We now discuss symmetry property which is desirable in many applications. We say that a filter or a sequence \(u =\{ u(k)\}_{k\in \mathbb{Z}}: \mathbb{Z} \rightarrow \mathbb{C}\) has symmetry if
$$\displaystyle{ u(c - k) =\epsilon u(k),\qquad \forall \;k \in \mathbb{Z} }$$
(1.2.17)
with \(c \in \mathbb{Z}\) and ε ∈ {−1, 1}. A filter u is symmetric about the point \(\frac{c} {2}\) if (1.2.17) holds with ε = 1, and antisymmetric about the point \(\frac{c} {2}\) if (1.2.17) holds with ε = −1. We call \(\frac{c} {2}\) the symmetry center of the filter u, which is simply the center of its filter support \(\mathop{\mathrm{fsupp}}\nolimits (u)\). Recall that \(\mathop{\mathrm{fsupp}}\nolimits (u) = [m,n]\) if u vanishes outside [m, n] and u(m)u(n) ≠ 0. It is often convenient to use a symmetry operatorS to record the symmetry type of a filter having symmetry. For this purpose, we define
$$\displaystyle{ [\mathsf{S}\,\widehat{u}](\xi ):= \frac{\widehat{u}(\xi )} {\widehat{u}(-\xi )},\qquad \xi \in \mathbb{R}. }$$
(1.2.18)
Now it is straightforward to see that (1.2.17) holds if and only if \([\mathsf{S}\,\widehat{u}](\xi ) =\epsilon e^{-ic\xi }\). It is easy to see that \([\mathsf{S}\,\widehat{u(\cdot - m)}](\xi ) = [\mathsf{S}\,\widehat{u}](\xi )e^{-i2m\xi }\) for any integer m. Consequently, up to an integer shift, there are essentially four types of symmetries \(\mathsf{S}\,\widehat{u}(\xi ) =\epsilon e^{-ic\xi }\) with c ∈ {0, 1} and ε ∈ {−1, 1}.
Since in this book we address both real-valued and complex-valued filters, there is a closely related notion of symmetry for complex-valued filters. We say that a filter or a sequence \(u =\{ u(k)\}_{k\in \mathbb{Z}}: \mathbb{Z} \rightarrow \mathbb{C}\) has complex symmetry if
$$\displaystyle{ u(c - k) =\epsilon \overline{u(k)},\qquad \forall \;k \in \mathbb{Z} }$$
(1.2.19)
with \(c \in \mathbb{Z}\) and ε ∈ {−1, 1}. That is, u(k) = εu(c + k) for all \(k \in \mathbb{Z}\). Define a complex symmetry operator\(\mathbb{S}\) by
$$\displaystyle{ [\mathbb{S}\widehat{u}](\xi ):= \frac{\widehat{u}(\xi )} {\overline{\widehat{u}(\xi )}},\qquad \xi \in \mathbb{R}. }$$
(1.2.20)
Then a filter u has complex symmetry in (1.2.19) if and only if \([\mathbb{S}\widehat{u}](\xi ) =\epsilon e^{-ic\xi }\). It is trivial to see that a filter u is real-valued if and only if \(\overline{\widehat{u}(\xi )} =\widehat{ u}(-\xi )\). Therefore, for a real-valued filter u, there is no difference between symmetry and complex symmetry since \(\mathsf{S}\,\widehat{u} = \mathbb{S}\widehat{u}\). We say that u has essential complex symmetry if (1.2.19) holds with \(c \in \mathbb{Z}\) and \(\epsilon \in \mathbb{T}:=\{\zeta \in \mathbb{C}\;:\; \vert \zeta \vert = 1\}\).

In the following, we make some remarks on the relation between linear-phase moments and symmetry. Note that a filter u has one linear-phase moment is equivalent to saying that \(\widehat{u}(0) = 1\).

Proposition 1.2.8

Suppose that\(u \in l_{0}(\mathbb{Z})\)has m but not m + 1 linear-phase moments with phase\(c \in \mathbb{R}\). If m > 1, then the phase c is uniquely determined by u through
$$\displaystyle{ c = i\widehat{u}'(0) =\sum _{k\in \mathbb{Z}}u(k)k. }$$
(1.2.21)
Moreover,
  1. (i)

    if u has symmetry: u(c u k) = u(k) for all\(k \in \mathbb{Z}\)for some\(c_{u} \in \mathbb{Z}\), then c = c u ∕2 (that is, the phase c agrees with the symmetry center c u ∕2 of u) and m must be an even integer;

     
  2. (ii)

    if u has complex symmetry:\(u(c_{u} - k) = \overline{u(k)}\)for all\(k \in \mathbb{Z}\)for some\(c_{u} \in \mathbb{Z}\), then c = c u ∕2.

     

Proof

(1.2.21) follows directly from the definition of linear-phase moments in (1.2.15).

For (i), we have \([\mathsf{S}\,\widehat{u}](\xi ) = e^{-ic_{u}\xi }\). Then it follows from (1.2.15) that we must have \(e^{-ic_{u}\xi } = \frac{\widehat{u}(\xi )} {\widehat{u}(-\xi )} = e^{-i2c\xi } +\mathbb{ O}(\vert \xi \vert ^{m})\) as ξ → 0. Since m > 1, we must have c = c u ∕2. Note that \(\mathsf{S}\,\widehat{u}(\xi ) = e^{-ic_{u}\xi }\) and c u = 2c imply \(\widehat{u}(\xi )e^{ic\xi } =\widehat{ u}(-\xi )e^{-ic\xi }\), from which we see that
$$\displaystyle{ [\widehat{u}(\cdot )e^{ic\cdot }]^{(\,j)}(0) = 0,\qquad \mbox{ for all positive odd integers}\;\,j. }$$
(1.2.22)
On the other hand, the definition of linear-phase moments in (1.2.15) is equivalent to
$$\displaystyle{\widehat{u}(\xi )e^{ic\xi } = 1 +\mathbb{ O}(\vert \xi \vert ^{m}),\qquad \xi \rightarrow 0.}$$
Since u has m but not m + 1 linear-phase moments with phase c, it now follows from (1.2.22) that m must be an even integer.

For (ii), we have \([\mathbb{S}\widehat{u}](\xi ) = e^{-ic_{u}\xi }\). Then it follows from (1.2.15) that we must have \(e^{-ic_{u}\xi } = \frac{\widehat{u}(\xi )} {\overline{\widehat{u}(\xi )}} = e^{-i2c\xi } +\mathbb{ O}(\vert \xi \vert ^{m})\) as ξ → 0. Thus, c = c u ∕2 holds. □

In the following we explain the relation between complex symmetry and linear phase of a filter.

Theorem 1.2.9

Let\(u \in l_{0}(\mathbb{Z})\)and ξ 0 ∈ (−π, π) such that\(\widehat{u}(\xi _{0})\neq 0\). Write\(\widehat{u}(\xi _{0}) = M_{0}e^{-i\theta _{0}}\)for some\(M_{0},\theta _{0} \in \mathbb{R}\). Then there exist unique real-valued continuous functions\(M,\theta: (-\pi,\pi ) \rightarrow \mathbb{R}\)such that
$$\displaystyle{ \widehat{u}(\xi ) = M(\xi )e^{-i\theta (\xi )}\quad \forall \;\xi \in (-\pi,\pi )\quad \mathit{\mbox{ with}}\quad M(\xi _{ 0}) = M_{0},\;\theta (\xi _{0}) =\theta _{0}. }$$
(1.2.23)
Moreover, the filter u has essential complex symmetry (that is, e id u has complex symmetry for some\(d \in \mathbb{R}\)) if and only if θ(ξ) = + λ, ξ ∈ (−π, π), where\(\lambda \in \mathbb{R}\)and\(c = \mathit{\mbox{ phase}}(u):= \mathit{\mbox{ Re}}(\sum _{k\in \mathbb{Z}}u(k)k)\). In addition, if\(\widehat{u}(0) = 1\), then d = 0; if u is real-valued, then\(d \in \pi \mathbb{Z}\).

Proof

We first define a function \(m: (-\pi,\pi ) \rightarrow \mathbb{N}_{0}\) such that m(ξ) denotes the number of all zeros, counting multiplicity, of \(\widehat{u}\) on the open interval between ξ and ξ 0. Define \(M: (-\pi,\pi ) \rightarrow \mathbb{R}\) by
$$\displaystyle{M(\xi ) = \vert \widehat{u}(\xi )\vert (-1)^{m(\xi )} \frac{M_{0}} {\vert M_{0}\vert }.}$$
Then it is pretty straightforward to conclude that M is a real-valued continuous function. Moreover, \(\frac{\widehat{u}(\xi )} {M(\xi )}\) is a continuous function on (−π, π) with all singularities removable. This can be seen as follows. Considering the Taylor expansion of \(\widehat{u}\) near ξ 1, we have
$$\displaystyle{\widehat{u}(\xi ) = C_{\xi _{1}}(\xi -\xi _{1})^{n} +\mathbb{ O}(\vert \xi -\xi _{ 1}\vert ^{n+1}),\qquad \xi \rightarrow \xi _{ 1}}$$
for some \(C_{\xi _{1}}\neq 0\) and \(n \in \mathbb{N}_{0}\). Therefore,
$$\displaystyle\begin{array}{rcl} \frac{\widehat{u}(\xi )} {M(\xi )}& =& \frac{\vert M_{0}\vert C_{\xi _{1}}(\xi -\xi _{1})^{n}} {M_{0}\,\vert C_{\xi _{1}}\vert \,\vert \xi -\xi _{1}\vert ^{n}(-1)^{m(\xi )}} +\mathbb{ O}(\vert \xi -\xi _{1}\vert ) {}\\ & =& \frac{C_{\xi _{1}}} {\vert C_{\xi _{1}}\vert }\frac{\vert M_{0}\vert } {M_{0}} \frac{(\xi -\xi _{1})^{n}} {\vert \xi -\xi _{1}\vert ^{n}(-1)^{m(\xi )}} +\mathbb{ O}(\vert \xi -\xi _{1}\vert ),\;\xi \rightarrow 0. {}\\ \end{array}$$
By the definition of the function m(ξ), we see that \(\frac{(\xi -\xi _{1})^{n}} {\vert \xi -\xi _{1}\vert ^{n}(-1)^{m(\xi )}}\) is a constant function in a neighborhood of ξ 1. Therefore, \(\frac{\widehat{u}(\xi )} {M(\xi )}\) is a continuous function at ξ = ξ 1 (that is, the singularity at ξ 1 is removable). Consequently, by \(\vert \frac{\widehat{u}(\xi )} {M(\xi )}\vert = 1\), define \(\theta (\xi ) = i\ln \frac{\widehat{u}(\xi )} {M(\xi )},\xi \in (-\pi,\pi )\), where ln denotes a branch of the natural log function with θ(ξ 0) = θ 0. Therefore, θ is a real-valued continuous function such that (1.2.23) holds.

We now show that a filter u has essential complex symmetry if and only if it has linear phase. Suppose that e id u has complex symmetry for some \(d \in \mathbb{R}\), that is, \([\mathbb{S}(e^{id}\widehat{u})](\xi ) =\epsilon e^{-i2c\xi }\) for some ε ∈ {−1, 1} and \(c \in \tfrac{1} {2}\mathbb{Z}\). Then \(\widehat{u}(\xi )e^{i(c\xi +d)} =\epsilon \overline{\widehat{u}(\xi )e^{i(c\xi +d)}}\). If ε = 1, then \(\widehat{u}(\xi )e^{i(c\xi +d)} \in \mathbb{R}\) for all ξ ∈ (−π, π). Consequently, we must have \(M(\xi ) =\widehat{ u}(\xi )e^{i(c\xi +d)}\) or \(-\widehat{u}(\xi )e^{i(c\xi +d)}\) for all ξ ∈ (−π, π). This implies that we must have θ(ξ) = + d + κπ for all ξ ∈ (−π, π) for some integer \(\kappa \in \mathbb{Z}\). Hence, θ is a linear function on (−π, π). If ε = −1, then \(i\widehat{u}(\xi )e^{i(c\xi +d)} \in \mathbb{R}\) and we have \(M(\xi ) = i\widehat{u}(\xi )e^{i(c\xi +d)}\) or \(-i\widehat{u}(\xi )e^{i(c\xi +d)}\) for all ξ ∈ (−π, π). A similar argument shows that \(\theta (\xi ) = c\xi + d + \tfrac{\pi } {2}+\kappa \pi\) for all ξ ∈ (−π, π) for some integer κ. Hence, θ is a linear function on (−π, π). Therefore, if a filter u has essential complex symmetry, then it must have linear phase on (−π, π).

Conversely, suppose that θ is a linear function. Then θ(ξ) = + λ for some \(c,\lambda \in \mathbb{R}\). Since both M and θ are real-valued, we deduce that
$$\displaystyle{\mathbb{S}\widehat{u}(\xi ) = \frac{\widehat{u}(\xi )} {\overline{\widehat{u}(\xi )}} = \frac{M(\xi )e^{-i\theta (\xi )}} {M(\xi )e^{i\theta (\xi )}} = e^{-i2\theta (\xi )} = e^{-i2(c\xi +\lambda )}.}$$
The above identity implies that e u has complex symmetry \(\mathbb{S}(e^{i\lambda }\widehat{u}) = e^{-i2c\xi }\). Thus, if the phase θ of u is a linear function on (−π, π), then the filter u must have essential complex symmetry. □

If M 0 > 0 and \(\widehat{u}(\xi )\neq 0\) for all ξ ∈ (−π, π), from the proof of Theorem 1.2.9 we see that \(M(\xi ) = \vert \widehat{u}(\xi )\vert\) for all ξ ∈ (−π, π). For filters u such that \(\widehat{u}(0) \in \mathbb{R}\setminus \{0\}\), without further mention in this book, we always take \(\xi _{0} = 0,M_{0} =\widehat{ u}(0)\), and θ 0 = 0 in Theorem 1.2.9. We call M(ξ) the default magnitude function of \(\widehat{u}\) and θ(ξ) the default phase function of \(\widehat{u}\). For this case, u has complex symmetry if and only if u has linear phase θ(ξ) = for ξ ∈ (−π, π).

1.2.4 An Example

We complete this section by presenting an example. According to item (5) of Theorem 1.2.5, a natural filter having m sum rules and the shortest possible filter support is
$$\displaystyle{ \widehat{a_{m}^{B}}(\xi ):= 2^{-m}(1 + e^{-i\xi })^{m},\qquad m \in \mathbb{N}, }$$
(1.2.24)
which is called the B-spline filter (or mask) of order m in the literature of wavelet analysis and approximation theory. Note that \(\widehat{a_{m}^{B}}(\xi ) = 2^{-m}\sum _{j=0}^{m}\binom{m}{j}e^{-ij\xi }\) and a m B has the symmetry type \([\mathsf{S}\,\widehat{a_{m}^{B}}](\xi ) = e^{-im\xi }\), where
$$\displaystyle{ \binom{m}{0}:= 1,\qquad \binom{m}{j}:= \frac{m!} {j!(m - j)!}\quad \mbox{ with}\quad j!:= 1 \cdot 2\cdots (\,j - 1)j. }$$
(1.2.25)
Hence, we have
$$\displaystyle{\widehat{a_{m}^{B}}^{(n)}(0) = (-i)^{n}2^{-m}\sum _{ j=0}^{m}\binom{m}{j}j^{n},\qquad n \in \mathbb{N}_{ 0}.}$$
Let us consider \(a_{4}^{B} =\{\underline{ \frac{\mathbf{1}} {\mathbf{16}}}, \frac{1} {4}, \frac{3} {8}, \frac{1} {4}, \frac{1} {16}\}_{[0,4]}\). Then
$$\displaystyle{\widehat{a_{4}^{B}}(0) = 1,\quad \widehat{a_{ 4}^{B}}'(0) = -2i,\quad \widehat{a_{ 4}^{B}}''(0) = -5,\quad \widehat{a_{ 4}^{B}}^{'''}(0) = 14i.}$$
For any \(\mathsf{p} \in \mathbb{P}_{3}\), by Lemma 1.2.1 and (1.2.5), we have
$$\displaystyle{[\mathsf{p} {\ast} a_{4}^{B}](x) = \mathsf{p}(x) - 2\mathsf{p}'(x) + \frac{5} {2}\mathsf{p}''(x) -\frac{7} {3}\mathsf{p}'''(x).}$$
By Theorem 1.2.2,
$$\displaystyle{[\mathcal{T}_{a_{4}^{B}}\mathsf{p}](x) = 2\mathsf{p}(2x) + 4\mathsf{p}'(2x) + 5\mathsf{p}''(2x) + \frac{14} {3} \mathsf{p}'''(2x).}$$
By Theorem 1.2.4 and (1.2.11),
$$\displaystyle{[\mathcal{S}_{a_{4}^{B}}\mathsf{p}](x) = \mathsf{p}(x/2) -\mathsf{p}'(x/2) + \frac{5} {8}\mathsf{p}''(x/2) - \frac{7} {24}\mathsf{p}'''(x/2).}$$
Let \(c:=\sum _{k\in \mathbb{Z}}a_{4}^{B}(k)k = 2\), which is also the symmetry center of a 4 B . Then a 4 B has no more than two linear-phase moments with the phase c = 2. Moreover, for p(x) = t 0 + t 1x with \(t_{0},t_{1} \in \mathbb{C}\), we have
$$\displaystyle\begin{array}{rcl} & & [\mathsf{p} {\ast} a_{4}^{B}](x) = (t_{ 0} + t_{1}x) - 2t_{1} = (t_{0} - 2t_{1}) + t_{1}x = \mathsf{p}(\cdot - 2), {}\\ & & [\mathcal{T}_{a_{4}^{B}}\mathsf{p}](x) = 2(t_{0} + 2t_{1}x) + 4t_{1} = (2t_{0} + 4t_{1}) + 4t_{1}x = 2\mathsf{p}(2x + 2), {}\\ & & [\mathcal{S}_{a_{4}^{B}}\mathsf{p}](x) = (t_{0} + t_{1}x/2) - t_{1} = (t_{0} - t_{1}) + t_{1}x/2 = \mathsf{p}(2^{-1}(x - 2)) = \mathsf{p}(2^{-1}x - 1).{}\\ \end{array}$$

1.3 Multilevel Discrete Framelet Transforms and Stability

To extract the multiscale structure embedded in signals, a multilevel discrete framelet transform is used in applications by recursively applying one-level discrete framelet transforms on selected sequences of framelet coefficients at the immediate higher scale level. In this section we discuss a (standard) multilevel discrete framelet transform, study its stability in the space \(l_{2}(\mathbb{Z})\), and introduce the notion of discrete affine systems in \(l_{2}(\mathbb{Z})\).

1.3.1 Multilevel Discrete Framelet Transforms

A standard multilevel discrete framelet transform is obtained by recursively performing one-level discrete framelet transforms on only one selected sequence of framelet coefficients. Certainly one may select several or even all the sequences of framelet coefficients for further decomposition, but we have more or less the same algorithm as we shall see in Sect. 1.3.4. The framelet coefficients and their associated filters in such selected sequences for further decomposition are called parent (or low-pass) framelet coefficients and parent (or low-pass) filters or masks (since they are often low-pass filters), respectively. In this book we use a or its indexed version to denote a low-pass (or parent) filter and use v or its indexed version to denote low-pass (or parent) framelet coefficients. The framelet coefficients and their associated filters in other not-selected sequences for decomposition are called child (or high-pass) framelet coefficients and child (or high-pass) filters (since they are often high-pass filters), respectively. In this book we use b or its indexed version to denote a high-pass (or child) filter and use w or its indexed version to denote high-pass (or child) framelet coefficients.

Readers may notice that our definitions of low-pass and high-pass filters are different from those used in the literature of engineering, where a low-pass filter u means \(\widehat{u}(0)\neq 0\) and \(\widehat{u}(\pi ) = 0\) while a high-pass filer v means \(\widehat{v}(0) = 0\) and \(\widehat{v}(\pi )\neq 0\). Such a purposely misuse of the notion of low-pass and high-pass filters in this book is not serious but convenient for our discussion, since quite often only wavelet coefficients associated with low-pass filters in the sense of engineering are selected for further decomposition.

Let \(\tilde{a},\tilde{b}_{1},\ldots,\tilde{b}_{s}\) be filters for decomposition. For a positive integer J, a J-level discrete framelet decomposition is given by
$$\displaystyle{ v_{j}:= \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{a}}v_{j-1},\quad w_{\ell,j}:= \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{b}_{\ell}}v_{j-1},\qquad \ell = 1,\ldots,s,\;\quad j = 1,\ldots,J, }$$
(1.3.1)
where \(v_{0}: \mathbb{Z} \rightarrow \mathbb{C}\) is an input signal. The filter \(\tilde{a}\) is often called a dual low-pass filter and the filters \(\tilde{b}_{1},\ldots,\tilde{b}_{s}\) are called dual high-pass filters. After a J-level discrete framelet decomposition, the original input signal v 0 is decomposed into one sequence v J of low-pass framelet coefficients and sJ sequences w , j of high-pass framelet coefficients for = 1, , s and j = 1, , J. Such framelet coefficients are often processed for various purposes. One of the most commonly employed operations is thresholding so that the low-pass framelet coefficients v J and high-pass framelet coefficients w , j become \(\mathring{v}_{J}\) and \(\mathring{w}_{\ell,j}\), respectively. More precisely, \(\mathring{w}_{\ell,j}(k) =\eta (w_{\ell,j}(k)),k \in \mathbb{Z}\), where \(\eta: \mathbb{C} \rightarrow \mathbb{C}\) is a thresholding function. For example, for a given threshold value λ > 0, the hard thresholding function η λ hard and soft-thresholding function η λ soft are defined to be
$$\displaystyle{ \eta _{\lambda }^{hard}(z) = \left \{\begin{array}{l@{\quad }l} z,\quad &\text{if }\vert z\vert \geqslant \lambda; \\ 0,\quad &\text{otherwise} \end{array} \right.\quad \mbox{ and}\quad \eta _{\lambda }^{soft}(z) = \left \{\begin{array}{l@{\quad }l} z -\lambda \frac{z} {\vert z\vert },\quad &\text{if }\vert z\vert \geqslant \lambda; \\ 0, \quad &\text{otherwise.} \end{array} \right. }$$
(1.3.2)
Quantization is another commonly employed operation after or without thresholding. For example, for a given quantization level q > 0, the quantization function \(\mathcal{Q}: \mathbb{R} \rightarrow q\mathbb{Z}\) is defined to be \(\mathcal{Q}(x):= q\lfloor \tfrac{x} {q} + \tfrac{1} {2}\rfloor\), \(x \in \mathbb{R}\), where ⌊⋅ ⌋ is the floor function such that ⌊x⌋ = n if \(n\leqslant x <n + 1\) for an integer n. See Fig. 1.2 for illustration.
Fig. 1.2

The hard thresholding function η λ hard , the soft thresholding function η λ soft , and the quantization function, respectively. Both thresholding and quantization operations are often used to process framelet coefficients in a discrete framelet transform

Let a, b 1, , b s be filters for reconstruction. Now a J-level discrete framelet reconstruction is
$$\displaystyle{\mathring{v}_{j-1}:= \frac{\sqrt{2}} {2} \mathcal{S}_{a}\mathring{v}_{j}+\frac{\sqrt{2}} {2} \sum _{\ell=1}^{s}\mathcal{S}_{ b_{\ell}}\mathring{w}_{\ell,j},\qquad j = J,\ldots,1. }$$
(1.3.3)
The filter a is often called a primal low-pass filter and the filters b 1, , b s are called primal high-pass filters. To analyze a multilevel discrete framelet transform, we rewrite the J-level discrete framelet decomposition employing the filter bank \(\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\) by using a J-level discrete framelet analysis operator \(\widetilde{\mathcal{W}}_{J}: l(\mathbb{Z}) \rightarrow (l(\mathbb{Z}))^{1\times (sJ+1)}\) as follows:
$$\displaystyle{ \widetilde{\mathcal{W}}_{J}v_{0}:= (w_{1,1},\ldots,w_{s,1},\ldots,w_{1,J},\ldots,w_{s,J},v_{J}), }$$
(1.3.4)
where w , j and v J are defined in (1.3.1). Similarly, a J-level discrete framelet synthesis operator \(\mathcal{V}_{J}: (l(\mathbb{Z}))^{1\times (sJ+1)} \rightarrow l(\mathbb{Z})\) employing the filter bank {a; b 1, , b s } is defined by
$$\displaystyle{ \mathcal{V}_{J}(\mathring{w}_{1,1},\ldots,\mathring{w}_{s,1},\ldots,\mathring{w}_{1,J},\ldots,\mathring{w}_{s,J},\mathring{v}_{J}) =\mathring{v}_{0}, }$$
(1.3.5)
where \(\mathring{v}_{0}\) is computed via the recursive formulas in (1.3.3). Note that
$$\displaystyle{\widetilde{\mathcal{W}}_{J} = (\mathrm{Id}_{(l(\mathbb{Z}))^{1\times s(J-1)}} \otimes \widetilde{\mathcal{W}})\cdots (\mathrm{Id}_{(l(\mathbb{Z}))^{1\times s}} \otimes \widetilde{\mathcal{W}})\widetilde{\mathcal{W}}}$$
and
$$\displaystyle{\mathcal{V}_{J} = \mathcal{V}(\mathrm{Id}_{(l(\mathbb{Z}))^{1\times s}} \otimes \mathcal{V})\cdots (\mathrm{Id}_{(l(\mathbb{Z}))^{1\times s(J-1)}} \otimes \mathcal{V}).}$$
Due to (1.3.1) and (1.3.3), a multilevel discrete framelet transform implemented using the recursive cascade structure is often called a fast framelet transform (FFrT). Due to Proposition 1.1.2, a fast framelet transform with s = 1 is called a fast wavelet transform (FWT). We shall denote a J-level discrete framelet analysis operator employing the filter bank {a; b 1, , b s } by \(\mathcal{W}_{J}\) and a J-level discrete framelet synthesis operator employing the filter bank \(\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\) by \(\widetilde{\mathcal{V}}_{J}\). When J = 1, these operators become the analysis \(\mathcal{W}\) and synthesis operators \(\mathcal{V}\) that we discussed in Sect. 1.1.
If \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})\) is a dual framelet filter bank, i.e., it satisfies the perfect reconstruction condition:
$$\displaystyle{ \left [\begin{array}{*{10}c} \widehat{\tilde{a}}(\xi ) & \widehat{\tilde{b}_{1}}(\xi ) &\cdots & \widehat{\tilde{b}_{s}}(\xi ) \\ \widehat{\tilde{a}}(\xi +\pi )&\widehat{\tilde{b}_{1}}(\xi +\pi )&\cdots &\widehat{\tilde{b}_{s}}(\xi +\pi ) \end{array} \right ]\left [\begin{array}{*{10}c} \widehat{a}(\xi ) & \widehat{b_{1}}(\xi ) &\cdots & \widehat{b_{s}}(\xi ) \\ \widehat{a}(\xi +\pi )&\widehat{b_{1}}(\xi +\pi )&\cdots &\widehat{b_{s}}(\xi +\pi ) \end{array} \right ]^{\star } = I_{ 2}, }$$
(1.3.6)
then Theorem 1.1.1 tells us that \(\mathcal{V}_{J}\widetilde{\mathcal{W}}_{J} =\mathrm{ Id}_{l(\mathbb{Z})}\) for all \(J \in \mathbb{N}\), that is, all the J-level discrete framelet transforms have the perfect reconstruction property. Observe that \(\mathcal{W}_{J}^{\star } = \mathcal{V}_{J}\) and \(\widetilde{\mathcal{W}}_{J}^{\star } =\widetilde{ \mathcal{V}}_{J}\). Thus \(\mathcal{V}_{J}\widetilde{\mathcal{W}}_{J} =\mathrm{ Id}_{l(\mathbb{Z})}\) if and only if \(\widetilde{\mathcal{V}}_{J}\mathcal{W}_{J} =\mathrm{ Id}_{l(\mathbb{Z})}\). See Fig. 1.3 for a diagram of a 2-level discrete framelet transform with a pair of filter banks \(\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\) and {a; b 1, , b s }.
Fig. 1.3

Diagram of a two-level discrete framelet transform employing filter banks \(\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\) and {a; b 1, , b s }. Note that \(\frac{\sqrt{2}} {2} \mathcal{T}_{\tilde{b}_{\ell}}v = \sqrt{2}(v {\ast}\tilde{ b}_{\ell}^{\star })\! \downarrow \! 2\) and \(\frac{\sqrt{2}} {2} \mathcal{S}_{b_{\ell}}v = \sqrt{2}(v\! \uparrow \! 2) {\ast} b_{\ell}\) for = 1, , s

1.3.2 Stability of Multilevel Discrete Framelet Transforms

In this section all our input signals and domains of the analysis/synthesis operators are from the space \(l_{2}(\mathbb{Z})\). A key property of a multilevel framelet transform is its stability. A filter bank {a; b 1, , b s } is said to have stability in \(l_{2}(\mathbb{Z})\) if there exist positive constants C 1 and C 2 such that
$$\displaystyle{ C_{1}\|v\|_{l_{2}(\mathbb{Z})}^{2}\leqslant \|\mathcal{W}_{ J}v\|_{(l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2}\leqslant C_{ 2}\|v\|_{l_{2}(\mathbb{Z})}^{2},\qquad \forall \,v \in l_{ 2}(\mathbb{Z}),J \in \mathbb{N}. }$$
(1.3.7)
A filter bank {a; b 1, , b s } having stability in \(l_{2}(\mathbb{Z})\) is called a framelet filter bank in \(l_{2}(\mathbb{Z})\). The inequalities (1.3.7) imply \(\|\mathcal{W}_{J}\|^{2}\leqslant C_{2}\) for all \(J \in \mathbb{N}\). By (1.3.7), the l 2-norm of framelet coefficients provides an equivalent norm for the sequence space \(l_{2}(\mathbb{Z})\).

For stability of a multilevel discrete framelet transform, we have

Theorem 1.3.1

Let\((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})\)be a dual framelet filter bank. Let\(\mathcal{W}_{J},\widetilde{\mathcal{W}}_{J}\)be its associated J-level discrete framelet analysis operators and\(\mathcal{V}_{J},\widetilde{\mathcal{V}}_{J}\)be its associated J-level discrete framelet synthesis operators. Let C 1, C 2be positive numbers. Then the following statements are equivalent:
  1. (1)
    Both filter banks\(\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\)and {a; b 1, , b s } have stability in the space\(l_{2}(\mathbb{Z})\)satisfying (1.3.7) and
    $$\displaystyle{ C_{2}^{-1}\|v\|_{ l_{2}(\mathbb{Z})}^{2}\leqslant \|\widetilde{\mathcal{W}}_{ J}v\|_{(l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2}\leqslant C_{ 1}^{-1}\|v\|_{ l_{2}(\mathbb{Z})}^{2},\quad \forall \,v \in l_{ 2}(\mathbb{Z}),J \in \mathbb{N}. }$$
    (1.3.8)
     
  2. (2)

    \(\|\mathcal{W}_{J}\|^{2}\leqslant C_{2}\) and \(\|\widetilde{\mathcal{W}}_{J}\|^{2}\leqslant C_{1}^{-1}\) for all \(J \in \mathbb{N}\) .

     
  3. (3)

    \(\|\mathcal{V}_{J}\|^{2}\leqslant C_{2}\) and \(\|\widetilde{\mathcal{V}}_{J}\|^{2}\leqslant C_{1}^{-1}\) for all \(J \in \mathbb{N}\) .

     
  4. (4)

    \(\|\mathcal{V}_{J}\|^{2}\leqslant C_{2}\) and \(\|\widetilde{\mathcal{W}}_{J}\|^{2}\leqslant C_{1}^{-1}\) for all \(J \in \mathbb{N}\) .

     
  5. (5)

    \(\|\mathcal{W}_{J}\|^{2}\leqslant C_{2}\) and \(\|\widetilde{\mathcal{V}}_{J}\|^{2}\leqslant C_{1}^{-1}\) for all \(J \in \mathbb{N}\) .

     
If in addition s = 1, then each of the above statements is further equivalent to
  1. (6)
    For all \(w \in (l_{2}(\mathbb{Z}))^{1\times (sJ+1)}\) and \(J \in \mathbb{N}\) ,
    $$\displaystyle{ C_{1}\|w\|_{(l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2}\leqslant \|\mathcal{V}_{ J}w\|_{l_{2}(\mathbb{Z})}^{2}\leqslant C_{ 2}\|w\|_{(l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2}, }$$
    (1.3.9)
    $$\displaystyle{ C_{2}^{-1}\|w\|_{ (l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2}\leqslant \|\widetilde{\mathcal{V}}_{ J}w\|_{l_{2}(\mathbb{Z})}^{2}\leqslant C_{ 1}^{-1}\|w\|_{ (l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2}. }$$
    (1.3.10)
     

Proof

Note that
$$\displaystyle{\mathcal{V}_{J} = \mathcal{W}_{J}^{\star },\quad \widetilde{\mathcal{V}}_{ J} =\widetilde{ \mathcal{W}}_{J}^{\star },\quad \|\mathcal{W}_{ J}^{\star }\| =\| \mathcal{W}_{ J}\|,\quad \|\widetilde{\mathcal{W}}_{J}^{\star }\| =\|\widetilde{ \mathcal{W}}_{ J}\|.}$$
We trivially have (1)⇒(2) ⇔ (3) ⇔ (4) ⇔ (5). We now prove that (2) and (3) together imply (1). Since \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})\) is a dual framelet filter bank, we have \(\widetilde{\mathcal{V}}_{J}\mathcal{W}_{J} = \mathcal{V}_{J}\widetilde{\mathcal{W}}_{J} =\mathrm{ Id}_{l_{2}(\mathbb{Z})}\). By item (3),
$$\displaystyle{\|v\|_{l_{2}(\mathbb{Z})}^{2} =\|\widetilde{ \mathcal{V}}_{ J}\mathcal{W}_{J}v\|_{l_{2}(\mathbb{Z})}^{2}\leqslant \|\widetilde{\mathcal{V}}_{ J}\|^{2}\|\mathcal{W}_{ J}v\|_{l_{2}(\mathbb{Z})}^{2}\leqslant C_{ 1}^{-1}\|\mathcal{W}_{ J}v\|_{(l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2},}$$
which is simply the left-hand inequality of (1.3.7). By item (2), (1.3.7) holds. The inequalities in (1.3.8) can be proved similarly.
Since \(v = \mathcal{V}_{J}\widetilde{\mathcal{W}}_{J}v\), replacing \(\|v\|_{l_{2}(\mathbb{Z})}\) in (1.3.8) by \(\|\mathcal{V}_{J}\widetilde{\mathcal{W}}_{J}v\|_{l_{2}(\mathbb{Z})}\), we deduce that
$$\displaystyle{ C_{1}\|\widetilde{\mathcal{W}}_{J}v\|_{(l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2}\leqslant \|\mathcal{V}_{ J}\widetilde{\mathcal{W}}_{J}v\|_{l_{2}(\mathbb{Z})}^{2}\leqslant C_{ 2}\|\widetilde{\mathcal{W}}_{J}v\|_{(l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2}. }$$
(1.3.11)
If s = 1, by Proposition 1.1.2, then \(\widetilde{\mathcal{W}}_{J}\) is onto and hence, (1.3.9) follows directly from (1.3.11). The inequalities in (1.3.10) can be proved similarly. □

For \(\mathcal{V}_{J}\) and \(\widetilde{\mathcal{V}}_{J}\), generally we can only have (1.3.11) and its duality part by replacing \(\widetilde{\mathcal{W}}_{J}\) and \(\mathcal{V}_{J}\) in (1.3.11) with \(\mathcal{W}_{J}\) and \(\widetilde{\mathcal{V}}_{J}\), respectively. For s > 1, both (1.3.9) and (1.3.10) cannot hold, since by Proposition 1.1.2, there exists \(w \in l_{0}(\mathbb{Z})\setminus \{0\}\) such that \(\mathcal{V}_{J}w = 0\). The stability of a multilevel discrete framelet transform implies that a small change of an input signal v induces a small change of all framelet coefficients, and a small perturbation of all framelet coefficients results in a small perturbation of a reconstructed signal. The notion of stability of a multilevel discrete framelet transform can be extended to other (weighted) sequence spaces and is closely related to refinable functions. The study of stability of multilevel discrete framelet transforms is an important part of mathematical analysis of wavelets and framelets. We shall devote Chaps.  5 and  6 of this book to address such issues.

1.3.3 Discrete Affine Systems in \(l_{2}(\mathbb{Z})\)

In this section we shall introduce the notion of discrete affine systems. A multilevel discrete framelet transform can be fully expressed through discrete affine systems. To do so, let us first generalize the definition of the subdivision operator and the transition operator. For a nonzero integer d and a finitely supported sequence u, the subdivision operator \(\mathcal{S}_{u,\mathsf{d}}: l(\mathbb{Z}) \rightarrow l(\mathbb{Z})\) and the transition operator \(\mathcal{T}_{u,\mathsf{d}}: l(\mathbb{Z}) \rightarrow l(\mathbb{Z})\) are defined to be
$$\displaystyle{ [\mathcal{S}_{u,\mathsf{d}}v](n):= \vert \mathsf{d}\vert \sum _{k\in \mathbb{Z}}v(k)u(n -\mathsf{d}k),\qquad n \in \mathbb{Z}, }$$
(1.3.12)
$$\displaystyle{ [\mathcal{T}_{u,\mathsf{d}}v](n):= \vert \mathsf{d}\vert \sum _{k\in \mathbb{Z}}v(k)\overline{u(k -\mathsf{d}n)} = \vert \mathsf{d}\vert \langle v,u(\cdot -\mathsf{d}n)\rangle,\qquad n \in \mathbb{Z} }$$
(1.3.13)
for \(v \in l(\mathbb{Z})\). For \(v \in l_{0}(\mathbb{Z})\), one can check that \(\widehat{\mathcal{S}_{u,\mathsf{d}}v}(\xi ) = \vert \mathsf{d}\vert \widehat{v}(\mathsf{d}\xi )\widehat{u}(\xi )\) and
$$\displaystyle{\widehat{\mathcal{T}_{u,\mathsf{d}}v}(\xi ) =\sum _{ \gamma =0}^{\vert \mathsf{d}\vert -1}\widehat{v}(\tfrac{\xi +2\pi \gamma } {\mathsf{d}} )\overline{\widehat{u}(\tfrac{\xi +2\pi \gamma } {\mathsf{d}} )}.}$$
Moreover, \(\mathcal{S}_{u,\mathsf{d}}v = \vert \mathsf{d}\vert u {\ast} (v\! \uparrow \!\mathsf{d})\), \(\mathcal{T}_{u,\mathsf{d}}v = \vert \mathsf{d}\vert (u^{\star } {\ast} v)\! \downarrow \!\mathsf{d}\), \(\mathcal{S}_{u,\mathsf{d}}^{\star } = \mathcal{T}_{u,\mathsf{d}}\), and
$$\displaystyle{ \langle \mathcal{S}_{u,\mathsf{d}}v,w\rangle =\langle \mathcal{T}_{u,\mathsf{d}}^{\star }v,w\rangle =\langle v,\mathcal{T}_{ u,\mathsf{d}}w\rangle,\qquad \forall \;v,w \in l_{2}(\mathbb{Z}). }$$
(1.3.14)

To understand the operators \(\widetilde{\mathcal{W}}_{J}\) and \(\mathcal{V}_{J}\) in a J-level discrete framelet transform, we need the following auxiliary result.

Lemma 1.3.2

For \(\mathsf{d}_{1},\mathsf{d}_{2} \in \mathbb{Z}\setminus \{0\}\) and \(u_{1},u_{2} \in l_{0}(\mathbb{Z})\) ,
$$\displaystyle{ \mathcal{S}_{u_{1},\mathsf{d}_{1}}\mathcal{S}_{u_{2},\mathsf{d}_{2}}v = \mathcal{S}_{u_{1}{\ast}(u_{2}\uparrow \,\mathsf{d}_{1}),\mathsf{d}_{1}\mathsf{d}_{2}}v = \vert \mathsf{d}_{1}\mathsf{d}_{2}\vert u_{1} {\ast} (u_{2}\! \uparrow \!\mathsf{d}_{1}) {\ast} (v\! \uparrow \!\mathsf{d}_{1}\mathsf{d}_{2}) }$$
(1.3.15)
and
$$\displaystyle{ \mathcal{T}_{u_{2},\mathsf{d}_{2}}\mathcal{T}_{u_{1},\mathsf{d}_{1}}v = \mathcal{T}_{u_{1}{\ast}(u_{2}\uparrow \mathsf{d}_{1}),\mathsf{d}_{1}\mathsf{d}_{2}}v = \vert \mathsf{d}_{1}\mathsf{d}_{2}\vert (u_{1}^{\star } {\ast} (u_{ 2}^{\star }\! \uparrow \!\mathsf{d}_{ 1}) {\ast} v)\! \downarrow \!\mathsf{d}_{1}\mathsf{d}_{2}. }$$
(1.3.16)

Proof

By \(\widehat{\mathcal{S}_{u,\mathsf{d}}v}(\xi ) = \vert \mathsf{d}\vert \widehat{v}(\mathsf{d}\xi )\widehat{u}(\xi )\), the Fourier series of the sequence \(\mathcal{S}_{u_{1},\mathsf{d}_{1}}\mathcal{S}_{u_{2},\mathsf{d}_{2}}v\) is
$$\displaystyle{\vert \mathsf{d}_{1}\vert \widehat{u_{1}}(\xi )\widehat{\mathcal{S}_{u_{2},\mathsf{d}_{2}}v}(\mathsf{d}_{1}\xi ) = \vert \mathsf{d}_{1}\mathsf{d}_{2}\vert \widehat{u_{1}}(\xi )\widehat{u_{2}}(\mathsf{d}_{1}\xi )\widehat{v}(\mathsf{d}_{1}\mathsf{d}_{2}\xi ) = \vert \mathsf{d}_{2}\vert \widehat{\mathcal{S}_{u_{1},\mathsf{d}_{1}}u_{2}}(\xi )\widehat{v}(\mathsf{d}_{1}\mathsf{d}_{2}\xi ).}$$
Therefore, (1.3.15) holds. By duality in (1.3.14) and (1.3.15), we have
$$\displaystyle\begin{array}{rcl} \langle w,\mathcal{T}_{u_{2},\mathsf{d}_{2}}\mathcal{T}_{u_{1},\mathsf{d}_{1}}v\rangle & =& \langle \mathcal{S}_{u_{2},\mathsf{d}_{2}}w,\mathcal{T}_{u_{1},\mathsf{d}_{1}}v\rangle =\langle \mathcal{S}_{u_{1},\mathsf{d}_{1}}\mathcal{S}_{u_{2},\mathsf{d}_{2}}w,v\rangle {}\\ & =& \langle \mathcal{S}_{u_{1}{\ast}(u_{2}\uparrow \mathsf{d}_{1}),\mathsf{d}_{1}\mathsf{d}_{2}}w,v\rangle =\langle w,\mathcal{T}_{u_{1}{\ast}(u_{2}\uparrow \mathsf{d}_{1}),\mathsf{d}_{1}\mathsf{d}_{2}}v\rangle, {}\\ \end{array}$$
from which we see that (1.3.16) holds. The identities in (1.3.15) and (1.3.16) can also be seen as follows:
$$\displaystyle\begin{array}{rcl} & & \mathcal{S}_{u_{1},\mathsf{d}_{1}}\mathcal{S}_{u_{2},\mathsf{d}_{2}}v = \vert \mathsf{d}_{1}\mathsf{d}_{2}\vert u_{1} {\ast} ((u_{2} {\ast} (v\! \uparrow \!\mathsf{d}_{2}))\! \uparrow \!\mathsf{d}_{1}) = \vert \mathsf{d}_{1}\mathsf{d}_{2}\vert u_{1} {\ast} (u_{2}\! \uparrow \!\mathsf{d}_{1}) {\ast} (v\! \uparrow \!\mathsf{d}_{1}\mathsf{d}_{2}), {}\\ & & \mathcal{T}_{u_{2},\mathsf{d}_{2}}\mathcal{T}_{u_{1},\mathsf{d}_{1}}v = \vert \mathsf{d}_{1}\mathsf{d}_{2}\vert (u_{2}^{\star } {\ast} ((u_{ 1}^{\star } {\ast} v)\! \downarrow \!\mathsf{d}_{ 1}))\! \downarrow \!\mathsf{d}_{2} = \vert \mathsf{d}_{1}\mathsf{d}_{2}\vert (u_{1}^{\star } {\ast} (u_{ 2}\! \uparrow \!\mathsf{d}_{1}) {\ast} v)\! \downarrow \!\mathsf{d}_{1}\mathsf{d}_{2},{}\\ \end{array}$$
where we used (uv)​ d = (ud) ∗ (vd) and u ∗ (vd) = ((ud) ∗ v)​ d. □
Define filters \(a_{j},\tilde{a}_{j},b_{\ell,j},\tilde{b}_{\ell,j}\) with \(j \in \mathbb{N}_{0}\) by
$$\displaystyle{ \widehat{a_{j}}(\xi ):=\widehat{ a}(\xi )\widehat{a}(2\xi )\cdots \widehat{a}(2^{j-1}\xi ),\qquad \widehat{\tilde{a}_{ j}}(\xi ):=\widehat{\tilde{ a}}(\xi )\widehat{\tilde{a}}(2\xi )\cdots \widehat{\tilde{a}}(2^{j-1}\xi ), }$$
(1.3.17)
$$\displaystyle\begin{array}{rcl} & & \widehat{b_{\ell,j}}(\xi ):=\widehat{ a}(\xi )\widehat{a}(2\xi )\cdots \widehat{a}(2^{j-2}\xi )\widehat{b_{\ell}}(2^{j-1}\xi ),\qquad {}\\ & & \widehat{\tilde{b}_{\ell,j}}(\xi ):=\widehat{\tilde{ a}}(\xi )\widehat{\tilde{a}}(2\xi )\cdots \widehat{\tilde{a}}(2^{j-2}\xi )\widehat{\tilde{b}_{\ell}}(2^{j-1}\xi ) {}\\ \end{array}$$
with the convention that \(a_{0} =\tilde{ a}_{0} = b_{\ell,0} =\tilde{ b}_{\ell,0}:=\boldsymbol{\delta }\). In other words,
$$\displaystyle{a_{j} = a {\ast} (a\! \uparrow \! 2) {\ast}\cdots {\ast} (a\! \uparrow \! 2^{j-1})\quad \mbox{ and}\quad \tilde{a}_{ j}:=\tilde{ a} {\ast} (\tilde{a}\! \uparrow \! 2) {\ast}\cdots {\ast} (\tilde{a}\! \uparrow \! 2^{j-1}).}$$
From the definition of framelet coefficients w , j in (1.3.1), noting that \(\mathcal{T}_{\tilde{u}} = \mathcal{T}_{\tilde{u},2}\) and v 0 = v, we see that
$$\displaystyle{v_{j} = \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{a}}v_{j-1} = \cdots = (\tfrac{\sqrt{2}} {2} )^{j}\mathcal{T}_{\tilde{ a}}^{j}v_{ 0} = (\tfrac{\sqrt{2}} {2} )^{j}\mathcal{T}_{\tilde{ a}{\ast}(\tilde{a}\uparrow 2){\ast}\cdots {\ast}(\tilde{a}\uparrow 2^{j-1}),2^{j}}v =\langle v,\tilde{a}_{j;\cdot }\rangle }$$
and
$$\displaystyle\begin{array}{rcl} w_{\ell,j}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{b}_{\ell}}v_{j-1} = (\tfrac{\sqrt{2}} {2} )^{j}\mathcal{T}_{\tilde{ b}_{\ell}}\mathcal{T}_{\tilde{a}}^{j-1}v_{ 0} {}\\ & =& (\tfrac{\sqrt{2}} {2} )^{j}\mathcal{T}_{\tilde{ a}{\ast}(\tilde{a}\uparrow 2){\ast}\cdots {\ast}(\tilde{a}\!\uparrow \!2^{j-2}){\ast}(\tilde{b}_{\ell}\ \uparrow \ 2^{j-1}),2^{j}}v =\langle v,\tilde{b}_{\ell,j;\cdot }\rangle, {}\\ \end{array}$$
where \(\tilde{a}_{j;k}\) and \(\tilde{b}_{\ell,j;k}\) are defined to be
$$\displaystyle{\tilde{a}_{j;k}:= 2^{j/2}\tilde{a}_{ j}(\cdot - 2^{j}k),\quad \tilde{b}_{\ell,j;k}:= 2^{j/2}\tilde{b}_{\ell,j}(\cdot - 2^{j}k),\qquad j \in \mathbb{N}_{ 0},k \in \mathbb{Z}.}$$
Similarly, we deduce that
$$\displaystyle{\mathcal{V}_{J}(0,\ldots,0,v_{J}) = (\tfrac{\sqrt{2}} {2} )^{J}\mathcal{S}_{ a}^{J}v_{ J} = (\tfrac{\sqrt{2}} {2} )^{J}\mathcal{S}_{ a{\ast}(a\uparrow 2){\ast}\cdots {\ast}(a\uparrow 2^{J-1}),2^{J}}v_{J} =\sum _{k\in \mathbb{Z}}v_{J}(k)a_{J;k}}$$
and
$$\displaystyle\begin{array}{rcl} & & \mathcal{V}_{J}(0,\ldots,0,w_{\ell,j},0,\ldots,0) = (\tfrac{\sqrt{2}} {2} )^{j}\mathcal{S}_{ a}^{j-1}\mathcal{S}_{ b_{\ell}}w_{\ell,j} {}\\ & & \phantom{\mathcal{V}_{J}(0,\ldots,0,}= (\tfrac{\sqrt{2}} {2} )^{j}\mathcal{S}_{ a{\ast}(\tilde{a}\uparrow 2){\ast}\cdots {\ast}(a\uparrow 2^{j-2}){\ast}(b_{\ell}\uparrow 2^{j-1}),2^{j}}w_{\ell,j} =\sum _{k\in \mathbb{Z}}w_{\ell,j}(k)b_{\ell,j;k}, {}\\ \end{array}$$
where a j; k and b , j; k are defined to be
$$\displaystyle{a_{j;k}:= 2^{j/2}a_{ j}(\cdot - 2^{j}k),\quad b_{\ell,j;k}:= 2^{j/2}b_{\ell,j}(\cdot - 2^{j}k),\qquad j \in \mathbb{N}_{ 0},k \in \mathbb{Z}.}$$
Now a J-level discrete framelet transform employing a dual framelet filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\), {a; b 1, , b s }) can be equivalently rewritten as
$$\displaystyle{ v =\sum _{k\in \mathbb{Z}}\langle v,\tilde{a}_{J;k}\rangle a_{J;k} +\sum _{ j=1}^{J}\sum _{ \ell=1}^{s}\sum _{ k\in \mathbb{Z}}\langle v,\tilde{b}_{\ell,j;k}\rangle b_{\ell,j;k}. }$$
(1.3.18)

By employing the dilation factor 2, a multilevel discrete framelet transform provides a multiscale representation of a signal, which is the key to extract the multiscale structure in a signal. The representation in (1.3.18) also shows that the stability of a multilevel discrete framelet transform in the space \(l_{2}(\mathbb{Z})\) is closely related to the asymptotic behavior of the sequences a J (and \(\tilde{a}_{J}\)) in (1.3.17) as J, which is in turn closely related to the behavior of the frequency-based refinable function \(\boldsymbol{\varphi }^{a}(\xi ):=\prod _{ j=1}^{\infty }\widehat{a}(2^{-j}\xi )\) for \(\xi \in \mathbb{R}\). Roughly speaking, \(2^{J}a_{J}(k) = \mathcal{S}_{a}^{J}\boldsymbol{\delta }(k) \approx \phi ^{a}(2^{-J}k),k \in \mathbb{Z}\) as J, where \(\phi ^{a}(x):= \frac{1} {2\pi }\int _{\mathbb{R}}\boldsymbol{\varphi }^{a}(\xi )e^{i\xi x}d\xi\) is the inverse Fourier transform of \(\boldsymbol{\varphi }^{a}\). We shall address the stability issue and refinable functions in Chaps.  5 and  6 of this book.

The above discussion motivates us to define discrete affine systems as follows:
$$\displaystyle{ \begin{array}{rl} \mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b_{1},\ldots,b_{s}\}):=&\{a_{J;k}\;:\; k \in \mathbb{Z}\} \\ & \cup \{ b_{\ell,j;k}\;:\;\ell= 1,\ldots,s,j = 1,\ldots,J,k \in \mathbb{Z}\} \end{array} }$$
(1.3.19)
and similarly
$$\displaystyle{\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}):=\{\tilde{ a}_{J;k}\;:\; k \in \mathbb{Z}\} \cup \{\tilde{ b}_{\ell,j;k}\;:\;\ell= 1,\ldots,s,j = 1,\ldots,J,k \in \mathbb{Z}\}.}$$
Under the convention that
$$\displaystyle{\sim \,:\mathop{ \mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b_{1},\ldots,b_{s}\}) \rightarrow \mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\})\quad \mbox{ with}\quad u\mapsto \tilde{u},}$$
that is, \((u,\tilde{u}\)) is always regarded as a pair together, the representation of \(v \in l_{2}(\mathbb{Z})\) in (1.3.18) can be rewritten as
$$\displaystyle{ v =\sum _{u\in \mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b_{1},\ldots,b_{s}\})}\langle v,\tilde{u}\rangle u,\qquad v \in l_{2}(\mathbb{Z}),J \in \mathbb{N}. }$$
(1.3.20)
Therefore, the stability of a filter bank {a; b 1, , b s } in \(l_{2}(\mathbb{Z})\) as defined in (1.3.7) simply means
$$\displaystyle{ C_{1}\|v\|_{l_{2}(\mathbb{Z})}^{2}\leqslant \sum _{ u\in \mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b_{1},\ldots,b_{s}\})}\vert \langle v,u\rangle \vert ^{2}\leqslant C_{ 2}\|v\|_{l_{2}(\mathbb{Z})}^{2},\qquad \forall \,v \in l_{ 2}(\mathbb{Z}) }$$
(1.3.21)
for all \(J \in \mathbb{N}\). This is equivalent to saying that \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b_{1},\ldots,b_{s}\})\) is a frame in \(l_{2}(\mathbb{Z})\) with uniform lower and upper frame bounds for all \(J \in \mathbb{N}\). It is also easy to prove that {a; b 1, , b s } is a tight framelet filter bank if and only if (1.3.21) holds with C 1 = C 2 = 1. Furthermore, {a; b} is an orthogonal wavelet filter bank if and only if \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b\})\) is an orthonormal basis for \(l_{2}(\mathbb{Z})\) for every \(J \in \mathbb{N}\) (see Exercises 1.24–1.27).
We complete this section on stability of a multilevel discrete framelet transform by presenting an example to illustrate the elements in a discrete affine system. Let {a; b} be an orthogonal wavelet filter bank given by
$$\displaystyle{ \begin{array}{rl} &a =\{ \tfrac{1+\sqrt{3}} {8},\underline{ \tfrac{\mathbf{3}+\sqrt{\mathbf{3}}} {\mathbf{8}} }, \tfrac{3-\sqrt{3}} {8}, \tfrac{1-\sqrt{3}} {8} \}_{[-1,2]},\qquad \\ &b =\{ \tfrac{1-\sqrt{3}} {8},\underline{ \tfrac{\sqrt{\mathbf{3}}\mathbf{-3}} {\mathbf{8}} }, \tfrac{3+\sqrt{3}} {8},-\tfrac{1+\sqrt{3}} {8} \}_{[-1,2]}. \end{array} }$$
(1.3.22)
Note that each \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b\})\) is an orthonormal basis of \(l_{2}(\mathbb{Z})\). Some generators of the discrete affine system \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b\})\) are presented in Fig. 1.4.
Fig. 1.4

{a; b} is the orthogonal wavelet filter bank given in (1.3.22). Some generators of the discrete affine systems \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b\})\), which are orthonormal bases of \(l_{2}(\mathbb{Z})\). (a) a 1; 0. (b) a 2; 0. (c) a 3; 0. (d) a 4; 0. (e) b 1; 0. (f) b 2; 0. (g) b 3; 0. (h) b 4; 0

1.3.4 Nonstationary and Undecimated Discrete Framelet Transforms

For some applications, a standard discrete framelet transform is often modified to achieve better performance. Here we discuss some of them such as nonstationary multilevel discrete framelet transforms, framelet packets, and undecimated discrete framelet transforms.

Let us first discuss a nonstationary multilevel discrete framelet transform which includes wavelet packets and undecimated discrete framelet transforms as special cases. The key idea of a nonstationary multilevel discrete framelet transform is to use possibly different filter banks at every scale level. Let
$$\displaystyle{ \big(\{\tilde{a}_{j,1},\ldots,\tilde{a}_{j,r_{j}};\tilde{b}_{j,1},\ldots,\tilde{b}_{j,s_{j}}\},\{a_{j,1},\ldots,a_{j,r_{j}};b_{j,1},\ldots,b_{j,s_{j}}\}\big),\;1\leqslant j\leqslant J }$$
(1.3.23)
be a sequence of filter banks. A J-level nonstationary discrete framelet decomposition with the J-level nonstationary filter bank in (1.3.23) is given by
$$\displaystyle\begin{array}{rcl} & & v_{j:k_{1},k_{2},\ldots,k_{j}}:= \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{a}_{j,k_{j}}}v_{j-1;k_{1},\ldots,k_{j-1}},\qquad k_{j} = 1,\ldots,r_{j}, {}\\ & & w_{\ell,j:k_{1},k_{2},\ldots,k_{j}}:= \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{b}_{j,\ell}}v_{j-1:k_{1},\ldots,k_{j-1}},\qquad \ell = 1,\ldots,s_{j}\quad \mbox{ and}\quad k_{j} = 1,\ldots,r_{j}, {}\\ \end{array}$$
for j = 1, , J, where v 0;  (i.e., v 0) is an input signal and we used the convention that the subscript index chain k m , , k n is empty if m > n. A J-level nonstationary discrete framelet reconstruction with the J-level nonstationary filter bank in (1.3.23) is given by
$$\displaystyle{\mathring{v}_{j-1:k_{1},k_{2},\ldots,k_{j-1}}:= \frac{\sqrt{2}} {2} \sum _{k_{j}=1}^{r_{j} }\mathcal{S}_{a_{j,k_{ j}}}v_{j:k_{1},k_{2},\ldots,k_{j}}+\frac{\sqrt{2}} {2} \sum _{\ell=1}^{s_{j} }\mathcal{S}_{b_{j,\ell}}w_{\ell,j:k_{1},k_{2},\ldots,k_{j}}}$$
for j = J, , 1. If all the pairs in (1.3.23) are dual framelet filter banks, then the above nonstationary J-level discrete framelet transform has the perfect reconstruction property. The word nonstationary refers to the fact that the filter bank at the scale level j in (1.3.23) depends on the scale level j. A standard multilevel discrete framelet transform in Sect. 1.3.1 uses the same filter bank for all scale levels with r j = 1 and therefore, it is formally called a (stationary) discrete framelet transform. If all s j are zero (that is, no child filters), then it corresponds to a (nonstationary) framelet packet. Furthermore, if r j = 2 and s j = 0, then it is called a (nonstationary) wavelet packet.

Observe that
$$\displaystyle{ \begin{array}{rl} &\mathcal{S}_{u}(v(\cdot - n)) = [\mathcal{S}_{u}v](\cdot - 2n),\qquad \mathcal{S}_{u(\cdot -n)}v = [\mathcal{S}_{u}v](\cdot - n),\qquad n \in \mathbb{Z}, \\ &\mathcal{T}_{u}(v(\cdot - 2n)) = [\mathcal{T}_{u}v](\cdot - n),\qquad \mathcal{T}_{u(\cdot +2n)}v = [\mathcal{T}_{u}v](\cdot - n),\qquad n \in \mathbb{Z}. \end{array} }$$
(1.3.24)
Hence, if we shift an input signal v or a filter u by an integer, then its output under the subdivision operator is a shifted version of \(\mathcal{S}_{u}v\). But for the transition operator, \(\mathcal{T}_{u}(v(\cdot - n))\) or \(\mathcal{T}_{u(\cdot +n)}v\) is generally no longer a shifted version of \(\mathcal{T}_{u}v\) for an odd integer n. This shift sensitivity of framelet coefficients with respect to a shift of an input signal is not desirable in some applications such as signal denoising, since a simple shift of a noise wouldn’t change the characteristics of a noise. To overcome this difficulty, a simple solution is to consider both sequences \(\mathcal{T}_{u}v\) and \(\mathcal{T}_{u}(v(\cdot - 1))\). Since \(\mathcal{T}_{u}(v(\cdot - 1)) = \mathcal{T}_{u(\cdot +1)}v\), we end up with a discrete framelet transform by considering two sequences \(\mathcal{T}_{u}v\) and \(\mathcal{T}_{u(\cdot +1)}v\) instead of just one sequence \(\mathcal{T}_{u}v\) of framelet coefficients.
Suppose that we have a J-level nonstationary filter bank in (1.3.23). To achieve shift invariance of framelet coefficients, we end up with another J-level nonstationary discrete framelet transform with nonstationary filter banks
$$\displaystyle{ \begin{array}{rl} &\big(\tfrac{\sqrt{2}} {2} \big\{\tilde{a}_{j,1},\tilde{a}_{j,1}(\cdot + 1),\ldots,\tilde{a}_{j,r_{j}},\tilde{a}_{j,r_{j}}(\cdot + 1);\tilde{b}_{j,1},\tilde{b}_{j,1}(\cdot + 1),\ldots,\tilde{b}_{j,s_{j}},\tilde{b}_{j,s_{j}}(\cdot + 1)\big\}, \\ &\tfrac{\sqrt{ 2}} {2} \big\{a_{j,1},a_{j,1}(\cdot + 1),\ldots,a_{j,r_{j}},a_{j,r_{j}}(\cdot + 1);b_{j,1},b_{j,1}(\cdot + 1),\ldots,b_{j,s_{j}},b_{j,s_{j}}(\cdot + 1)\big\}\big)\end{array} }$$
(1.3.25)
for j = 1, , J. Simply speaking, the above new nonstationary filter banks are obtained by replacing each filter \(\tilde{u}\) in (1.3.23) with \(\frac{\sqrt{2}} {2} \tilde{u}\) and \(\frac{\sqrt{ 2}} {2} \tilde{u}(\cdot + 1)\). Note that
$$\displaystyle{\mathcal{T}_{\frac{\sqrt{2}} {2} \tilde{u}}v = \sqrt{2}(\tilde{u}^{\star } {\ast} v)(2\cdot )\quad \mbox{ and}\quad \mathcal{T}_{\frac{\sqrt{2}} {2} \tilde{u}(\cdot +1)}v = \sqrt{2}(\tilde{u}^{\star } {\ast} v)(2 \cdot -1).}$$
Consequently, the two sequences \(\mathcal{T}_{\frac{\sqrt{2}} {2} \tilde{u}}v\) and \(\mathcal{T}_{\frac{\sqrt{2}} {2} \tilde{u}(\cdot +1)}v\) putting together in a disjoint way are simply the sequence \(\sqrt{ 2}\tilde{u}^{\star } {\ast} v\). Similarly, it is easy to verify that
$$\displaystyle{ \mathcal{S}_{\frac{\sqrt{2}} {2} u}(w(2\cdot )) + \mathcal{S}_{\frac{\sqrt{2}} {2} u(\cdot +1)}(w(2 \cdot -1)) = \sqrt{2}u {\ast} w. }$$
(1.3.26)
In other words, the new nonstationary discrete framelet transform is undecimated by removing the downsampling (that is, decimation) and upsampling operations in the original discrete framelet transform. Consequently, the J-level nonstationary discrete framelet transform with the new filter bank in (1.3.25) is called a J-level undecimated discrete nonstationary framelet transform employing the filter bank in (1.3.23). By the above discussion, the seemingly complicated undecimated nonstationary discrete framelet transform with the filter bank in (1.3.23) in fact has a very simple structure as follows. A J-level undecimated nonstationary discrete framelet decomposition with the filter bank in (1.3.23) becomes
$$\displaystyle\begin{array}{rcl} & & v_{j:k_{1},k_{2},\ldots,k_{j}}:= (\tilde{a}_{j,k_{j}}^{\star }\! \uparrow \! 2^{j}) {\ast} v_{ j-1:k_{1},\ldots,k_{j-1}},\qquad k_{j} = 1,\ldots,r_{j}\quad \mbox{ and}\quad j = 1,\ldots,J, {}\\ & & w_{\ell,j:k_{1},k_{2},\ldots,k_{j}}:= (\tilde{b}_{j,\ell}^{\star }\! \uparrow \! 2^{j}) {\ast} v_{ j-1:k_{1},\ldots,k_{j-1}},\qquad \ell = 1,\ldots,s_{j}\quad \mbox{ and}\quad j = 1,\ldots,J. {}\\ \end{array}$$
A J-level undecimated nonstationary discrete framelet reconstruction with the filter bank in (1.3.23) becomes
$$\displaystyle{\mathring{v}_{j-1:k_{1},k_{2},\ldots,k_{j-1}}:=\sum _{ k_{j}=1}^{r_{j} }(a_{j,k_{j}}\! \uparrow \! 2^{j}){\ast}\mathring{v}_{ j:k_{1},k_{2},\ldots,k_{j}}+\sum _{\ell=1}^{s_{j} }(b_{j,\ell}\! \uparrow \! 2^{j}){\ast}\mathring{w}_{\ell,j:k_{1},k_{2},\ldots,k_{j}}}$$
for j = J, , 1. To illustrate, we present a J-level undecimated (stationary) discrete framelet transform employing a stationary filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\), {a; b 1, , b s }). A J-level undecimated (stationary) discrete framelet decomposition is given by
$$\displaystyle{v_{j}:= (\tilde{a}^{\star }\! \uparrow \! 2^{j}) {\ast} v_{ j-1},\qquad w_{\ell,j}:= (\tilde{b}_{\ell}^{\star }\! \uparrow \! 2^{j}) {\ast} v_{ j-1},\qquad \ell = 1,\ldots,s,\quad j = 1,\ldots,J,}$$
where \(v_{0}: \mathbb{Z} \rightarrow \mathbb{C}\) is an input signal. A J-level undecimated (stationary) discrete framelet reconstruction is given by
$$\displaystyle{\mathring{v}_{j-1}:= (a\! \uparrow \! 2^{j}){\ast}\mathring{v}_{ j}+\sum _{\ell=1}^{s}(b_{\ell}\! \uparrow \! 2^{j}){\ast}\mathring{w}_{\ell,j},\qquad j = J,\ldots,1.}$$
We observe that the above J-level undecimated (stationary) discrete framelet transform has the perfect reconstruction property if and only if
$$\displaystyle{ \widehat{\tilde{a}}(\xi )\overline{\widehat{a}(\xi )} +\widehat{\tilde{ b}_{1}}(\xi )\overline{\widehat{b_{1}}(\xi )} + \cdots +\widehat{\tilde{ b}_{s}}(\xi )\overline{\widehat{b_{s}}(\xi )} = 1. }$$
(1.3.27)
If (1.3.27) holds, then we have the following signal representation similar to (1.3.18):
$$\displaystyle{v =\sum _{k\in \mathbb{Z}}\big\langle v,\tilde{a}_{J}(\cdot - k)\big\rangle a_{J}(\cdot - k) +\sum _{ j=0}^{J-1}\sum _{ \ell=1}^{s}\sum _{ k\in \mathbb{Z}}\langle v,\tilde{b}_{\ell,j}(\cdot - k)\rangle b_{\ell,j}(\cdot - k)}$$
whose underlying discrete affine system for reconstruction is
$$\displaystyle{\{a_{J}(\cdot - k)\;:\; k \in \mathbb{Z}\} \cup \{ b_{\ell,j}(\cdot - k)\;:\; j = 1,\ldots,J,\ell= 1,\ldots,s,k \in \mathbb{Z}\}.}$$
See Fig. 1.5 for a diagram of a two-level undecimated (stationary) discrete framelet transform.
Fig. 1.5

Diagram of a two-level discrete undecimated framelet transform using filter banks \(\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\) and {a; b 1, , b s }

1.4 The Oblique Extension Principle (OEP)

In this section we introduce a generalized discrete framelet transform based on the oblique extension principle (OEP). Let us first explain our motivation for generalizing the standard discrete framelet transform described in Sects. 1.1 and 1.3.

1.4.1 Oblique Extension Principle

As we discussed in Sect. 1.2, to have sparse representations for smooth signals, it is important for high-pass filters to possess high vanishing moments and for low-pass filters to have high sum rules. However, vanishing moments of high-pass filters put some necessary constraints on low-pass filters in a dual framelet filter bank, as shown by the following result.

Lemma 1.4.1

Let\((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})\)be a dual framelet filter bank. If all primal high-pass filters b 1, , b s have\(\tilde{m}\)vanishing moments and all dual high-pass filters\(\tilde{b}_{1},\ldots,\tilde{b}_{s}\)have m vanishing moments, where m and\(\tilde{m}\)are nonnegative integers satisfying\(m +\tilde{ m}> 0\), then
  1. (i)

    the primal low-pass filter a must have m sum rules:\(\widehat{a}(\xi +\pi ) =\mathbb{ O}(\vert \xi \vert ^{m})\), ξ → 0;

     
  2. (ii)

    the dual low-pass filter\(\tilde{a}\)must have\(\tilde{m}\)sum rules:\(\widehat{\tilde{a}}(\xi +\pi ) =\mathbb{ O}(\vert \xi \vert ^{\tilde{m}})\), ξ → 0;

     
  3. (iii)
    \(\widehat{\tilde{a}}\overline{\widehat{a}}\)(or equivalently\(\tilde{a} {\ast} a^{\star }\)) has\(m +\tilde{ m}\)linear-phase moments with phase 0:
    $$\displaystyle{ 1 -\widehat{\tilde{ a}}(\xi )\overline{\widehat{a}(\xi )} =\mathbb{ O}(\vert \xi \vert ^{m+\tilde{m}}),\qquad \xi \rightarrow 0. }$$
    (1.4.1)
     

Proof

Since \(\widehat{b_{\ell}}(\xi ) =\mathbb{ O}(\vert \xi \vert ^{\tilde{m}})\) and \(\widehat{\tilde{b}_{\ell}}(\xi ) =\mathbb{ O}(\vert \xi \vert ^{m})\) as ξ → 0 for = 1, , s, it is straightforward to deduce from (1.3.6) that (1.4.1) holds. Thus, item (iii) holds. In particular, by \(m +\tilde{ m}> 0\), we have \(\widehat{\tilde{a}}(0)\overline{\widehat{a}(0)} = 1\). By (1.3.6), we have
$$\displaystyle{\widehat{\tilde{a}}(\xi )\overline{\widehat{a}(\xi +\pi )} =\widehat{\tilde{ b}_{1}}(\xi )\overline{\widehat{b_{1}}(\xi +\pi )} + \cdots +\widehat{\tilde{ b}_{s}}(\xi )\overline{\widehat{b_{s}}(\xi +\pi )} =\mathbb{ O}(\vert \xi \vert ^{m}),\qquad \xi \rightarrow 0.}$$
Since \(\widehat{\tilde{a}}(0)\neq 0\), we must have \(\widehat{a}(\xi +\pi ) =\mathbb{ O}(\vert \xi \vert ^{m})\) as ξ → 0. That is, item (i) holds. By the same argument, we see that item (ii) holds. □
Since (1.4.1) implies \(\widehat{\tilde{a}}(0)\overline{\widehat{a}(0)} = 1\), we often normalize a low-pass filter a by \(\widehat{a}(0) = 1\). Let \(\widehat{a}(\xi ) =\widehat{ a_{m}^{B}}(\xi ) = 2^{-m}(1 + e^{-i\xi })^{m}\) and \(\widehat{\tilde{a}}(\xi ) =\widehat{ a_{\tilde{m}}^{B}}(\xi ) = 2^{-\tilde{m}}(1 + e^{-i\xi })^{\tilde{m}}\) be two B-spline low-pass filters in (1.2.24). Then \(\widehat{\tilde{a}}(\xi )\overline{\widehat{a}(\xi )} = 2^{-m-\tilde{m}}(1 + e^{-i\xi })^{\tilde{m}}(1 + e^{i\xi })^{m}\) and it is not difficult to check that
$$\displaystyle{\widehat{\tilde{a}}(0)\overline{\widehat{a}(0)} = 1,\quad \big[\widehat{\tilde{a}}\overline{\widehat{a}}\big]'(0) = \tfrac{i(m-\tilde{m})} {2},\quad \big[\widehat{\tilde{a}}\overline{\widehat{a}}\big]''(0) = \tfrac{(m-\tilde{m})^{2}+m+\tilde{m}} {4}.}$$
Note that \(\big[\widehat{\tilde{a}}\overline{\widehat{a}}\big]'(0) = 0\) if and only if \(m =\tilde{ m}\). Regardless of the choices of the positive integers m and \(\tilde{m}\), the above identities imply that (1.4.1) cannot be true if \(m +\tilde{ m}> 2\). More generally, let \(n,\tilde{n}\) be any integers and define \(\widehat{a}(\xi ):= e^{-in\xi }\widehat{a_{m}^{B}}(\xi )\) and \(\widehat{\tilde{a}}(\xi ):= e^{-i\tilde{n}\xi }\widehat{a_{\tilde{m}}^{B}}(\xi )\). Then we can show (see Exercise 1.28) that the relation \(\widehat{\tilde{a}}(\xi )\overline{\widehat{a}(\xi )} = 1 + O(\vert \xi \vert ^{3}),\xi \rightarrow 0\) can never be true, regardless of the choices of \(m,n,\tilde{m},\tilde{n}\). Consequently, for any dual framelet filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})\) with B-spline low-pass filters a and \(\tilde{a}\), some primal high-pass filters b 1, , b s and some dual high-pass filters \(\tilde{b}_{1},\ldots,\tilde{b}_{s}\) must have no more than one vanishing moment. Therefore, a more general filter bank is needed in order to improve vanishing moments of high-pass filters derived from B-spline low-pass filters.

The main goal of the following oblique extension principle (OEP) is to increase vanishing moments of high-pass filters derived from a given pair of low-pass filters.

Theorem 1.4.2 (Oblique Extension Principle)

Let \(\varTheta,\tilde{a},\tilde{b}_{1},\ldots,\tilde{b}_{s},a,b_{1},\ldots,b_{s} \in l_{0}(\mathbb{Z})\) be finitely supported sequences on \(\mathbb{Z}\) . Then the following statements are equivalent:
  1. (i)
    The filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\) has the following generalized perfect reconstruction property: for all \(v \in l(\mathbb{Z})\) ,
    $$\displaystyle{ \varTheta ^{\star } {\ast} v = \frac{1} {2}\mathcal{S}_{a}(\varTheta ^{\star } {\ast}\mathcal{T}_{\tilde{ a}}v) + \frac{1} {2}\sum _{\ell=1}^{s}\mathcal{S}_{ b_{\ell}}\mathcal{T}_{\tilde{b}_{\ell}}v. }$$
    (1.4.2)
     
  2. (ii)

    The identity in (1.4.2) holds for all\(v \in l_{0}(\mathbb{Z})\).

     
  3. (iii)

    The identity in (1.4.2) holds for the two particular sequences\(v =\boldsymbol{\delta }\)andδ(⋅ − 1).

     
  4. (iv)
    The following perfect reconstruction condition holds: for all \(\xi \in \mathbb{R}\) ,
    $$\displaystyle{ \widehat{\varTheta }(2\xi )\widehat{\tilde{a}}(\xi )\overline{\widehat{a}(\xi )} +\widehat{\tilde{ b}_{1}}(\xi )\overline{\widehat{b_{1}}(\xi )} + \cdots +\widehat{\tilde{ b}_{s}}(\xi )\overline{\widehat{b_{s}}(\xi )} =\widehat{\varTheta } (\xi ), }$$
    (1.4.3)
    $$\displaystyle{ \widehat{\varTheta }(2\xi )\widehat{\tilde{a}}(\xi )\overline{\widehat{a}(\xi +\pi )} +\widehat{\tilde{ b}_{1}}(\xi )\overline{\widehat{b_{1}}(\xi +\pi )} + \cdots +\widehat{\tilde{ b}_{s}}(\xi )\overline{\widehat{b_{s}}(\xi +\pi )} = 0. }$$
    (1.4.4)
     

Proof

Taking Fourier series on both sides of (1.4.2), we see that (1.4.2) is equivalent to
$$\displaystyle\begin{array}{rcl} \overline{\widehat{\varTheta }(\xi )}\widehat{v}(\xi )& =& \widehat{v}(\xi )\Big[\overline{\widehat{\varTheta }(2\xi )}\,\overline{\widehat{\tilde{a}}(\xi )}\widehat{a}(\xi ) +\sum _{ \ell=1}^{s}\overline{\widehat{\tilde{b}_{\ell}}(\xi )}\widehat{b_{\ell}}(\xi )\Big] {}\\ & & \qquad +\widehat{ v}(\xi +\pi )\Big[\overline{\widehat{\varTheta }(2\xi )}\,\overline{\widehat{\tilde{a}}(\xi +\pi )}\widehat{a}(\xi ) +\sum _{ \ell=1}^{s}\overline{\widehat{\tilde{b}_{\ell}}(\xi +\pi )}\widehat{b_{\ell}}(\xi )\Big]. {}\\ \end{array}$$
All the claims follow from the same argument as in the proof of Theorem 1.1.1. □

1.4.2 OEP-Based Tight Framelet Filter Banks

For OEP-based filter banks, we have the following result generalizing Theorem 1.1.4:

Theorem 1.4.3

Let\(\theta,a,b_{1},\ldots,b_{s} \in l_{0}(\mathbb{Z})\)be sequences on\(\mathbb{Z}\). Then
$$\displaystyle{\|\theta {\ast}\mathcal{T}_{a}v\|_{l_{2}(\mathbb{Z})}^{2} +\| \mathcal{T}_{ b_{1}}v\|_{l_{2}(\mathbb{Z})}^{2} + \cdots +\| \mathcal{T}_{ b_{s}}v\|_{l_{2}(\mathbb{Z})}^{2} = 2\|\theta {\ast} v\|_{ l_{2}(\mathbb{Z})}^{2},\qquad \forall \;v \in l_{ 2}(\mathbb{Z}),}$$
if and only if the filter bank {a; b 1, , b s } Θ satisfies
$$\displaystyle{ \left [\begin{array}{*{10}c} \widehat{b_{1}}(\xi ) &\cdots & \widehat{b_{s}}(\xi ) \\ \widehat{b_{1}}(\xi +\pi )&\cdots &\widehat{b_{s}}(\xi +\pi ) \end{array} \right ]\left [\begin{array}{*{10}c} \widehat{b_{1}}(\xi ) &\cdots & \widehat{b_{s}}(\xi ) \\ \widehat{b_{1}}(\xi +\pi )&\cdots &\widehat{b_{s}}(\xi +\pi ) \end{array} \right ]^{\star } = \mathcal{M}_{ a,\varTheta }(\xi ), }$$
(1.4.5)
where
$$\displaystyle{ \mathcal{M}_{a,\varTheta }(\xi ):= \left [\begin{array}{*{10}c} \widehat{\varTheta }(\xi ) -\widehat{\varTheta } (2\xi )\vert \widehat{a}(\xi )\vert ^{2} & -\widehat{\varTheta }(2\xi )\widehat{a}(\xi )\overline{\widehat{a}(\xi +\pi )} \\ -\widehat{\varTheta }(2\xi )\widehat{a}(\xi +\pi )\overline{\widehat{a}(\xi )} &\widehat{\varTheta }(\xi +\pi ) -\widehat{\varTheta } (2\xi )\vert \widehat{a}(\xi +\pi )\vert ^{2} \end{array} \right ] }$$
(1.4.6)
and
$$\displaystyle{ \varTheta:=\theta {\ast}\theta ^{\star },\quad \mathit{\mbox{ that is}},\quad \widehat{\varTheta }(\xi ):= \vert \widehat{\theta }(\xi )\vert ^{2}. }$$
(1.4.7)

Proof

Note that Θ = Θ and \(\|\theta {\ast}v\|_{l_{2}(\mathbb{Z})}^{2} = \frac{1} {2\pi }\int _{-\pi }^{\pi }\vert \widehat{\theta }(\xi )\vert ^{2}\vert \widehat{v}(\xi )\vert ^{2}d\xi =\langle \varTheta ^{\star } {\ast} v,v\rangle\). By the relation of \(\mathcal{S}_{a}\) and \(\mathcal{T}_{a}\) in Lemma 1.1.3,
$$\displaystyle{\|\theta {\ast}\mathcal{T}_{a}v\|_{l_{2}(\mathbb{Z})}^{2} =\langle \theta {\ast}\mathcal{T}_{ a}v,\theta {\ast}\mathcal{T}_{a}v\rangle =\langle \varTheta ^{\star } {\ast}\mathcal{T}_{ a}v,\mathcal{T}_{a}v\rangle =\langle \mathcal{S}_{a}(\varTheta ^{\star } {\ast}\mathcal{T}_{ a}v),v\rangle.}$$
All claims follow from the same proof as in Theorem 1.1.4. □

A filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\) satisfying the perfect reconstruction condition in (1.4.3) and (1.4.4) is called an (OEP-based) dual framelet filter bank. Similarly, an OEP-based filter bank {a; b 1, , b s } Θ satisfying the perfect reconstruction condition in (1.4.5) is called a tight framelet filter bank. From the perfect reconstruction condition in (1.4.3) and (1.4.4), it is straightforward to see that \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\) is a dual framelet filter bank if and only if \((\{a;b_{1},\ldots,b_{s}\},\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\})_{\varTheta ^{\star }}\) is a dual framelet filter bank.

Here is an example using B-spline filters. We shall provide many examples in Chap.  3 to illustrate how the oblique extension principle can be used to improve vanishing moments of high-pass filters.

Example 1.4.1

{a; b 1, b 2} Θ is a tight framelet filter bank, where
$$\displaystyle\begin{array}{rcl} a& =& \{\tfrac{1} {4},\underline{ \tfrac{\mathbf{1}} {\mathbf{2}}}, \tfrac{1} {4}\}_{[-1,1]},\qquad b_{1} =\{ -\tfrac{1} {4},\underline{ \tfrac{\mathbf{1}} {\mathbf{2}}},-\tfrac{1} {4}\}_{[-1,1]},\qquad b_{2} = \tfrac{\sqrt{6}} {24} \{1,\underline{\mathbf{2}},-6,2,1\}_{[-1,3]}, {}\\ \varTheta & =& \{-\tfrac{1} {6},\underline{ \tfrac{\mathbf{4}} {\mathbf{3}}},-\tfrac{1} {6}\}_{[-1,1]}. {}\\ \end{array}$$
In comparison with Example 1.1.3 where one of the two high-pass filters has only one vanishing moment, both high-pass filters b 1 and b 2 here have two vanishing moments.

The following well-known Fejér-Riesz Lemma will be needed later.

Lemma 1.4.4 (the Fejér-Riesz Lemma)

Let\(\widehat{\varTheta }\)be a 2π-periodic trigonometric polynomial with real coefficients (or with complex coefficients) such that\(\widehat{\varTheta }(\xi )\geqslant 0\)for all\(\xi \in \mathbb{R}\). Then there exists a 2π-periodic trigonometric polynomial\(\widehat{\theta }\)with real coefficients (or with complex coefficients) such that\(\vert \widehat{\theta }(\xi )\vert ^{2} =\widehat{\varTheta } (\xi )\)for all\(\xi \in \mathbb{R}\). Moreover, we can further require\(\widehat{\theta }(0) = \sqrt{\widehat{\varTheta }(0)}\).

Proof

Since \(\widehat{\varTheta }(\xi ) =\sum _{k\in \mathbb{Z}}\varTheta (k)e^{-ik\xi }\), we define a Laurent polynomial Open image in new window . Let Z be the set of all the roots, counting multiplicity, of Open image in new window . Since \(\overline{\widehat{\varTheta }(\xi )} =\widehat{\varTheta } (\xi )\) implies Open image in new window , we see that Z is invariant under the mapping \(z\mapsto \bar{z}^{-1}\). Denote \(\mathbb{T}:=\{ z \in \mathbb{C}\;:\; \vert z\vert = 1\}\). Then \(\mathbb{T}\) is the invariant set of the mapping \(z\mapsto \bar{z}^{-1}\). Since Open image in new window for all \(z \in \mathbb{T}\), we see that any point in \(\mathbb{T} \cap Z\) has an even multiplicity in Z. Consequently, Z can be described as \(\{\zeta,\bar{\zeta }^{-1}\;:\;\zeta \in Y \}\) for a unique subset Y of \(Z \cap \{ z \in \mathbb{C}\;:\; \vert z\vert \leqslant 1\}\). Define \(\boldsymbol{\theta }(z):=\prod _{\zeta \in Y }(z-\zeta )\). Then Open image in new window for some c > 0. Set \(\widehat{\theta }(\xi ):= c\boldsymbol{\theta }(e^{-i\xi })\). Now it is straightforward to check that \(\vert \widehat{\theta }(\xi )\vert ^{2} =\widehat{\varTheta } (\xi )\) for all \(\xi \in \mathbb{R}\).

We now show that if \(\widehat{\varTheta }\) has real coefficients, then so does \(\widehat{\theta }\). Since \(\widehat{\varTheta }\) has real coefficients, we have Open image in new window , from which we see that Z is invariant under the mapping \(z\mapsto \bar{z}\). Note that the real line \(\mathbb{R}\) is the invariant set of the mapping \(z\mapsto \bar{z}\). Now we see that the set Y can be described as \(\{x,\bar{x}\;:\; x \in Y,\mathop{\mathrm{Im}}\nolimits (x)> 0\} \cup \{ y \in Y \;:\;\mathop{ \mathrm{Im}}\nolimits (y) = 0\}\). From the definition of \(\boldsymbol{\theta }\), now we see that \(\boldsymbol{\theta }\) must have real coefficients and therefore, \(\widehat{\theta }\) has real coefficients. □

Up to a factor e i(+c) for some integer n and \(c \in \mathbb{R}\), the 2π-periodic trigonometric polynomial \(\widehat{\theta }\) constructed in the proof of the above Fejér-Riesz Lemma is unique and there are many numerical algorithms in the literature to compute \(\widehat{\theta }\). However, other than the choice in the proof of Lemma 1.4.4, there are many other choices of pairing the roots of Open image in new window which lead to different 2π-periodic trigonometric polynomials \(\widehat{\theta }\) (with real coefficients) satisfying \(\vert \widehat{\theta }(\xi )\vert ^{2} =\widehat{\varTheta } (\xi )\). For this reason, \(\widehat{\theta }\) constructed in the proof of Lemma 1.4.4 is often called the canonical choice.

We now show that a filter Θ in every tight framelet filter bank must take the special form in (1.4.7). Recall that an r × r matrix U of complex numbers is positive semidefinite, denoted by \(U\geqslant 0\), if \(\bar{x}^{\mathsf{T}}Ux\geqslant 0\) for all \(x \in \mathbb{C}^{r}\).

Lemma 1.4.5

Let\(a,\varTheta \in l_{0}(\mathbb{Z})\setminus \{0\}\)and\(\mathcal{M}_{a,\varTheta }\)be defined in (1.4.6). Then
$$\displaystyle{ \mathcal{M}_{a,\varTheta }(\xi )\geqslant 0\qquad \forall \;\xi \in \mathbb{R}, }$$
(1.4.8)
if and only if for all\(\xi \in \mathbb{R}\),\(\widehat{\varTheta }(\xi )\geqslant 0\)and
$$\displaystyle{ \begin{array}{rll} \det (\mathcal{M}_{a,\varTheta }(\xi )) =&\widehat{\varTheta }(\xi )\widehat{\varTheta }(\xi +\pi ) \\ & -\widehat{\varTheta } (2\xi )\big[\widehat{\varTheta }(\xi +\pi )\vert \widehat{a}(\xi )\vert ^{2} +\widehat{\varTheta } (\xi )\vert \widehat{a}(\xi +\pi )\vert ^{2}\big]\geqslant 0.\end{array} }$$
(1.4.9)

Proof

The inequality (1.4.8) implies \(\mathcal{M}_{a,\varTheta }^{\star }(\xi ) = \mathcal{M}_{a,\varTheta }(\xi )\), \(\det (\mathcal{M}_{a,\varTheta }(\xi ))\geqslant 0\), and
$$\displaystyle{ \widehat{\varTheta }(\xi ) -\widehat{\varTheta } (2\xi )\vert \widehat{a}(\xi )\vert ^{2}\geqslant 0\qquad \forall \,\xi \in \mathbb{R}, }$$
(1.4.10)
which is the (1, 1)-entry of \(\mathcal{M}_{a,\varTheta }\). By \(\overline{[\mathcal{M}_{a,\varTheta }(\xi )]_{1,2}} = [\mathcal{M}_{a,\varTheta }(\xi )]_{2,1}\), we must have \(\overline{\widehat{\varTheta }(\xi )} =\widehat{\varTheta } (\xi )\) since \(\widehat{a}\) is not identically zero. Hence, \(\widehat{\varTheta }(\xi ) \in \mathbb{R}\) for all \(\xi \in \mathbb{R}\).
Suppose \(\widehat{\varTheta }(\xi ) <0\) for ξ ∈ (c, d) for some c < d. Then (1.4.10) implies \(\widehat{\varTheta }(\xi ) <0\) for all ξ ∈ (2 j c, 2 j d) and \(j \in \mathbb{N}\). For \(j \in \mathbb{N}\) such that 2 j (dc) > 2π, since \(\widehat{\varTheta }\) is 2π-periodic, this leads to \(\widehat{\varTheta }(\xi ) <0\) for all \(\xi \in \mathbb{R}\). Now by (1.4.9) and \(\widehat{\varTheta }(\xi ) <0\),
$$\displaystyle{ \big[\widehat{\varTheta }(\xi ) -\widehat{\varTheta } (2\xi )\vert \widehat{a}(\xi )\vert ^{2}\big]\widehat{\varTheta }(\xi +\pi )\geqslant \widehat{\varTheta }(2\xi )\widehat{\varTheta }(\xi )\vert \widehat{a}(\xi +\pi )\vert ^{2}\geqslant 0 }$$
(1.4.11)
for all \(\xi \in \mathbb{R}\). Since \(\widehat{\varTheta }(\xi ) <0\) for all \(\xi \in \mathbb{R}\), the above inequality and (1.4.10) imply
$$\displaystyle{ \widehat{\varTheta }(\xi ) -\widehat{\varTheta } (2\xi )\vert \widehat{a}(\xi )\vert ^{2} = 0\qquad \forall \,\xi \in \mathbb{R}. }$$
(1.4.12)
From (1.4.11) and (1.4.12), we must have \(\widehat{\varTheta }(2\xi )\widehat{\varTheta }(\xi )\vert \widehat{a}(\xi +\pi )\vert ^{2} = 0\) contradicting our assumption \(a,\varTheta \in l_{0}(\mathbb{Z})\setminus \{0\}\). This proves that \(\widehat{\varTheta }(\xi )\geqslant 0\) for all \(\xi \in \mathbb{R}\).

Conversely, if \(\widehat{\varTheta }(\xi )\geqslant 0\) and (1.4.9) holds, then (1.4.11) holds. By \(\widehat{\varTheta }(\xi )\geqslant 0\), we conclude that (1.4.10) holds. That is, the (1, 1)-entry of \(\mathcal{M}_{a,\varTheta }\) must be nonnegative. Since (1.4.9) holds, by a standard result from linear algebra, (1.4.8) must hold. □

As a direct consequence of Lemma 1.4.5, we have

Corollary 1.4.6

If {a; b 1, , b s } Θ is a tight framelet filter bank, then\(\widehat{\varTheta }(\xi )\geqslant 0\)for all\(\xi \in \mathbb{R}\)and consequently, there exists\(\theta \in l_{0}(\mathbb{Z})\)such that\(\vert \widehat{\theta }(\xi )\vert ^{2} =\widehat{\varTheta } (\xi )\)and\(\widehat{\theta }(0) = \sqrt{\widehat{\varTheta }(0)}\).

Proof

By (1.4.5), we have \(\mathcal{M}_{a,\varTheta }(\xi )\geqslant 0\) for all \(\xi \in \mathbb{R}\). If a = 0, then \(\mathcal{M}_{a,\varTheta }(\xi )\geqslant 0\) directly implies \(\widehat{\varTheta }(\xi )\geqslant 0\). If \(a \in l_{0}(\mathbb{Z})\setminus \{0\}\), then it follows from Lemma 1.4.5 that \(\widehat{\varTheta }(\xi )\geqslant 0\). By the Fejér-Riesz Lemma in Lemma 1.4.4, (1.4.7) holds for some \(\theta \in l_{0}(\mathbb{Z})\) with \(\widehat{\theta }(0) = \sqrt{\widehat{\varTheta }(0)}\). □

1.4.3 OEP-Based Filter Banks with One Pair of High-Pass Filters

Let \(u \in l_{0}(\mathbb{Z})\) be a finitely supported sequence. Recall that \(\mathop{\mathrm{fsupp}}\nolimits (u) = [m,n]\) is the filter support of u if u vanishes outside [m, n] and u(m)u(n) ≠ 0. If u is the zero sequence, by default \(\mathop{\mathrm{fsupp}}\nolimits (u) =\emptyset\), the empty set. For simplicity, we also set \(\mathop{\mathrm{fsupp}}\nolimits (\widehat{u}) =\mathop{ \mathrm{fsupp}}\nolimits (u)\) and \(\mathop{\mathrm{len}}\nolimits (u):= n - m\).

For some applications, the number of high-pass filters is preferred to be as small as possible. As demonstrated by the following result, the number of high-pass filters in an (OEP-based) dual framelet filter bank can seldom be s = 1; otherwise, it is essentially a usual biorthogonal wavelet filter bank.

Theorem 1.4.7

Let\((\{\tilde{a};\tilde{b}\},\{a;b\})_{\varTheta }\)be a dual framelet filter bank such that Θ is not identically zero. Then there exists a nonzero number\(\lambda \in \mathbb{C}\setminus \{0\}\)such that
$$\displaystyle{ \widehat{\varTheta }(2\xi ) =\lambda \widehat{\varTheta } (\xi )\widehat{\varTheta }(\xi +\pi ),\qquad \forall \;\xi \in \mathbb{R} }$$
(1.4.13)
and
$$\displaystyle{ \left [\begin{array}{*{10}c} \widehat{\mathring{a}}(\xi ) & \widehat{\mathring{b}}(\xi )\\ \widehat{\mathring{a}}(\xi +\pi ) &\widehat{\mathring{b}}(\xi +\pi ) \end{array} \right ]\left [\begin{array}{*{10}c} \widehat{a}(\xi ) & \widehat{b}(\xi )\\ \widehat{a}(\xi +\pi ) &\widehat{b}(\xi +\pi ) \end{array} \right ]^{\star } = \left [\begin{array}{*{10}c} 1&0 \\ 0&1 \end{array} \right ], }$$
(1.4.14)
where all the above filters are finitely supported and are given by
$$\displaystyle{ \widehat{\mathring{a}}(\xi ):=\widehat{\tilde{ a}}(\xi )\lambda \widehat{\varTheta }(\xi +\pi )\qquad \mathit{\mbox{ and}}\qquad \widehat{\mathring{b}}(\xi ):=\widehat{\tilde{ b}}(\xi )/\widehat{\varTheta }(\xi ). }$$
(1.4.15)
That is,\((\{\mathring{a};\mathring{b}\},\{a;b\})\)is a biorthogonal wavelet filter bank. Moreover, (1.4.14) implies that
$$\displaystyle{ \begin{array}{rl} &\widehat{\mathring{b}}(\xi ) = \overline{c^{-1}}e^{i(2n-1)\xi }\overline{\widehat{a}(\xi +\pi )},\quad \widehat{b}(\xi ) = ce^{i(2n-1)\xi }\overline{\widehat{\mathring{a}}(\xi +\pi )},\;\; \\ &\qquad \qquad \mathit{\mbox{ for some}}\;\;c \in \mathbb{C}\setminus \{0\},\;n \in \mathbb{Z}. \end{array} }$$
(1.4.16)
If {a; b} Θ is a tight framelet filter bank, then Θ = θθfor some\(\theta \in l_{0}(\mathbb{Z})\)and\(\{\breve{a};\breve{b}\}\)is an orthogonal wavelet filter bank, where\(\breve{a},\breve{b} \in l_{0}(\mathbb{Z})\)are given by
$$\displaystyle{ \widehat{\breve{a}}(\xi ):=\widehat{ a}(\xi )\sqrt{\lambda }\widehat{\theta }(\xi +\pi ),\qquad \widehat{\breve{b}}(\xi ):=\widehat{ b}(\xi )/\widehat{\theta }(\xi ). }$$
(1.4.17)

Proof

(1.4.3) and (1.4.4) with s = 1 can be rewritten as
$$\displaystyle{ \left [\begin{array}{*{10}c} \widehat{\tilde{a}}(\xi ) & \widehat{\tilde{b}}(\xi )\\ \widehat{\tilde{a}}(\xi +\pi ) &\widehat{\tilde{b}}(\xi +\pi ) \end{array} \right ]\left [\begin{array}{*{10}c} \widehat{\varTheta }(2\xi )&0\\ 0 &1 \end{array} \right ]\left [\begin{array}{*{10}c} \widehat{a}(\xi ) & \widehat{b}(\xi )\\ \widehat{a}(\xi +\pi ) &\widehat{b}(\xi +\pi ) \end{array} \right ]^{\star } = \left [\begin{array}{*{10}c} \widehat{\varTheta }(\xi )& 0 \\ 0 &\widehat{\varTheta }(\xi +\pi ) \end{array} \right ]. }$$
(1.4.18)
Taking determinant on both sides of (1.4.18), we have
$$\displaystyle{\widehat{\varTheta }(2\xi )\tilde{\boldsymbol{\theta }}(\xi )\overline{\boldsymbol{\theta }(\xi )} =\widehat{\varTheta } (\xi )\widehat{\varTheta }(\xi +\pi ),}$$
where
$$\displaystyle\begin{array}{rcl} & & \tilde{\boldsymbol{\theta }}(\xi ):= e^{i\xi }\big(\widehat{\tilde{a}}(\xi )\widehat{\tilde{b}}(\xi +\pi ) -\widehat{\tilde{ a}}(\xi +\pi )\widehat{\tilde{b}}(\xi )\big), {}\\ & & \boldsymbol{\theta }(\xi ):= e^{i\xi }\big(\widehat{a}(\xi )\widehat{b}(\xi +\pi ) -\widehat{ a}(\xi +\pi )\widehat{b}(\xi )\big). {}\\ \end{array}$$
Since \(\widehat{\varTheta }\) is not identically zero and \(\mathop{\mathrm{len}}\nolimits (\widehat{\varTheta }(2\cdot )) = 2\mathop{\mathrm{len}}\nolimits (\widehat{\varTheta }) =\mathop{ \mathrm{len}}\nolimits (\widehat{\varTheta }\widehat{\varTheta }(\cdot +\pi ))\), we see that (1.4.13) must hold for some \(\lambda \in \mathbb{C}\setminus \{0\}\). Thus, \(\lambda \tilde{\boldsymbol{\theta }}(\xi )\overline{\boldsymbol{\theta }(\xi )} = 1\). Since \(\boldsymbol{\theta }\) and \(\tilde{\boldsymbol{\theta }}\) are π-periodic trigonometric polynomials, this identity forces \(\boldsymbol{\theta }(\xi ) = -ce^{i2n\xi }\) for some \(c \in \mathbb{C}\setminus \{0\}\) and some \(n \in \mathbb{Z}\). By (1.4.18), a direct calculation shows that (1.4.14) must hold with \(\widehat{\mathring{a}}\) and \(\widehat{\mathring{b}}\) in (1.4.15). By (1.4.14), we deduce that
$$\displaystyle{\left [\begin{array}{*{10}c} \widehat{\mathring{a}}(\xi ) & \widehat{\mathring{b}}(\xi )\\ \widehat{\mathring{a}}(\xi +\pi ) &\widehat{\mathring{b}}(\xi +\pi ) \end{array} \right ] = \left [\begin{array}{*{10}c} \overline{\widehat{a}(\xi )}&\overline{\widehat{a}(\xi +\pi )} \\ \overline{\widehat{b}(\xi )} & \overline{\widehat{b}(\xi +\pi )} \end{array} \right ]^{-1}= \frac{1} {e^{i\xi }\overline{\boldsymbol{\theta }(\xi )}}\left [\begin{array}{*{10}c} \overline{\widehat{b}(\xi +\pi )}&-\overline{\widehat{a}(\xi +\pi )}\\ -\overline{\widehat{b}(\xi ) } & \overline{\widehat{a}(\xi ) } \end{array} \right ].}$$
Plugging \(\boldsymbol{\theta }(\xi ) = -ce^{i2n\xi }\) into the above identity and comparing the entries of the matrices on both sides, we conclude that (1.4.16) holds. Consequently, \(\mathring{b}\) must be a finitely supported sequence. By (1.4.15), \(\mathring{a}\) is a finitely supported sequence.

Suppose that \(\tilde{a} = a\) and \(\tilde{b} = b\). By Lemma 1.4.5 and (1.4.13), we see that λ > 0 and Θ = θθ for some \(\theta \in l_{0}(\mathbb{Z})\). It is obvious from (1.4.15) that \(\breve{b} \in l_{0}(\mathbb{Z})\). Using (1.4.13) and (1.4.17), we can directly check that \(\{\breve{a};\breve{b}\}\) is an orthogonal wavelet filter bank. □

1.4.4 OEP-Based Multilevel Discrete Framelet Transforms

We now discuss a multilevel discrete framelet transform employing a dual framelet filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\), {a; b 1, , b s }) Θ . A J-level discrete framelet decomposition is exactly the same as the one in (1.3.1). A J-level discrete framelet reconstruction, which will be described as follows, is a slight modification to the J-level discrete framelet reconstruction in (1.3.3). For given low-pass framelet coefficients \(\mathring{v}_{J}\) and high-pass framelet coefficients \(\mathring{w}_{\ell,j},\ell= 1,\ldots,s\) and j = 1, , J, a J-level discrete framelet reconstruction is
$$\displaystyle{ \breve{v}_{J}:=\varTheta ^{\star }{\ast}\mathring{v}_{ J}, }$$
(1.4.19)
$$\displaystyle{ \breve{v}_{j-1}:= \frac{\sqrt{2}} {2} \mathcal{S}_{a}\breve{v}_{j} + \frac{\sqrt{2}} {2} \sum _{\ell=1}^{s}\mathcal{S}_{ b_{\ell}}\mathring{w}_{\ell,j},\qquad j = J,\ldots,1, }$$
(1.4.20)
$$\displaystyle{ \mbox{ recover}\;\mathring{v}_{0}\;\text{from }\breve{v}_{0}\text{ via the relation}\;\mathring{v}_{0} =\varTheta ^{\star } {\ast}\breve{ v}_{ 0}. }$$
(1.4.21)
If \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\) is a dual framelet filter bank satisfying (1.4.3) and (1.4.4), then Theorem 1.4.2 tells us that its associated J-level discrete framelet transform has the perfect reconstruction property. See Fig. 1.6 for a diagram of a two-level discrete framelet transform employing a dual framelet filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\). Using discrete affine systems, a J-level discrete framelet transform using an OEP-based dual framelet filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\), {a; b 1, , b s }) Θ can be equivalently expressed as
$$\displaystyle{v {\ast}\varTheta ^{\star } =\sum _{ k\in \mathbb{Z}}\Big(\sum _{m\in \mathbb{Z}}\langle v,\tilde{a}_{J;m}\rangle \varTheta ^{\star }(k - m)\Big)a_{ J;k} +\sum _{ j=1}^{J}\sum _{ \ell=1}^{s}\sum _{ k\in \mathbb{Z}}\langle v,\tilde{b}_{\ell,j;k}\rangle b_{\ell,j;k}.}$$
Fig. 1.6

Diagram of a two-level discrete framelet transform implemented using (1.1.29) employing a dual framelet filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\)

In the above multilevel discrete framelet transform, there is a deconvolution in (1.4.21) to recover \(\mathring{v}_{0}\) from \(\breve{v}_{0}\) if \(\widehat{\varTheta }\) is not a nonzero monomial. We can easily avoid this troubling de-convolution by the following argument. Let
$$\displaystyle{(\{\tilde{a}_{m};\tilde{b}_{m,1},\ldots,\tilde{b}_{m,s_{m}}\},\{a_{m};b_{m,1},\ldots,b_{m,s_{m}}\})_{\varTheta _{m}},}$$
m = 1, , n be a family of dual framelet filter banks. For any input signal v, let \(\breve{v}_{0,m}\) be the reconstructed signal using the dual framelet filter bank \((\{\tilde{a}_{m};\tilde{b}_{m,1},\ldots,\tilde{b}_{m,s_{m}}\}\), \(\{a_{m};b_{m,1},\ldots,b_{m,s_{m}}\})_{\varTheta _{m}}\). Suppose that there exist \(\tilde{\varTheta }_{1},\ldots,\tilde{\varTheta }_{n} \in l_{0}(\mathbb{Z})\) such that
$$\displaystyle{ \widehat{\tilde{\varTheta }_{1}}(\xi )\overline{\widehat{\varTheta _{1}}(\xi )} + \cdots +\widehat{\tilde{\varTheta } _{n}}(\xi )\overline{\widehat{\varTheta _{n}}(\xi )} = 1. }$$
(1.4.22)
Avoiding de-convolution in (1.4.21), we can recover \(\mathring{v}_{0}\) via the following formula:
$$\displaystyle{\mathring{v}_{0} =\tilde{\varTheta } _{1} {\ast}\breve{ v}_{0,1} + \cdots +\tilde{\varTheta } _{n} {\ast}\breve{ v}_{0,n}.}$$
As shown by many examples in Chap.  3, quite often n = 2 is sufficient to achieve (1.4.22).

1.5 Discrete Framelet Transforms for Signals on Bounded Intervals

In this section we present several algorithms to implement a discrete framelet transform and its variants to deal with signals on a bounded interval.

1.5.1 Boundary Effect in a Standard Discrete Framelet Transform

The input signals to discrete framelet transforms described in Sect. 1.1 have support on the integer lattice \(\mathbb{Z}\), which has infinite length and no boundaries. However, signals in applications often have finite length and can be modeled by \(v^{b} =\{ v^{b}(k)\}_{k=0}^{N-1}: [0,N - 1] \cap \mathbb{Z} \rightarrow \mathbb{C}\), where the superscript b over v is used to emphasize that v b is a signal on a bounded interval. To apply a discrete framelet transform discussed in Sects. 1.1 and 1.3, we have to extend signals from the bounded interval [0, N − 1] to the integer lattice \(\mathbb{Z}\).

Since a J-level discrete framelet transform employs one-level discrete framelet transforms recursively, it suffices for us to discuss how to implement one-level discrete framelet transforms for signals on the interval [0, N − 1]. Let \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\), {u 0, , u s }) be a dual framelet filter bank satisfying the perfect reconstruction condition in (1.1.12) and (1.1.13). In order to modify a one-level discrete framelet transform for signals on a bounded interval, one often first shifts the filters in a given dual framelet filter bank \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\) properly so that the data structure for storing framelet coefficients is simple. This can be easily done, according to the following result.

Proposition 1.5.1

Let\((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\)be a dual framelet filter bank. Suppose that n 0, , n s are integers such that
$$\displaystyle{ n_{1} - n_{0},\;\ldots,\;n_{s} - n_{0}\quad \mathit{\mbox{ are even integers}}. }$$
(1.5.1)
Then the shifted filter bank\((\{\tilde{u}_{0}(\cdot - n_{0}),\ldots,\tilde{u}_{s}(\cdot - n_{s})\},\{u_{0}(\cdot - n_{0}),\ldots,u_{s}(\cdot - n_{s})\})\)is also a dual framelet filter bank.

Proof

Since \(\widehat{[u_{\ell}(\cdot - n_{\ell})]}(\xi ) = e^{-in_{\ell}\xi }\widehat{u_{\ell}}(\xi )\) for = 1, , s and (1.5.1) holds, for ω ∈ {0, 1}, we have
$$\displaystyle\begin{array}{rcl} \widehat{[\tilde{u}_{\ell}(\cdot - n_{\ell})]}(\xi )\overline{\widehat{[u_{\ell}(\cdot - n_{\ell})]}(\xi +\pi \omega )}& =& e^{i\pi n_{\ell}\omega }\widehat{\tilde{u}_{\ell}}(\xi )\overline{\widehat{u_{\ell}}(\xi +\pi \omega )} {}\\ & =& e^{i\pi n_{0}\omega }\widehat{\tilde{u}_{\ell}}(\xi )\overline{\widehat{u_{\ell}}(\xi +\pi \omega )}, {}\\ \end{array}$$
where we used (1.5.1) in the last identity. We conclude that (1.1.12) and (1.1.13) are satisfied if \(\tilde{u}_{0},\ldots,\tilde{u}_{s},u_{0},\ldots,u_{s}\) are replaced by \(\tilde{u}_{0}(\cdot - n_{0}),\ldots,\tilde{u}_{s}(\cdot - n_{s}),u_{0}(\cdot - n_{0}),\ldots,u_{s}(\cdot - n_{s})\), respectively. By Theorem 1.1.1, the shifted filter bank \((\{\tilde{u}_{0}(\cdot - n_{0}),\ldots,\tilde{u}_{s}(\cdot - n_{s})\},\{u_{0}(\cdot - n_{0}),\ldots,u_{s}(\cdot - n_{s})\})\) also satisfies the perfect reconstruction condition and therefore is a dual framelet filter bank. □

For a finitely supported filter/sequence \(u \in l_{0}(\mathbb{Z})\), recall that its filter support [n , n +] is defined to be u(k) = 0 for all \(k \in \mathbb{Z}\setminus [n_{-},n_{+}]\), but u(n )u(n +) ≠  0. That is, the filter support of a sequence u is the smallest interval (with integer endpoints) outside which u vanishes. Obviously, if all n 0, , n s are even integers, then the condition in (1.5.1) is automatically satisfied. Due to (1.3.24), for a pair \((\tilde{u}:=\tilde{ u}_{\ell},u:= u_{\ell})\) in practical implementation, we often replace \((\tilde{u},u)\) by \((\tilde{u}(\cdot - 2n),u(\cdot - 2n))\), where n is an appropriate integer chosen in such a way that the filter support [n , n +] of the filter u(⋅ − 2n) contains 0; moreover, in implementation one often further assumes that its middle point (n + n +)∕2 is the smallest in modulus. In this section, for a filter bank \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\), we always assume that the filter support [n , n +] of u satisfies \(n_{-}\leqslant 0\) and \(n_{+}\geqslant 0\) for every filter u , = 0, , s.

We now discuss how to modify a discrete framelet transform for handling signals on \(\mathbb{Z}\) into a framelet algorithm for handling a signal v b = {v b (k)} k = 0 N−1 with N being a positive integer. First, one extends v b from \([0,N - 1] \cap \mathbb{Z}\) to a sequence v (which can be also explicitly denoted by v e ) on \(\mathbb{Z}\) by any method that the reader prefers. For example, the signal v b can be extended from the interval \([0,N - 1] \cap \mathbb{Z}\) to \(\mathbb{Z}\) by the simple zero-padding extension: v(k) = v b (k) for k = 0, , N − 1, and v(k) = 0 for \(k \in \mathbb{Z}\setminus [0,N - 1]\). To preserve the perfect reconstruction property for a given signal v b , a framelet algorithm shall be able to reconstruct all the original values v b (0), , v b (N − 1), while the artificial values outside the interval [0, N − 1] may or may not be preserved after reconstruction. This implies that we have to calculate \([\mathcal{S}_{u_{\ell}}\mathcal{T}_{\tilde{u}_{\ell}}v](n)\) for all n = 0, , N − 1 and all the values \([\mathcal{T}_{\tilde{u}_{\ell}}v](k)\), which are involved in the calculation of \([\mathcal{S}_{u_{\ell}}\mathcal{T}_{\tilde{u}_{\ell}}v](n)\) for n = 0, , N − 1, must be retained.

In the following, let us look at one typical pair \(\tilde{u}:=\tilde{ u}_{\ell}\) and u: = u such that the filter support [n , n +] of u satisfies \(n_{-}\leqslant 0\) and \(n_{+}\geqslant 0\). By the definition of the subdivision operator in (1.1.2), we have
$$\displaystyle{\frac{1} {2}[\mathcal{S}_{u}\mathcal{T}_{\tilde{u}}v](n) =\sum _{k\in \mathbb{Z}}[\mathcal{T}_{\tilde{u}}v](k)u(n-2k) =\sum _{ k=\lceil (n-n_{+})/2\rceil }^{\lfloor (n-n_{-})/2\rfloor }[\mathcal{T}_{\tilde{ u}}v](k)u(n-2k),\qquad n \in \mathbb{Z},}$$
where ⌊x⌋ = m is the floor function for \(m\leqslant x <m + 1\) and ⌈y⌉ = m is the ceiling function for \(m - 1 <y\leqslant m\) with \(m \in \mathbb{Z}\). Thus, to calculate \([\mathcal{S}_{u}\mathcal{T}_{\tilde{u}}v](n)\) for n = 0, , N − 1, we have to record all the framelet coefficients:
$$\displaystyle{ [\mathcal{T}_{\tilde{u}}v](k),\qquad k = \lceil \tfrac{-n_{+}} {2} \rceil,\ldots,\lfloor \tfrac{N-1-n_{-}} {2} \rfloor. }$$
(1.5.2)
Assuming that N is a positive even integer, by \(n_{-}\leqslant 0\) and \(n_{+}\geqslant 0\), we always have \(\lceil \tfrac{-n_{+}} {2} \rceil \leqslant 0\) and \(\lfloor \tfrac{N-1-n_{-}} {2} \rfloor \geqslant \frac{N} {2} - 1\). In other words, regardless of the filter support [n , n +] of the filter u, the framelet coefficients \(\{[\mathcal{T}_{\tilde{u}}v](k)\}_{k=0}^{\frac{N} {2} -1}\) must be recorded. Note that the must-be-recorded \(\{[\mathcal{T}_{\tilde{u}}v](k)\}_{k=0}^{\frac{N} {2} -1}\) has exactly half of the length N of the original signal v. This is the ideal situation, since the total number of framelet coefficients to be recorded will be (s + 1)N∕2; in particular, for the wavelet case s = 1, one is using a biorthogonal wavelet filter bank and the total number of recorded wavelet coefficients will be the same as the length of the original signal. This ideal situation is very convenient from the viewpoint of data structure for practical programming.
Now the extra work of a framelet algorithm for an arbitrary bounded signal v b on [0, N − 1] is to record the extra framelet coefficients:
$$\displaystyle{ [\mathcal{T}_{\tilde{u}}v](k),\qquad k = \lceil \tfrac{-n_{+}} {2} \rceil,\ldots,-1\quad \mbox{ and}\quad k = \tfrac{N} {2},\ldots,\lfloor \tfrac{N-1-n_{-}} {2} \rfloor. }$$
(1.5.3)
The ideal situation, that there are no extra framelet coefficients to be recorded, can happen if and only if \(0\leqslant n_{+}\leqslant 1\) and n = 0; in other words, u has a very short support and the only possible nonzero values of u are u(0) and u(1). Obviously, many filters have a much longer filter support than [0, 1] and we have to find a way to avoid directly recording the extra framelet coefficients in (1.5.3).

The main idea in an efficient framelet algorithm for signals on a bounded interval to handle the extra framelet coefficients in (1.5.3) is to use correlation: one hopes that the extra framelet coefficients in (1.5.3) are linked in some simple way to the must-be-recorded framelet coefficients \([\mathcal{T}_{\tilde{u}}v](k),k = 0,\ldots, \frac{N} {2} - 1\). In other words, if all the extra framelet coefficients in (1.5.3) are completely determined by the must-be-recorded framelet coefficients \([\mathcal{T}_{\tilde{u}}v](k),k = 0,\ldots, \frac{N} {2} - 1\), then there is no need to record the extra framelet coefficients explicitly, since they can be recovered from the must-be-recorded framelet coefficients. In the following, let us present two possible ways of achieving this goal: one is to explore the periodic structure, and the other is to take advantage of symmetries of filters.

For 0 < p < , we denote by \(l_{p}(\mathbb{Z})\) the space of all sequences \(v =\{ v(k)\}_{k\in \mathbb{Z}} \in l(\mathbb{Z})\) such that
$$\displaystyle{\|v\|_{l_{p}(\mathbb{Z})}^{p}:=\sum _{ k\in \mathbb{Z}}\vert v(k)\vert ^{p} <\infty.}$$
For p = , we use \(\|v\|_{l_{\infty }(\mathbb{Z})}:=\sup _{k\in \mathbb{Z}}\vert v(k)\vert <\infty\).

For the convenience of discussion on frequency-based framelet algorithms in the next section, from now on in this section, we assume that all filters \(u,\tilde{u} \in l_{1}(\mathbb{Z})\) instead of its subspace \(l_{0}(\mathbb{Z})\). Since v b = {v b (k)} k = 0 N−1 has only finitely many values, it is natural for us to assume that its extended sequence v belongs to \(l_{\infty }(\mathbb{Z})\) and consequently, it is natural to require that \(u \in l_{1}(\mathbb{Z})\) so that the convolution uv is well defined.

1.5.2 Discrete Framelet Transforms Using Periodic Extension

We first explore the periodic structure by the following result.

Proposition 1.5.2

Let\(u \in l_{1}(\mathbb{Z})\)be a filter and v b = {v b (k)} k = 0 N−1be an arbitrary input signal. Extend v b into an N-periodic sequence v on\(\mathbb{Z}\)as follows:
$$\displaystyle{ v(Nn + k):= v^{b}(k),\qquad k = 0,\ldots,N - 1\quad \mathit{\mbox{ and}}\quad n \in \mathbb{Z}. }$$
(1.5.4)
Then the following properties hold:
  1. (i)

    uv is an N-periodic sequence on\(\mathbb{Z}\);

     
  2. (ii)

    \(\mathcal{S}_{u}v\)is a 2N-periodic sequence on\(\mathbb{Z}\);

     
  3. (iii)

    If N is a positive even integer, then \(\mathcal{T}_{u}v\) is an \(\frac{N} {2}\) -periodic sequence on \(\mathbb{Z}\) ;

     
  4. (iv)
    If N is a positive odd integer, then\(\mathcal{T}_{u}v\)is an N-periodic sequence on\(\mathbb{Z}\)and is given by
    $$\displaystyle{[\mathcal{T}_{u}v](k) = 2(u^{\star } {\ast} v)(2k),\qquad k = 0,\ldots,N - 1.}$$
    That is,\(\mathcal{T}_{u}v\)is a simple rearrangement of the N-periodic sequence 2uv.
     

Proof

Since v is N-periodic, we have v(N + k) = v(k) and
$$\displaystyle{[u {\ast} v](N + n) =\sum _{k\in \mathbb{Z}}u(k)v(N + n - k) =\sum _{k\in \mathbb{Z}}u(k)v(n - k) = [u {\ast} v](n),\qquad n \in \mathbb{Z}.}$$
Hence, uv is also N-periodic and item (i) holds.

It is not difficult to see that v​ 2 is the same sequence as the sequence obtained by first upsampling v b then extending it into a 2N-periodic sequence. Now by \(\mathcal{S}_{u}v = 2u {\ast} (v\! \uparrow \! 2)\), we see that \(\mathcal{S}_{u}v\) is a 2N-periodic sequence.

Since \(\mathcal{T}_{u}v = 2(u^{\star } {\ast} v)\! \downarrow \! 2\) and N is a positive even integer, it is straightforward to check that \(\mathcal{T}_{u}v\) is an \(\frac{N} {2}\)-periodic sequence on \(\mathbb{Z}\). Item (iv) is left as Exercise 1.34. □

In the following we describe a one-level periodic discrete framelet transform for signals v b = {v b (k)} k = 0 N−1 with N being a positive even integer. Let \(\tilde{u}_{0},\ldots,\tilde{u}_{s} \in l_{1}(\mathbb{Z})\) be filters for decomposition. A one-level periodic discrete framelet decomposition is
$$\displaystyle{ w_{\ell}^{b} =\Big\{ w_{\ell}^{b}(k):= \tfrac{\sqrt{2}} {2} [\mathcal{T}_{\tilde{u}_{\ell}}v](k)\Big\}_{k=0}^{\tfrac{N} {2} -1},\qquad \ell = 0,\ldots,s, }$$
(1.5.5)
where v is the N-periodic extension of v b given in (1.5.4). Grouping all framelet coefficients in (1.5.5) together, we can define a periodic discrete framelet analysis operator \(\widetilde{\mathcal{W}}^{per}\) employing the filter bank \(\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\) as follows:
$$\displaystyle{ \begin{array}{rl} \widetilde{\mathcal{W}}^{per}(v^{b})& = (w_{0}^{b},w_{1}^{b},\ldots,w_{s}^{b})^{\mathsf{T}} \\ & = (w_{0}^{b}(0),\ldots,w_{0}^{b}(\tfrac{N} {2} - 1),\ldots,w_{s}^{b}(0),\ldots,w_{ s}^{b}(\tfrac{N} {2} - 1))^{\mathsf{T}}. \end{array} }$$
(1.5.6)
In other words, if we regard v b as an N × 1 column vector, then the coefficient matrix, still denoted by \(\widetilde{\mathcal{W}}^{per}\), of the linear operator \(\widetilde{\mathcal{W}}^{per}\) is an \(N(\tfrac{s+1} {2} ) \times N\) matrix and a one-level periodic discrete framelet decomposition in (1.5.5) simply becomes \(\widetilde{\mathcal{W}}^{per}v^{b}\), where v b here is regarded as an N × 1 column vector. We shall use \(\mathcal{W}^{per}\) to denote the associated periodic discrete framelet analysis operator employing the filter bank {u 0, , u s }.
Let \(u_{0},\ldots,u_{s} \in l_{1}(\mathbb{Z})\) be filters for reconstruction. For given sequences of framelet coefficients \(w_{0}^{b} =\{ w_{0}^{b}(k)\}_{k=0}^{\tfrac{N} {2} -1}\), , \(w_{s}^{b} =\{ w_{s}^{b}(k)\}_{k=0}^{\tfrac{N} {2} -1}\), a one-level periodic discrete framelet reconstruction is
$$\displaystyle{ v^{b} = \mathcal{V}^{per}(w_{ 0}^{b},\ldots,w_{ s}^{b}):=\Big\{ v^{b}(k):= \tfrac{\sqrt{2}} {2} \sum _{\ell=0}^{s}[\mathcal{S}_{ u_{\ell}}w_{\ell}](k)\Big\}_{k=0}^{N-1}, }$$
(1.5.7)
where w is the \(\frac{N} {2}\)-periodic extension of w b . If we still denote by \(\mathcal{V}^{per}\) the \(N \times N(\tfrac{s+1} {2} )\) coefficient matrix of the linear operator \(\mathcal{V}^{per}\), then a one-level periodic discrete framelet synthesis simply becomes \(\mathcal{V}^{per}(w_{0}^{b},\ldots,w_{s}^{b})^{\mathsf{T}}\). We shall use \(\widetilde{\mathcal{V}}^{per}\) to denote the associated periodic discrete framelet reconstruction operator employing the filter bank \(\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\). For a dual framelet filter bank \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\), we have \(\mathcal{V}^{per}\widetilde{\mathcal{W}}^{per} =\widetilde{ \mathcal{V}}^{per}\mathcal{W}^{per} = I_{N}\). In case that \(\tilde{u}_{0} = u_{0},\ldots,\tilde{u}_{s} = u_{s}\) (that is, a tight framelet filter bank), it is easy to see that \(\mathcal{V}^{per} = (\mathcal{W}^{per})^{\star } = (\overline{\mathcal{W}^{per}})^{\mathsf{T}}\). In particular, for an orthogonal wavelet filter bank {u 0, u 1}, \(\mathcal{V}^{per}\) and \(\mathcal{W}^{per}\) are N × N unitary matrices satisfying \((\mathcal{V}^{per})^{\star }\mathcal{V}^{per} = (\mathcal{W}^{per})^{\star }\mathcal{W}^{per} = I_{N}\) and \(\mathcal{V}^{per} = (\mathcal{W}^{per})^{\star } = (\mathcal{W}^{per})^{-1}\).
It is easy to see that the above described periodic discrete framelet transform can be straightforwardly modified for a dual framelet filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\}\), {a; b 1, , b s }) Θ such that the one-level periodic discrete framelet decomposition is the same as (1.5.5) and the one-level periodic discrete framelet reconstruction is
$$\displaystyle{\breve{v}^{b} =\Big\{\breve{ v}^{b}(k):= \tfrac{\sqrt{2}} {2} [\mathcal{S}_{a}(\varTheta ^{\star } {\ast} w_{ 0})](k) + \tfrac{\sqrt{2}} {2} \sum _{\ell=1}^{s}[\mathcal{S}_{ b_{\ell}}w_{\ell}](k)\Big\}_{k=0}^{N-1}}$$
and the reconstructed signal v b : = {v(k)} k = 0 N−1 is obtained from \(\breve{v}^{b}\) via \(v =\varTheta ^{\star } {\ast}\breve{ v}\), where \(\breve{v}\) is the N-periodic extension of \(\breve{v}^{b}\). As discussed at the end of Sect. 1.4, the deconvolution here can be easily avoided by combining more than one discrete framelet transforms employing several dual framelet filter banks.

We present an example to illustrate the periodic discrete framelet transform.

Example 1.5.1

Let v b be the same input signal as in (1.1.26), that is,
$$\displaystyle{ v^{b} =\{\underline{ \mathbf{-21}},-22,-23,-23,-25,38,36,34\}_{ [0,7]}. }$$
(1.5.8)
We apply the tight framelet filter bank in Example 1.1.3 to v b in (1.5.8). We extend v b to an 8-periodic sequence v on \(\mathbb{Z}\), given by
$$\displaystyle{v =\{\ldots,-25,38,36,34,\underline{\mathbf{-21},-22,-23,-23,-25,38,36,34},-21,-22,-23,\ldots \}.}$$
Then all sequences \(\mathcal{T}_{u_{0}}v,\mathcal{T}_{u_{1}}v,\mathcal{T}_{u_{2}}v\) are 4-periodic and
$$\displaystyle\begin{array}{rcl} w_{0}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{u_{0}}v = \tfrac{\sqrt{2}} {2} \{\ldots,-15,-\tfrac{91} {2},-\tfrac{35} {2},72,\underline{\mathbf{-15},-\tfrac{91} {2},-\tfrac{35} {2},72,}-15,-\tfrac{91} {2},-\tfrac{35} {2},\ldots \}, {}\\ w_{1}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{u_{1}}v =\{\ldots,-28,-\tfrac{1} {2}, \tfrac{61} {2},-2,\underline{\mathbf{-28},-\tfrac{1} {2}, \tfrac{61} {2},-2,}-28,-\tfrac{1} {2}, \tfrac{61} {2},-2,\ldots \}, {}\\ w_{2}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{u_{2}}v = \tfrac{\sqrt{2}} {2} \{\ldots,-27,-\tfrac{1} {2},-\tfrac{65} {2},0,\underline{\mathbf{-27},-\tfrac{1} {2},-\tfrac{65} {2},0,}-27,-\tfrac{1} {2},-\tfrac{65} {2},0,\ldots \}. {}\\ \end{array}$$
It is also easy to directly check that \(\tfrac{\sqrt{2}} {2} (\mathcal{S}_{u_{0}}w_{0} + \mathcal{S}_{u_{1}}w_{1} + \mathcal{S}_{u_{2}}w_{2}) = v\).

1.5.3 Discrete Framelet Transforms Using Symmetric Extension

The periodic extension in Proposition 1.6.2 is often used for a filter without any symmetry. Taking advantages of symmetry of a dual framelet filter bank, we now discuss a symmetric discrete framelet transform for signals on a bounded interval [0, N − 1]. Generally, we need to adapt a symmetric discrete framelet transform for different types of symmetries.

Recall that \(\mathsf{S}\,\widehat{u}(\xi ) =\widehat{ u}(\xi )/\widehat{u}(-\xi )\) records the symmetry type of a filter u and \(\mathbb{S}\widehat{u}(\xi ) =\widehat{ u}(\xi )/\overline{\widehat{u}(\xi )}\) records the complex symmetry type of u. The following result can be easily verified and will be needed later.

Proposition 1.5.3

If sequences u and v have symmetry such that
$$\displaystyle{[\mathsf{S}\,\widehat{u}](\xi ) =\epsilon _{u}e^{-ic_{u}\xi },\quad [\mathsf{S}\,\widehat{v}](\xi ) =\epsilon _{v}e^{-ic_{v}\xi }\quad \mathit{\mbox{ for some}}\quad c_{u},c_{v} \in \mathbb{Z},\epsilon _{u},\epsilon _{v} \in \{-1,1\},}$$
then both uv and\(\mathcal{S}_{u}v\)have symmetry satisfying
$$\displaystyle{[\mathsf{S}(\widehat{u {\ast} v})](\xi ) = [\mathsf{S}\,\widehat{u}](\xi )[\mathsf{S}\,\widehat{v}](\xi ) =\epsilon _{u}\epsilon _{v}e^{-i(c_{u}+c_{v})\xi }}$$
and
$$\displaystyle{[\mathsf{S}\,\widehat{\mathcal{S}_{u}v}](\xi ) = [\mathsf{S}\,\widehat{u}](\xi )[\mathsf{S}\,\widehat{v}](2\xi ) =\epsilon _{u}\epsilon _{v}e^{-i(c_{u}+2c_{v})\xi }.}$$
If in addition c v c u is an even integer, then\(\mathcal{T}_{u}v\)also has symmetry satisfying
$$\displaystyle{[\mathsf{S}(\widehat{\mathcal{T}_{u}v})](\xi ) = \overline{[\mathsf{S}\,\widehat{u}](\xi /2)}[\mathsf{S}\,\widehat{v}](\xi /2) =\epsilon _{u}\epsilon _{v}e^{-i\xi (c_{v}-c_{u})/2}.}$$
Moreover, all the identities and conclusions still hold if the symmetry operator\(\mathsf{S}\)is replaced by the complex symmetry operator\(\mathbb{S}\).

Symmetry of a filter bank is not only very much desired for better visual quality of reconstructed signals in a lot of applications, but also plays a critical role in the implementation of a symmetric discrete framelet transform for signals on a bounded interval, which we shall address in detail here. Due to the same behavior of the symmetry operator S and the complex symmetry operator \(\mathbb{S}\), in this section we only consider filters and signals with symmetry; the closely related case for complex symmetry can be deduced similarly and easily.

First, we discuss some natural conditions that we will put on a given dual framelet filter bank \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\), {u 0, , u s }) with each filter having symmetry. Let v, w be two arbitrary sequences with symmetry. It is pretty easy to see from Proposition 1.5.3 that vw also has symmetry and \(\mathsf{S}\widehat{v {\ast} w} = \mathsf{S}\,\widehat{v}\mathsf{S}\widehat{w}\). However, the sum v + w generally does not have any symmetry. If \(\mathsf{S}\,\widehat{v} = \mathsf{S}\widehat{w}\), that is, both v and w have the same symmetry type, then indeed v + w has symmetry and \(\mathsf{S}(\widehat{v + w}) = \mathsf{S}\,\widehat{v} = \mathsf{S}\widehat{w}\). Now assume that we have a dual framelet filter bank \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\) with each filter having symmetry such that
$$\displaystyle{ \mathsf{S}\widehat{u_{\ell}} =\epsilon _{u_{\ell}}e^{-ic_{u_{\ell}}\xi },\quad \mathsf{S}\widehat{\tilde{u}_{\ell}} =\epsilon _{\tilde{u}_{\ell}}e^{-ic_{\tilde{u}_{\ell}}\xi },\;\mbox{ where}\;\epsilon _{u_{\ell}},\epsilon _{\tilde{u}_{\ell}} \in \{-1,1\},\;\;c_{u_{\ell}},c_{\tilde{u}_{\ell}} \in \mathbb{Z}, }$$
(1.5.9)
for = 0, , s. By (1.1.12) and the above discussion on compatibility of symmetry types, it is natural for us to assume that
$$\displaystyle{\mathsf{S}\widehat{\tilde{u}_{0}}\mathsf{S}\overline{\widehat{u_{0}}} = \cdots = \mathsf{S}\widehat{\tilde{u}_{s}}\mathsf{S}\overline{\widehat{u_{s}}} = \mathsf{S}1 = 1.}$$
Since \(\mathsf{S}\overline{\widehat{u_{\ell}}} = \overline{\mathsf{S}\widehat{u_{\ell}}} = (\mathsf{S}\widehat{u_{\ell}})^{-1}\), by (1.5.9), the above relation is equivalent to assuming that
$$\displaystyle{ \epsilon _{\tilde{u}_{\ell}} =\epsilon _{u_{\ell}},\quad c_{\tilde{u}_{\ell}} = c_{u_{\ell}},\qquad \ell = 0,\ldots,s. }$$
(1.5.10)
Similarly, by (1.1.13), it is natural for us to assume that
$$\displaystyle{\mathsf{S}\widehat{\tilde{u}_{0}}\overline{\mathsf{S}\widehat{u_{0}}(\cdot +\pi )} = \cdots = \mathsf{S}\widehat{\tilde{u}_{s}}\overline{\mathsf{S}\widehat{u_{s}}(\cdot +\pi )}.}$$
Since \(\mathsf{S}(\widehat{u_{\ell}}(\xi +\pi )) = [\mathsf{S}\widehat{u_{\ell}}](\xi +\pi )\), by (1.5.9), the above relation is equivalent to assuming that
$$\displaystyle{\epsilon _{u_{\ell}}\epsilon _{\tilde{u}_{\ell}}e^{-i\pi c_{u_{\ell}} } =\epsilon _{u_{0}}\epsilon _{\tilde{u}_{0}}e^{-i\pi c_{u_{0}} }\quad \mbox{ and}\quad \quad c_{\tilde{u}_{\ell}} - c_{u_{\ell}} = c_{\tilde{u}_{0}} - c_{u_{0}},\qquad \ell = 0,\ldots,s.}$$
For a dual framelet filter bank \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\), {u 0, , u s }) with each filter having symmetry in (1.5.9), taking into account of (1.5.10), we always assume the following natural condition:
$$\displaystyle{ \epsilon _{\tilde{u}_{\ell}} =\epsilon _{u_{\ell}},\quad c_{\tilde{u}_{\ell}} = c_{u_{\ell}},\quad \mbox{ and}\quad c_{u_{\ell}} - c_{u_{0}} \in 2\mathbb{Z},\qquad \ell = 0,\ldots,s. }$$
(1.5.11)
After shifting the filters by even integers according to Proposition 1.5.1, we can further assume that \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\}\), {u 0, , u s }) with each filter having symmetry in (1.5.9) is normalized so that (1.5.11) is satisfied with all \(c_{u_{0}},\ldots,c_{u_{s}} \in \{-1,0,1,2\}\). By (1.5.11), we see that either \(c_{u_{0}},\ldots,c_{u_{s}} \in \{ 0,2\}\) or \(c_{u_{0}},\ldots,c_{u_{s}} \in \{-1,1\}\).
For a dual framelet filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\) with all involved filters having symmetry in (1.5.9) and \(\mathsf{S}\widehat{\varTheta }(\xi ) =\epsilon _{\varTheta }e^{-ic_{\varTheta }\xi }\), by the same argument, we have
$$\displaystyle{\epsilon _{\tilde{a}} =\epsilon _{a},\;\epsilon _{\tilde{b}_{\ell}} =\epsilon _{\varTheta }\epsilon _{b_{\ell}},\;c_{\tilde{a}} = c_{a} - c_{\varTheta },\;c_{\tilde{b}_{\ell}} = c_{b_{\ell}} + c_{\varTheta },\;c_{b_{\ell}} - c_{a} \in 2\mathbb{Z},\quad \ell = 1,\ldots,s.}$$

We say that an interval [m, n] is a control interval of \(v \in l_{0}(\mathbb{Z})\) if v is uniquely determined by {v(k)} k = m n through the periodicity and/or symmetry of v.

Building on Propositions 1.5.2 and 1.5.3, we have the following result on a symmetric discrete framelet decomposition for signals on a bounded interval.

Proposition 1.5.4

Let\(u \in l_{1}(\mathbb{Z})\)be a decomposition filter such that\(\mathsf{S}\,\widehat{u}(\xi ) =\epsilon e^{-ic\xi }\)for some ε ∈ {−1, 1} and\(c \in \mathbb{Z}\), that is,
$$\displaystyle{ u(c - k) =\epsilon u(k)\qquad \forall \;k \in \mathbb{Z}. }$$
(1.5.12)
Let v b = {v b (k)} k = 0 N−1be an arbitrary input signal. Extend v b , with both endpoints non-repeated (EN), into a (2N − 2)-periodic sequence v on\(\mathbb{Z}\)by
$$\displaystyle{ \begin{array}{rl} &v(k) = v^{b}(k),\qquad k = 0,\ldots,N - 1\quad \mathit{\mbox{ and}}\quad \\ &v(k) = v^{b}(2N - 2 - k),\qquad k = N,\ldots,2N - 3.\end{array} }$$
(1.5.13)
  1. (i)
    Then uv is a (2N − 2)-periodic sequence, has the following symmetries:
    $$\displaystyle{ [u^{\star } {\ast} v](-c - k) = [u^{\star } {\ast} v](2N - 2 - c - k) =\epsilon [u^{\star } {\ast} v](k),\qquad \forall \;k \in \mathbb{Z}, }$$
    (1.5.14)
    and\([-\lfloor \tfrac{c} {2}\rfloor,N - 1 -\lceil \tfrac{c} {2}\rceil ]\)is a control interval of uv.
     
  2. (ii)
    If c is an even integer, then\(\mathcal{T}_{u}v\)is an (N − 1)-periodic sequence, has the following symmetries:
    $$\displaystyle{ [\mathcal{T}_{u}v](-\tfrac{c} {2} - k) = [\mathcal{T}_{u}v](N - 1 -\tfrac{c} {2} - k) =\epsilon [\mathcal{T}_{u}v](k)\qquad \forall \;k \in \mathbb{Z}, }$$
    (1.5.15)
    and\([\lceil -\tfrac{c} {4}\rceil,\lfloor \frac{N-1} {2} -\tfrac{c} {4}\rfloor ]\)is a control interval of\(\mathcal{T}_{u}v\).
     

Proof

Since v is (2N − 2)-periodic, we see that v is symmetric about the points 0 and N − 1:
$$\displaystyle{ \quad v(-k) = v(2N - 2 - k) = v(k)\qquad \forall \;k \in \mathbb{Z}. }$$
(1.5.16)
Applying Proposition 1.5.2, we deduce that uv is (2N − 2)-periodic and \(\mathcal{T}_{u}v\) is (N − 1)-periodic. On the other hand, by Proposition 1.5.3 and (1.5.16), we have \(\mathsf{S}\widehat{u^{\star } {\ast} v}(\xi ) =\epsilon e^{-i\xi (c_{v}-c)}\) and \(\mathsf{S}\widehat{\mathcal{T}_{u}v}(\xi ) =\epsilon e^{-i\xi (c_{v}-c)/2}\) for c v = 0, N − 1. Hence, (1.5.14) and (1.5.15) hold true. □

Proposition 1.5.5

Let\(u \in l_{1}(\mathbb{Z})\)be a decomposition filter such that (1.5.12) holds for some ε ∈ {−1, 1} and\(c \in \mathbb{Z}\). Let v b = {v b (k)} k = 0 N−1be an input signal. Extend v b , with both endpoints repeated (ER), into a 2N-periodic sequence v on\(\mathbb{Z}\)by
$$\displaystyle{ \begin{array}{rl} &v(k) = v^{b}(k),\qquad k = 0,\ldots,N - 1\quad \mathit{\mbox{ and}}\quad \\ &v(k) = v^{b}(2N - 1 - k),\qquad k = N,\ldots,2N - 1.\end{array} }$$
(1.5.17)
  1. (i)
    Then uv is 2N-periodic, has the following symmetries:
    $$\displaystyle{[u^{\star } {\ast} v](-1 - c - k) = [u^{\star } {\ast} v](2N - 1 - c - k) =\epsilon [u^{\star } {\ast} v](k),\qquad \forall \;k \in \mathbb{Z},}$$
    and\([-\lfloor \tfrac{1+c} {2} \rfloor,N -\lceil \tfrac{1+c} {2} \rceil ]\)is a control interval of uv.
     
  2. (ii)
    If c is an odd integer, then \(\mathcal{T}_{u}v\) is an N-periodic sequence, has the following symmetries:
    $$\displaystyle{[\mathcal{T}_{u}v](-\tfrac{1+c} {2} - k) = [\mathcal{T}_{u}v](N -\tfrac{1+c} {2} - k) =\epsilon [\mathcal{T}_{u}v](k)\qquad \forall \;k \in \mathbb{Z},}$$
    and \([\lceil -\tfrac{1+c} {4} \rceil,\lfloor \frac{N} {2} -\frac{1+c} {4} \rfloor ]\) is a control interval of \(\mathcal{T}_{u}v\) .
     

Proof

Since v is 2N-periodic, (1.5.17) implies that v is symmetric about the points \(-\frac{1} {2}\) and \(N -\frac{1} {2}\):
$$\displaystyle{v(-1 - k) = v(2N - 1 - k) = v(k)\qquad \forall \;k \in \mathbb{Z}.}$$
Now the claims can be verified by a similar argument as in the proof of Proposition 1.5.4. □
For the convenience of the reader, the results in Propositions 1.5.4 and 1.5.5 are summarized in Tables 1.1 and 1.2.
Table 1.1

The analysis/decomposition filter u has symmetry \(\mathsf{S}\widehat{u}(\xi ) =\epsilon e^{-ic\xi }\), where ε ∈ {−1, 1} and c ∈ {0, 2}. v is a symmetric extension with both endpoints non-repeated (EN) from an input signal v b = {v b (k)} k = 0 N−1 in (1.5.13). For the control interval of \(\mathcal{T}_{u}v\), we assumed that N is an even integer

Filter u

uv with v extended by EN

\(\mathcal{T}_{u}v\) with v extended by EN

 

c = 0

ε = 1

(2N − 2)-periodic,

symmetric about 0 and N − 1,

a control interval [0, N − 1]

(N − 1)-periodic,

symmetric about 0 and \(\tfrac{N-1} {2}\),

a control interval \([0, \tfrac{N} {2} - 1]\)

 

c = 0

ε = −1

(2N − 2)-periodic,

antisymmetric about 0 and N − 1,

a control interval [0, N − 1],

[uv](0) = [uv](N − 1) = 0

(N − 1)-periodic,

antisymmetric about 0 and \(\tfrac{N-1} {2}\),

a control interval \([0, \tfrac{N} {2} - 1]\),

\([\mathcal{T}_{u}v](0) = 0\)

 

c = 2

ε = 1

(2N − 2)-periodic,

symmetric about − 1 and N − 2,

a control interval [−1, N − 2]

(N − 1)-periodic,

symmetric about \(-\tfrac{1} {2}\) and \(\tfrac{N} {2} - 1\),

a control interval \([0, \tfrac{N} {2} - 1]\)

 

c = 2

ε = −1

(2N − 2)-periodic,

antisymmetric about − 1 and N − 2,

a control interval [−1, N − 2],

[uv](−1) = [uv](N − 2) = 0

(N − 1)-periodic,

antisymmetric about \(-\tfrac{1} {2}\) and \(\tfrac{N} {2} - 1\),

a control interval \([0, \tfrac{N} {2} - 1]\),

\([\mathcal{T}_{u}v](\tfrac{N} {2} - 1) = 0\)

 
Table 1.2

The analysis/decomposition filter u has symmetry \(\mathsf{S}\widehat{u}(\xi ) =\epsilon e^{-ic\xi }\), where ε ∈ {−1, 1} and c ∈ {−1, 1}. v is a symmetric extension with both endpoints repeated (ER) from an input signal v b = {v b (k)} k = 0 N−1 in (1.5.17). For the control interval of \(\mathcal{T}_{u}v\), we assumed that N is an even integer

Filter u

uv with v extended by ER

\(\mathcal{T}_{u}v\) with v extended by ER

 

c = 1

ε = 1

2N-periodic,

symmetric about − 1 and N − 1,

a control interval [−1, N − 1]

N-periodic,

symmetric about \(-\tfrac{1} {2}\) and \(\tfrac{N-1} {2}\),

a control interval \([0, \tfrac{N} {2} - 1]\)

 

c = 1

ε = −1

2N-periodic,

antisymmetric about − 1 and N − 1,

a control interval [−1, N − 1],

[uv](−1) = [uv](N − 1) = 0

N-periodic,

antisymmetric about \(-\tfrac{1} {2}\) and \(\tfrac{N-1} {2}\),

a control interval \([0, \tfrac{N} {2} - 1]\)

 

c = −1

ε = 1

2N-periodic,

symmetric about 0 and N,

a control interval [0, N]

N-periodic,

symmetric about 0 and \(\tfrac{N} {2}\),

a control interval \([0, \tfrac{N} {2} ]\)

 

c = −1

ε = −1

2N-periodic,

antisymmetric about 0 and N,

a control interval [0, N],

[uv](0) = [uv](N) = 0

N-periodic,

antisymmetric about 0 and \(\tfrac{N} {2}\),

a control interval \([0, \tfrac{N} {2} ]\),

\([\mathcal{T}_{u}v](0) = [\mathcal{T}_{u}v](\tfrac{N} {2} ) = 0\)

 

For an input signal v b = {v b (k)} k = 0 N−1 with an even integer N, by Tables 1.1 and 1.2, except the two cases for c = −1, we only need to record \(\{\mathcal{T}_{u}v(k)\}_{k=0}^{\tfrac{N} {2} -1}\), which has exactly half of the length of v b . For the particular case that c = −1 and ε = 1, we have to record \(\{\mathcal{T}_{u}v(k)\}_{k=0}^{\tfrac{N} {2} }\), which has \(\tfrac{N} {2} + 1\) coefficients; while for the particular case that c = −1 and ε = −1, we only have to record \(\{\mathcal{T}_{u}v(k)\}_{k=1}^{\tfrac{N} {2} -1}\), which has \(\tfrac{N} {2} - 1\) coefficients. If the case c = 1 does not appear and we have only the case c = −1 in a dual framelet filter bank\((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\), then by Proposition 1.5.1 we may use the shifted dual framelet filter bank \((\{\tilde{u}_{0}(\cdot + 1),\ldots,\tilde{u}_{s}(\cdot + 1)\},\{u_{0}(\cdot + 1),\ldots,u_{s}(\cdot + 1)\})\). Note that all the filters in the shifted dual framelet filter bank have the symmetry center 1∕2.

We now discuss a symmetric discrete framelet reconstruction for signals on a bounded interval. Since the reconstruction filter u has the same symmetry type as the decomposition filter \(\tilde{u}\), for recorded framelet coefficients \(\{w^{b}(k)\}_{k=0}^{\tfrac{N} {2} -1}\), according to the cases c and ε, we extend w b into a sequence w on \(\mathbb{Z}\) according to the symmetries and periodicity of \(\mathcal{T}_{u}v\) in Tables 1.1 and 1.2. Then the reconstructed sequence \(\mathcal{S}_{u}w\) will have the corresponding same symmetry property as v in Tables 1.1 and 1.2, which can be easily verified by Propositions 1.5.2 and 1.5.3.

We present a few examples to illustrate the symmetric discrete framelet transforms.

Example 1.5.2

We apply the biorthogonal wavelet filter bank in Example 1.1.2 to v b in (1.5.8). Since \(\mathsf{S}\widehat{\tilde{u}_{0}} = 1\) and \(\mathsf{S}\widehat{\tilde{u}_{1}} = e^{-i2\xi }\), we extend v b according to Table 1.1 by both endpoints non-repeated (EN): v is 14-periodic, is symmetric about the points 0 and 7, and is given by
$$\displaystyle{ v =\{\ldots,-25,-23 - 23,-22,\underline{\mathbf{-21},-22,-23,-23,-25,38,36,34},36,38,-25,\ldots \}. }$$
(1.5.18)
Then \(\mathcal{T}_{\tilde{u}_{0}}v\) is 7-periodic and is symmetric about the points 0 and 7∕2:
$$\displaystyle{w_{0} = \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{u}_{0}}v = \tfrac{\sqrt{2}} {2} \{\ldots,-\tfrac{133} {4},-\tfrac{91} {2},\underline{\mathbf{-42},-\tfrac{91} {2},-\tfrac{133} {4}, \tfrac{349} {4} }, \tfrac{349} {4},-\tfrac{133} {4},-\tfrac{91} {2},-42,\ldots \},}$$
and \(\mathcal{T}_{\tilde{u}_{1}}v\) is 7-periodic and is symmetric about the points \(-\tfrac{1} {2}\) and 3:
$$\displaystyle{w_{1} = \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{u}_{1}}v = \tfrac{\sqrt{2}} {2} \{\ldots,-2, \tfrac{65} {2},1,0,\underline{\mathbf{0},1, \tfrac{65} {2},-2}, \tfrac{65} {2},1,0,0,1, \tfrac{65} {2},\ldots \}.}$$
Both the control intervals of \(\mathcal{T}_{\tilde{u}_{0}}v\) and \(\mathcal{T}_{\tilde{u}_{1}}v\) are underlined and have 4 coefficients. It is easy to directly check that \(\tfrac{\sqrt{2}} {2} (\mathcal{S}_{u_{0}}w_{0} + \mathcal{S}_{u_{1}}w_{1}) = v\).

Example 1.5.3

We apply the tight framelet filter bank in Example 1.1.3 to v b in (1.5.8). Since \(\mathsf{S}\widehat{u_{0}} = \mathsf{S}\widehat{u_{2}} = 1\) and \(\mathsf{S}\widehat{u_{1}} = -1\), we extend v b according to Table 1.1 with both endpoints non-repeated (EN): v is 14-periodic, is symmetric about both 0 and 7, and is given in (1.5.18). Then both \(\mathcal{T}_{u_{0}}v\) and \(\mathcal{T}_{u_{2}}v\) are 7-periodic and symmetric about the points 0 and 7∕2, and \(\mathcal{T}_{u_{1}}v\) is 7-periodic and antisymmetric about the points 0 and 7∕2:
$$\displaystyle\begin{array}{rcl} w_{0}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{u_{0}}v = \tfrac{\sqrt{2}} {2} \{\ldots,72,-\tfrac{35} {2},-\tfrac{91} {2},\underline{\mathbf{-43},-\tfrac{91} {2},-\tfrac{35} {2},72},72,-\tfrac{35} {2},-\tfrac{91} {2},-43,\ldots \}, {}\\ w_{1}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{u_{1}}v =\{\ldots,2,-\tfrac{61} {2}, \tfrac{1} {2},\underline{\mathbf{0},-\tfrac{1} {2}, \tfrac{61} {2},-2},2,-\tfrac{61} {2}, \tfrac{1} {2},0,-\tfrac{1} {2}, \tfrac{61} {2},\ldots \}, {}\\ w_{2}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{u_{2}}v = \tfrac{\sqrt{2}} {2} \{\ldots,0,-\tfrac{65} {2},-\tfrac{1} {2},\underline{\mathbf{1},-\tfrac{1} {2},-\tfrac{65} {2},0},0,-\tfrac{65} {2},-\tfrac{1} {2},1,-\tfrac{1} {2},\ldots \}. {}\\ \end{array}$$
It is easy to directly check that \(\tfrac{\sqrt{2}} {2} (\mathcal{S}_{u_{0}}w_{0} + \mathcal{S}_{u_{1}}w_{1} + \mathcal{S}_{u_{2}}w_{2}) = v\).

Example 1.5.4

We apply the dual framelet filter bank in Example 1.1.4 to v b in (1.5.8). Since \(\mathsf{S}\widehat{\tilde{u}_{0}} = e^{-i\xi }\), we extend v b according to Table 1.2 by both endpoints repeated (ER): v is 16-periodic, is symmetric about the points \(-\tfrac{1} {2}\) and \(7\tfrac{1} {2}\), and is given by
$$\displaystyle{v =\{\ldots,-23,-23,-22,-21,\underline{\mathbf{-21},-22,-23,-23,-25,38,36,34},34,36,38,\ldots \}.}$$
Then all \(\mathcal{T}_{\tilde{u}_{0}}v\), \(\mathcal{T}_{\tilde{u}_{1}}v\) and \(\mathcal{T}_{\tilde{u}_{2}}v\) are 8-periodic. \(\mathcal{T}_{\tilde{u}_{0}}v\) is symmetric about the points \(-\tfrac{1} {2}\) and \(\tfrac{7} {2}\), \(\mathcal{T}_{\tilde{u}_{1}}v\) is antisymmetric about the points 0 and 4, and \(\mathcal{T}_{\tilde{u}_{2}}v\) is antisymmetric about the points \(-\tfrac{1} {2}\) and \(\tfrac{7} {2}\):
$$\displaystyle\begin{array}{rcl} w_{0}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{u}_{0}}v = \tfrac{\sqrt{2}} {2} \{\ldots,70,13,-46,-43,\underline{\mathbf{-43},-46,13,70},70,13,-46,-43,\ldots \}, {}\\ w_{1}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{u}_{1}}v = \tfrac{\sqrt{2}} {2} \{\ldots,0,2,2,1,\underline{\mathbf{0},-1,-2,-2},0,2,2,1,0,\ldots \}, {}\\ w_{2}& =& \tfrac{\sqrt{2}} {2} \mathcal{T}_{\tilde{u}_{2}}v = \tfrac{\sqrt{2}} {2} \{\ldots,2,-63,0,1,\underline{\mathbf{-1},0,63,-2},2,-63,0,1,-1,0,\ldots \}. {}\\ \end{array}$$
It is easy to directly check that \(\tfrac{\sqrt{2}} {2} (\mathcal{S}_{u_{0}}w_{0} + \mathcal{S}_{u_{1}}w_{1} + \mathcal{S}_{u_{2}}w_{2}) = v\).

1.5.4 Symmetric Extension for Filter Banks Without Symmetry

On one hand, the implementation of a discrete framelet transform using symmetry extension in Sect. 1.5.3 is complicated by the many different symmetry patterns of the filters. On the other hand, many filter banks do not have any symmetry at all. In order to reduce artificial jumps near boundaries induced by periodic extension, if storage of framelet coefficients is not an issue, there are two ways to employ symmetric extension for general filter banks without symmetry. The first way is to extend the input signal by either EN or ER (or any other extension method) and then directly record all the framelet coefficients in (1.5.2) including the extra framelet coefficients in (1.5.3). The second way is to use the following simple algorithm.

Algorithm 1.5.6

Let v b = {v b (k)} k = 0 N−1be an arbitrary input signal.
  1. (S1)
    (Pre-processing) Extend the right-hand endpoint of v b by ER (often used) or EN to obtain another signal\(\mathring{v}^{b}\). More precisely, for ER extension,\(\mathring{v}^{b}(k) =\mathring{v}^{b}(2N - 1 - k):= v^{b}(k)\), k = 0, , N − 1, that is,
    $$\displaystyle{\mathring{v}^{b} =\{ v^{b}(0),\ldots,v^{b}(N-2),v^{b}(N-1),v^{b}(N-1),v^{b}(N-2),\ldots,v^{b}(0)\}_{ [0,2N-1]};}$$
    for EN extension,\(\mathring{v}^{b}(k) =\mathring{v}^{b}(2N - 2 - k):= v^{b}(k)\), k = 0, , N − 1, that is,
    $$\displaystyle{\mathring{v}^{b} =\{ v^{b}(0),\ldots,v^{b}(N - 2),v^{b}(N - 1),v^{b}(N - 2),\ldots,v^{b}(1)\}_{ [0,2N-3]}.}$$
     
  2. (S2)

    Apply a periodic discrete framelet transform to \(\mathring{v}^{b}\) . Denote the reconstructed signal of \(\mathring{v}^{b}\) by \(\mathring{v}^{r}\) .

     
  3. (S3)

    (After-processing) If ER is used in (S1), a reconstructed signal v r of v b is obtained by\(v^{r}(k) = \tfrac{\mathring{v}^{r}(k)+\mathring{v}^{r}(2N-1-k)} {2}\), k = 0, , N − 1. If EN is used in (S1), a reconstructed signal v r of v b is obtained by\(v^{r}(k) = \tfrac{\mathring{v}^{r}(k)+\mathring{v}^{r}(2N-2-k)} {2},k = 0,\ldots,N - 1\).

     

1.6 Discrete Framelet Transforms Implemented in the Frequency Domain​​ 

In this section we provide an equivalent implementation in the frequency domain of the framelet algorithms for signals on bounded intervals in Sect. 1.5.

Filters having infinite support are also used in applications. Such filters \(u =\{ u(k)\}_{k\in \mathbb{Z}}\) are often given in the frequency domain such that \(\widehat{u}\) has an explicit expression, while its time domain form \(\{u(k)\}_{k\in \mathbb{Z}}\) is only implicitly given by \(u(k) = \frac{1} {2\pi }\int _{-\pi }^{\pi }\widehat{u}(\xi )e^{-ik\xi }d\xi,k \in \mathbb{Z}\) and lacks an explicit expression. On the other hand, for certain circumstances, it is easier to design a filter in the frequency domain, that is, to design \(\widehat{u}\), rather than its time domain form \(\{u(k)\}_{k\in \mathbb{Z}}\). This can be seen from the perfect reconstruction condition in (1.1.12) and (1.1.13) which are expressed in the frequency domain. As a consequence, it is important to have an equivalent implementation in the frequency domain of the framelet algorithms described in Sect. 1.5 for signals on bounded intervals.

A periodic discrete framelet transform can be implemented using discrete Fourier transform (DFT). For v b = {v b (k)} k = 0 N−1, its N-point discrete Fourier transform is another N-point sequence \(\{\widehat{v^{b}}(\frac{2\pi n} {N} )\}_{n=0}^{N-1}\) on the interval \([0,N - 1] \cap \mathbb{Z}\), where \(\widehat{v^{b}}(\xi ):=\sum _{ k=0}^{N-1}v^{b}(k)e^{-ik\xi }\) for \(\xi \in \mathbb{R}\). That is, if we regard v b as a sequence on \(\mathbb{Z}\) by the simple zero-padding extension, then the N-point discrete Fourier transform of v b = {v b (k)} k = 0 N−1 is obtained by sampling the Fourier series \(\widehat{v^{b}}(\xi )\) at \(\xi = \frac{2\pi n} {N}\) for n = 0, , N − 1. It is well known that the original signal v b = {v b (k)} k = 0 N−1 can be recovered from its N-point discrete Fourier transform via the inverse discrete Fourier transform:
$$\displaystyle{ v^{b}(k) = \frac{1} {N}\sum _{n=0}^{N-1}\widehat{v^{b}}(\tfrac{2\pi n} {N} )e^{i\tfrac{2\pi nk} {N} },\qquad k = 0,\ldots,N - 1. }$$
(1.6.1)
For N-point signals, both the discrete Fourier transform and its inverse can be implemented by fast Fourier transform (FFT) with computational complexity \(\mathbb{O}(N\log N)\). See Appendix A for basic properties of the discrete Fourier transform.

The periodic discrete framelet transform in Proposition 1.5.2 can be implemented in the frequency domain using DFT as follows.

Proposition 1.6.1

Let\(u \in l_{1}(\mathbb{Z})\)be a filter and v b = {v b (k)} k = 0 N−1be an arbitrary input signal. Extend v b into an N-periodic sequence v on\(\mathbb{Z}\)as in (1.5.4). Then uv is N-periodic and the following properties hold:
  1. (i)
    the N-point discrete Fourier transform of {[uv](k)} k = 0 N−1is\(\{\widehat{u}(\frac{2\pi n} {N} )\widehat{v^{b}}(\frac{2\pi n} {N} )\}_{n=0}^{N-1}\). Therefore, {[uv](k)} k = 0 N−1can be obtained by the inverse discrete Fourier transform (see (1.6.1)) of\(\{\widehat{u}(\frac{2\pi n} {N} )\widehat{v^{b}}(\frac{2\pi n} {N} )\}_{n=0}^{N-1}\)as follows:
    $$\displaystyle{[u {\ast} v](k) = \frac{1} {N}\sum _{n=0}^{N-1}\widehat{u}(\tfrac{2\pi n} {N} )\widehat{v^{b}}(\tfrac{2\pi n} {N} )e^{i\tfrac{2\pi nk} {N} },\qquad k \in \mathbb{Z};}$$
     
  2. (ii)
    the 2N-point discrete Fourier transform of\(\{[\mathcal{S}_{u}v](k)\}_{k=0}^{2N-1}\)is given by
    $$\displaystyle{ \sum _{n=0}^{2N-1}[\mathcal{S}_{ u}v](k)e^{-i\tfrac{2\pi kn} {2N} } = 2\widehat{u}( \tfrac{\pi n}{N})\widehat{v^{b}}(\tfrac{2\pi n}{N}),\qquad n \in \mathbb{Z}; }$$
    (1.6.2)
     
  3. (iii)
    if N is a positive even integer, then the\(\frac{N} {2}\)-point discrete Fourier transform of\(\{[\mathcal{T}_{u}v](k)\}_{k=0}^{N/2-1}\)is given by: for n = 0, , N∕2 − 1,
    $$\displaystyle{ \sum _{k=0}^{\tfrac{N} {2} -1}[\mathcal{T}_{u}v](k)e^{-i \tfrac{2\pi kn} {N/2} } = \overline{\widehat{u}(\tfrac{2\pi n} {N} )}\widehat{v^{b}}(\tfrac{2\pi n} {N} ) + \overline{\widehat{u}(\tfrac{2\pi n} {N} +\pi )}\widehat{v^{b}}(\tfrac{2\pi n} {N} +\pi ). }$$
    (1.6.3)
     

Proof

By Proposition 1.5.2, uv is N-periodic. Item (i) is a basic property of discrete Fourier transform, see Appendix A for details. By \(\mathcal{S}_{u}v = 2u {\ast} (v\! \uparrow \! 2)\) and item (i), we see that \(\mathcal{S}_{u}v\) is (2N)-periodic and (1.6.2) holds. By item (i), we have
$$\displaystyle{[u^{\star } {\ast} v](k) = \frac{1} {N}\sum _{n=0}^{N-1}\overline{\widehat{u}(\tfrac{2\pi n} {N} )}\widehat{v^{b}}(\tfrac{2\pi n} {N} )e^{i\tfrac{2\pi nk} {N} },\qquad k = 0,\ldots,N - 1.}$$
Since \(\mathcal{T}_{u}v = 2(u^{\star } {\ast} v)\! \downarrow \! 2\) and N is even, it is straightforward to check that \(\mathcal{T}_{u}v\) is \(\frac{N} {2}\)-periodic and
$$\displaystyle\begin{array}{rcl} & & \sum _{k=0}^{\tfrac{N} {2} -1}[\mathcal{T}_{u}v](k)e^{-i \tfrac{2\pi kn} {N/2} } =\sum _{ k=0}^{\tfrac{N} {2} -1}2(u^{\star } {\ast} v)(2k)e^{-i\tfrac{4\pi kn} {N} } {}\\ & & = 2\sum _{k=0}^{\tfrac{N} {2} -1} \frac{1} {N}\sum _{m=0}^{N-1}\overline{\widehat{u}(\tfrac{2\pi m} {N} )}\widehat{v^{b}}(\tfrac{2\pi m} {N} )e^{i\tfrac{4\pi (m-n)k} {N} } = \frac{2} {N}\sum _{m=0}^{N-1}\overline{\widehat{u}(\tfrac{2\pi m} {N} )}\widehat{v^{b}}(\tfrac{2\pi m} {N} )\sum _{k=0}^{\tfrac{N} {2} -1}e^{i\tfrac{4\pi (m-n)k} {N} }.{}\\ \end{array}$$
Note that for \(n = 0,\ldots, \tfrac{N} {2} - 1\), we have
$$\displaystyle{\sum _{k=0}^{\tfrac{N} {2} -1}e^{i\tfrac{4\pi (m-n)k} {N} } = \left \{\begin{array}{@{}l@{\quad }l@{}} \frac{N} {2},\quad &m = n\quad \mbox{ or}\quad n + \frac{N} {2}, \\ 0, \quad &m \in \{ 0,\ldots,N - 1\}\setminus \{n,n + \frac{N} {2} \}. \end{array} \right.}$$
Now we can easily see that (1.6.3) holds. □

As a direct consequence of Propositions 1.1.1 and 1.5.2, we have the following result on the perfect reconstruction property of a periodic discrete framelet transform for signals on a bounded interval stated right after Proposition 1.5.2.

Proposition 1.6.2

Let\(N \in 2\mathbb{N}\). The one-level periodic discrete framelet transform has the perfect reconstruction property (i.e.,\(\mathcal{V}^{per}\widetilde{\mathcal{W}}^{per}v^{b} = v^{b}\)for all v b = {v b (k)} k = 0 N−1) if and only if (1.1.12) and (1.1.13) hold for all\(\xi = 0, \frac{2\pi } {N},\ldots, \frac{2\pi (N-1)} {N}\).

Due to many different types of symmetries, it is a little bit more complicated to implement a symmetric discrete framelet transform in the frequency domain. Since both the transition operator and subdivision operator use convolution operation as the core operation, it is essential for us to discuss the convolution operation in a symmetric discrete framelet transform.

Let v b = {v b (k)} k = 0 N−1 be an input signal. Let \(u \in l_{1}(\mathbb{Z})\) be a filter with \(\mathsf{S}\,\widehat{u}(\xi ) =\varepsilon e^{ic\xi }\) for ɛ ∈ {−1, 1} and \(c \in \mathbb{Z}\). Define \(\mathbf{u}_{c}(\xi ):= e^{ic\xi /2}\widehat{u}(\xi )\). Then u c (ξ) = ɛu c (−ξ). Moreover, if the filter u is real-valued, then \(\sqrt{\varepsilon }\mathbf{u}_{c}(\xi )\) is real-valued for all \(\xi \in \mathbb{R}\).

Let v be the sequence in (1.5.13) extending v b with both endpoints non-repeated (EN). Using (2N − 2)-point discrete Fourier transform of the signal {v(k)} k = 2−N N−1 and its inverse Fourier transform, the (2N − 2)-periodic sequence uv is computed via
$$\displaystyle{ [u{\ast}v](k) = \frac{1} {2N - 2}\sum _{n=2-N}^{N-1}\widehat{u}\big( \tfrac{\pi n} {N-1}\big)\Big(\sum _{m=2-N}^{N-1}v(m)e^{-i \tfrac{\pi nm} {N-1} }\Big)e^{i \tfrac{\pi kn} {N-1} },\quad k \in \mathbb{Z}. }$$
(1.6.4)
To compute (1.6.4) efficiently, we use variants of discrete cosine transforms (DCTs) and discrete sine transforms (DSTs). Since v extends v b as in (1.5.13), the DCT I (Discrete Cosine Transform of Type I) of the N-point {v b (k)} k = 0 N−1 is defined by
$$\displaystyle{\widehat{v}^{DCT_{I} }(n):=\! \frac{1} {2}\sum _{m=2-N}^{N-1}v(m)e^{-i \tfrac{\pi nm} {N-1} }\! =\! \frac{1} {2}v^{b}(0)+\frac{(-1)^{n}} {2} v^{b}(N-1)\!+\!\!\sum _{ m=1}^{N-2}v^{b}(m)\cos ( \tfrac{\pi nm} {N-1}).}$$
Note that \(\widehat{v}^{DCT_{I}}(-n) =\widehat{ v}^{DCT_{I}}(n)\) for all \(n \in \mathbb{Z}\). If ɛ = 1, the identity (1.6.4) becomes
$$\displaystyle\begin{array}{rcl} & & [u {\ast} v](k) = \frac{1} {N - 1}\sum _{n=2-N}^{N-1}\mathbf{u}_{ c}( \tfrac{\pi n} {N-1})\widehat{v}^{DCT_{I} }(n)e^{i\tfrac{\pi (k-c/2)n} {N-1} } = \frac{1} {N - 1}\mathbf{u}_{c}(0)\widehat{v}^{DCT_{I} }(0) {}\\ & & \quad + \frac{e^{i\pi (k-c/2)}} {N - 1} \mathbf{u}_{c}(\pi )\widehat{v}^{DCT_{I} }(N - 1) + \frac{2} {N - 1}\sum _{n=1}^{N-2}\mathbf{u}_{ c}( \tfrac{\pi n} {N-1})\widehat{v}^{DCT_{I} }(n)\cos (\tfrac{\pi (k-c/2)n} {N-1} ), {}\\ \end{array}$$
which is the N-point DCT I of \(\{ \tfrac{2} {N-1}\mathbf{u}_{c}( \tfrac{\pi n} {N-1})\widehat{v}^{DCT_{I}}(n)\}_{ n=0}^{N-1}\) if c = 0. The above is the (N − 1)-point DCT III of \(\{ \tfrac{2} {N-1}\mathbf{u}_{c}( \tfrac{\pi n} {N-1})\widehat{v}^{DCT_{I}}(n)\}_{ n=0}^{N-2}\) if c = −1, since u c (π) = 0 if ɛ = 1 and c is odd. If ɛ = −1, then we have u c (0) = 0 and similarly (1.6.4) becomes
$$\displaystyle{[u{\ast}v](k) = \frac{e^{i\pi (k-\frac{c} {2} )}} {N - 1} \mathbf{u}_{c}(\pi )\widehat{v}^{DCT_{I} }(N-1)+ \frac{2} {N - 1}\sum _{n=1}^{N-2}i\mathbf{u}_{ c}( \tfrac{\pi n} {N-1})\widehat{v}^{DCT_{I} }(n)\sin (\tfrac{\pi (k-\frac{c} {2} )n} {N-1} ),}$$
which is linked to variants of discrete sine transforms.
Similarly, let v be the sequence in (1.5.17) extending v b with both endpoints repeated (ER). Using 2N-point discrete Fourier transform of {v(k)} k = −N N−1 and its inverse, the 2N-periodic sequence uv can be computed via
$$\displaystyle{ [u {\ast} v](k) = \frac{1} {2N}\sum _{n=-N}^{N-1}\widehat{u}( \tfrac{\pi n} {N})\Big(\sum _{m=-N}^{N-1}v(m)e^{-i\tfrac{\pi nm} {N} }\Big)e^{i\tfrac{\pi kn} {N} },\qquad k \in \mathbb{Z}. }$$
(1.6.5)
By (1.5.17), the (widely used) DCT II of the N-point {v b (k)} k = 0 N−1 is defined by
$$\displaystyle{\widehat{v}^{DCT_{II} }(n):= \frac{e^{-i \tfrac{n\pi } {2N} }} {2} \sum _{m=-N}^{N-1}v(m)e^{-i\tfrac{\pi nm} {N} } =\sum _{ m=0}^{N-1}v^{b}(m)\cos (\tfrac{\pi n(m+1/2)} {N} ).}$$
Note that \(\widehat{v}^{DCT_{II}}(-n) =\widehat{ v}^{DCT_{II}}(n)\) for all \(n \in \mathbb{Z}\) and \(\widehat{v}^{DCT_{II}}(N) = 0\). If ɛ = 1, then the identity (1.6.5) becomes
$$\displaystyle\begin{array}{rcl} [u {\ast} v](k)& =& \frac{1} {N}\sum _{n=-N}^{N-1}\mathbf{u}_{ c}( \tfrac{\pi n} {N})\widehat{v}^{DCT_{II} }(n)e^{i\tfrac{\pi (k+\frac{1-c} {2} )n} {N} } {}\\ & =& \frac{1} {N}\mathbf{u}_{c}(0)\widehat{v}^{DCT_{II} }(0) + \frac{2} {N}\sum _{n=1}^{N-1}\mathbf{u}_{ c}( \tfrac{\pi n} {N})\widehat{v}^{DCT_{II} }(n)\cos (\tfrac{\pi (k+\frac{1-c} {2} )n} {N} ), {}\\ \end{array}$$
which is the N-point DCT III of \(\{ \tfrac{2} {N}\widehat{u}( \frac{\pi n} {N})\widehat{v}^{DCT_{II}}(n)\}_{ n=0}^{N-1}\) if c = 0. If ɛ = 1 and c is odd, then u c (π) = 0 and the above is the (N + 1)-point DCT I of \(\{ \tfrac{2} {N}\widehat{u}( \frac{\pi n} {N})\widehat{v}^{DCT_{II}}(n)\}_{ n=0}^{N}\) if c = 1. If ɛ = −1, then we have u c (0) = 0 and similarly (1.6.4) becomes
$$\displaystyle{[u {\ast} v](k) = \frac{2} {N}\sum _{n=1}^{N-1}i\mathbf{u}_{ c}( \tfrac{\pi n} {N})\widehat{v}^{DCT_{II} }(n)\sin (\tfrac{\pi (k+\frac{1-c} {2} )n} {N} ),}$$
which is linked to variants of discrete sine transforms.

1.7 Exercises

  1. 1.1.

    Prove the following identities: For \(u,v,w \in l_{2}(\mathbb{Z})\) and \(n \in \mathbb{Z}\), 〈v, w〉 = [vw](0), [vw](n) = 〈v, w(⋅ − n)〉, uv = vu, 〈uv, w〉 = 〈v, uw〉, 〈vd, w〉 = 〈v, wd〉, where \(\mathsf{d} \in \mathbb{Z}\setminus \{0\}\) is a sampling factor.

     
  2. 1.2.
    Let \((\mathcal{H},\langle \cdot,\cdot \rangle )\) be an inner product space over the complex field \(\mathbb{C}\), e.g., \(\mathcal{H} = l_{2}(\mathbb{Z})\). Let \(T: \mathcal{H}\rightarrow \mathcal{H}\) be a linear mapping. Prove the polarization identity:
    $$\displaystyle\begin{array}{rcl} \langle Tv,w\rangle & =& \frac{1} {4}\big[\langle T(v + w),v + w\rangle -\langle T(v - w),v - w\rangle {}\\ & & \quad + i\langle T(v + iw),v + iw\rangle - i\langle T(v - iw),v - iw\rangle \big],\quad v,w \in \mathcal{H}. {}\\ \end{array}$$
     
  3. 1.3.

    Prove Proposition 1.1.5.

     
  4. 1.4.
    Prove that the perfect reconstruction condition in (1.3.6) is equivalent to
    $$\displaystyle{\left [\begin{array}{*{10}c} \widehat{\tilde{a}^{[0]}}(\xi )&\widehat{\tilde{b}_{ 1}^{[0]}}(\xi )&\cdots &\widehat{\tilde{b}_{ s}^{[0]}}(\xi ) \\ \widehat{\tilde{a}^{[1]}}(\xi )&\widehat{\tilde{b}_{1}^{[1]}}(\xi )&\cdots &\widehat{\tilde{b}_{s}^{[1]}}(\xi )\end{array} \right ]\left [\begin{array}{*{10}c} \widehat{a^{[0]}}(\xi )&\widehat{b_{ 1}^{[0]}}(\xi )&\cdots &\widehat{b_{ s}^{[0]}}(\xi ) \\ \widehat{a^{[1]}}(\xi )&\widehat{b_{1}^{[1]}}(\xi )&\cdots &\widehat{b_{s}^{[1]}}(\xi )\end{array} \right ]^{\star } = \frac{1} {2}I_{2}.}$$
     
  5. 1.5.

    Prove the Leibniz differentiation formula: \([\mathbf{f}\mathbf{g}]^{(n)}=\sum _{j=0}^{n} \frac{n!} {j!(n-j)!}\mathbf{f}^{(\,j)}(\cdot )\mathbf{g}^{(n-j)}(\cdot ).\)

     
  6. 1.6.

    Let \(\partial _{1}:= \frac{\partial } {\partial \xi _{1}}\) and \(\partial _{2}:= \frac{\partial } {\partial \xi _{2}}\). Using directional derivatives to prove \([\mathbf{f}(\xi )\mathbf{g}(\xi )]^{(n)} = [(\partial _{1} + \partial _{2})^{n}(\mathbf{f}(\xi _{1})\mathbf{g}(\xi _{2}))]\vert _{\xi _{1}=\xi,\xi _{2}=\xi }\) and use it to prove Exercise 1.5.

     
  7. 1.7.

    Prove the generalized product rule for differentiation in (1.2.3).

     
  8. 1.8.

    Prove the identity in (1.2.4).

     
  9. 1.9.

    For \(u =\{ u(k)\}_{k\in \mathbb{Z}} \in l_{0}(\mathbb{Z})\), (1.2.15) holds  ⇔  \(\sum _{k\in \mathbb{Z}}u(k)k^{j} = c^{j}\) for all j = 0, , m  ⇔  \(\sum _{k\in \mathbb{Z}}u(k)(k - c)^{j} =\boldsymbol{\delta } (\,j)\), j = 0, , m.

     
  10. 1.10.

    For \(0 <p\leqslant \infty\) and \(v \in l(\mathbb{Z})\), define \(\|v\|_{l_{p}(\mathbb{Z})}:= (\sum _{k\in \mathbb{Z}}\vert v(k)\vert ^{p})^{1/p}\). Prove that \(\|u + v\|_{l_{p}(\mathbb{Z})}^{\min (\,p,1)}\leqslant \|u\|_{l_{p}(\mathbb{Z})}^{\min (\,p,1)} +\| v\|_{l_{p}(\mathbb{Z})}^{\min (\,p,1)}\) and \(\|v\|_{l_{p}(\mathbb{Z})}\leqslant \|v\|_{l_{q}(\mathbb{Z})}\)\(\forall 0\! <\! q\leqslant p\leqslant \infty\).

     
  11. 1.11.
    For a linear operator \(T: l_{p}(\mathbb{Z}) \rightarrow l_{p}(\mathbb{Z})\), its operator norm is defined to be \(\|T\|:=\sup \{\| Tv\|_{l_{p}(\mathbb{Z})}\;:\;\| v\|_{l_{p}(\mathbb{Z})}\leqslant 1\}\). For a filter \(u \in l_{0}(\mathbb{Z})\), prove that all the linear operators \(u{\ast}: l_{p}(\mathbb{Z}) \rightarrow l_{p}(\mathbb{Z})\), \(\mathcal{S}_{u}: l_{p}(\mathbb{Z}) \rightarrow l_{p}(\mathbb{Z})\), \(\mathcal{T}_{u}: l_{p}(\mathbb{Z}) \rightarrow l_{p}(\mathbb{Z})\) are well defined and bounded for all \(0 <p\leqslant \infty\). In particular, with q: = min( p, 1),
    $$\displaystyle\begin{array}{rcl} & & \|u {\ast} v\|_{l_{p}(\mathbb{Z})}\leqslant \|u\|_{l_{q}(\mathbb{Z})}\|v\|_{l_{p}(\mathbb{Z})},\qquad \|\mathcal{T}_{u}v\|_{l_{p}(\mathbb{Z})}\leqslant \|u\|_{l_{q}(\mathbb{Z})}\|v\|_{l_{p}(\mathbb{Z})}, {}\\ & & \|\mathcal{S}_{u}v\|_{l_{p}(\mathbb{Z})}\leqslant \|v\|_{l_{p}(\mathbb{Z})}\max (\|u^{[0]}\|_{ l_{q}(\mathbb{Z})},\|u^{[1]}\|_{ l_{q}(\mathbb{Z})}). {}\\ \end{array}$$
     
  12. 1.12.

    Prove that Proposition 1.1.2 is still true if \(l(\mathbb{Z})\) is replaced by \(l_{p}(\mathbb{Z})\),\(0 <p\leqslant \infty\).

     
  13. 1.13.

    Show that \(\mathcal{W}: l_{p}(\mathbb{Z}) \rightarrow (l_{p}(\mathbb{Z}))^{1\times (s+1)}\) in (1.1.19) and \(\mathcal{V}: (l_{p}(\mathbb{Z}))^{1\times (s+1)} \rightarrow l_{p}(\mathbb{Z})\) in (1.1.20) are well-defined bounded linear operators for \(0 <p\leqslant \infty\).

     
  14. 1.14.
    Let m be a nonnegative integer. Let u, v be functions which are m-times differentiable at the origin and satisfy u(0) = v(0) ≠ 0. Suppose that d and λ are real numbers such that d n u(0) ≠ λ n v(0) for all n = 1, , m. Show that there exists a finitely supported sequence \(\theta \in l_{0}(\mathbb{Z})\) satisfying
    $$\displaystyle{\widehat{\theta }(0) = 1\quad \mbox{ and}\quad \widehat{\theta }(\mathsf{d}\xi )\mathbf{u}(\xi ) =\widehat{\theta } (\lambda \xi )\mathbf{v}(\xi ) +\mathbb{ O}(\vert \xi \vert ^{m+1}),\quad \xi \rightarrow 0.}$$
    More precisely, \(\widehat{\theta }^{(\,j)}(0),0\leqslant j\leqslant m\) are uniquely determined by u( j)(0) and \(\mathbf{v}^{(\,j)}(0),0\leqslant j\leqslant m\) via the following recursive formula: For n = 1, , m,
    $$\displaystyle{\widehat{\theta }(0) = 1\quad \mbox{ and}\quad \widehat{\theta }^{(n)}(0) =\sum _{ j=0}^{n-1} \frac{n!} {j!(n - j)!} \frac{\lambda ^{j}\mathbf{v}^{(n-j)}(0) -\mathsf{d}^{\,j}\mathbf{u}^{(n-j)}(0)} {\mathsf{d}^{n}\mathbf{u}(0) -\lambda ^{n}\mathbf{v}(0)} \widehat{\theta }^{(\,j)}(0).}$$
     
  15. 1.15.
    Let \(u \in l_{0}(\mathbb{Z})\) and m be a positive integer. Show that the coefficient matrix of \(\mathcal{T}_{u}\vert _{\mathbb{P}_{m-1}}\) under the basis {1, x, , x m−1} of \(\mathbb{P}_{m-1}\) is a lower triangular matrix with its diagonal entries being \(2\overline{\widehat{u}(0)},2^{2}\overline{\widehat{u}(0)},\ldots,2^{m}\overline{\widehat{u}(0)}\). Moreover, if \(\widehat{u}(0)\neq 0\), then
    $$\displaystyle{\mathcal{T}_{u}[(\cdot )^{\,j}{\ast}\theta ] = 2^{j+1}\overline{\widehat{u}(0)}[(\cdot )^{\,j}{\ast}\theta ],\qquad j = 0,\ldots,m - 1,}$$
    that is, \((\cdot )^{\,j}{\ast}\theta \in \mathbb{P}_{m-1}\) is a nonzero eigenvector of \(\mathcal{T}_{u}: \mathbb{P}_{m-1} \rightarrow \mathbb{P}_{m-1}\) corresponding to the eigenvalue \(2^{n+1}\overline{\widehat{u}(0)}\) for all n = 0, , m − 1, where (⋅ )j is the polynomial sequence induced by x j and \(\theta \in l_{0}(\mathbb{Z})\) satisfies
    $$\displaystyle{\widehat{\theta }(0) = 1\quad \mbox{ and}\quad \widehat{\theta }(2\xi )\overline{\widehat{u}(0)} =\widehat{\theta } (\xi )\overline{\widehat{u}(\xi )} +\mathbb{ O}(\vert \xi \vert ^{m}),\qquad \xi \rightarrow 0.}$$
     
  16. 1.16.
    For \(\mathsf{p} \in \mathbb{P}_{m-1}\) and \(\widehat{v}(\xi ) =\mathbb{ O}(\vert \xi \vert ^{m})\) as ξ → 0, show that
    $$\displaystyle{\langle \mathcal{S}_{u}\mathsf{p},v\rangle =\Big [\mathsf{p}\Big(-i\frac{d} {d\xi }\Big)\big(\,\widehat{u}(\xi /2+\pi )\overline{\widehat{v}(\xi /2+\pi )}\big)\Big]\Big\vert _{\xi =0} =\Big [\mathsf{p}\Big(-\frac{i} {2} \frac{d} {d\xi }\Big)\big(\,\widehat{u}(\xi )\overline{\widehat{v}(\xi )}\big)\Big]\Big\vert _{\xi =\pi }.}$$
    Then use it to prove the equivalence between items (1) and (5) of Theorem 1.2.4.
     
  17. 1.17.

    Prove the identity in (1.3.26).

     
  18. 1.18.

    Let \(u,\tilde{u} \in l_{0}(\mathbb{Z})\) and \(m \in \mathbb{N}\). Prove that \(\tfrac{1} {2}\mathcal{S}_{u}\mathcal{T}_{\tilde{u}}\mathsf{p} = \mathsf{p}\) for all \(\mathsf{p} \in \mathbb{P}_{m-1}\) if and only if u has m sum rules and \(\overline{\widehat{u}(\xi )}\widehat{\tilde{u}}(\xi ) = 1 +\mathbb{ O}(\vert \xi \vert ^{m})\) as ξ → 0, that is, \(u^{\star } {\ast}\tilde{ u}\) has m linear-phase moments with phase 0.

     
  19. 1.19.
    Let \(u \in l_{0}(\mathbb{Z})\).
    1. a.

      Show that there exists a nontrivial sequence \(v \in l_{0}(\mathbb{Z})\) such that \(\mathcal{T}_{u}v = 0\).

       
    2. b.

      If \(\mathcal{S}_{u}v = 0\) for some \(v \in l_{0}(\mathbb{Z})\), prove that v(k) = 0 for all \(k \in \mathbb{Z}\).

       
    3. c.

      For \(u =\{ \frac{1} {2},\underline{\mathbf{0}}, \frac{1} {2}\}_{[-1,1]}\), find a nontrivial sequence \(v \in l(\mathbb{Z})\) such that\(\mathcal{S}_{u}v = 0\).

       
     
  20. 1.20.

    Let \(m \in \mathbb{N}\) and \(u \in l_{0}(\mathbb{Z})\) such that u has m sum rules. Show that the coefficient matrix of \(\mathcal{S}_{u}\) under the basis {1, x, , x m−1} of \(\mathbb{P}_{m-1}\) is a lower triangular matrix with its diagonal entries being \(\widehat{u}(0),2^{-1}\widehat{u}(0),\ldots,2^{1-m}\widehat{u}(0)\). Moreover, if \(\widehat{u}(0)\neq 0\), then \(\mathcal{S}_{u}[(\cdot )^{\,j}{\ast}\vartheta ] = 2^{-j}\widehat{u}(0)[(\cdot )^{\,j}{\ast}\vartheta ]\), j = 0, , m − 1, where \(\vartheta \in l_{0}(\mathbb{Z})\) satisfies \(\widehat{\vartheta }(0) = 1\) and \(\widehat{\vartheta }(2\xi )\widehat{u}(\xi ) =\widehat{\vartheta } (\xi )\widehat{u}(0) +\mathbb{ O}(\vert \xi \vert ^{m})\), ξ → 0.

     
  21. 1.21.

    Let \(\mathsf{p} \in \mathbb{P}\) be a polynomial and \(v \in l(\mathbb{Z})\) such that v(n) = p(n) for all \(n\geqslant M\) for some \(M \in \mathbb{N}\). Let \(u \in l_{0}(\mathbb{Z})\). If there exist \(N \in \mathbb{N}\) and \(\mathsf{q} \in \mathbb{P}\) such that \(\mathcal{S}_{u}v(n) = \mathsf{q}(n)\) for all \(n\geqslant N\), prove that \(\mathcal{S}_{u}\mathsf{p}(n) = \mathsf{q}(n)\) for all \(n \in \mathbb{Z}\).

     
  22. 1.22.

    Let \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\) be a dual framelet filter bank. If s > 1, by Proposition 1.1.2, then \(\mathcal{V}\) is not one-to-one. Explicitly construct \(w \in (l_{0}(\mathbb{Z}))^{1\times (s+1)}\) such that w is not identically zero but \(\mathcal{V}w = 0\), where \(\mathcal{V}\) is the discrete framelet synthesis operator defined in (1.1.8).

     
  23. 1.23.

    Let the J-level discrete framelet analysis operator \(\mathcal{W}_{J}\) and the J-level discrete framelet synthesis operator \(\mathcal{V}_{J}\) employing a filter bank {a; b 1, , b s } be defined in Sect. 1.3. Show that \(\mathcal{W}_{J}^{\star } = \mathcal{V}_{J}\) and \(\mathcal{V}_{J}^{\star } = \mathcal{W}_{J}\).

     
  24. 1.24.
    Prove that {a; b 1, , b s } is a tight framelet filter bank if and only if the discrete affine system \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b_{1},\ldots,b_{s}\})\) is a (normalized) tight frame for the space \(l_{2}(\mathbb{Z})\) for every \(J \in \mathbb{N}\), that is,
    $$\displaystyle{\|v\|_{l_{2}(\mathbb{Z})}^{2} =\sum _{ u\in \mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b_{1},\ldots,b_{s}\})}\vert \langle v,u\rangle \vert ^{2},\qquad \forall \;v \in l_{ 2}(\mathbb{Z}).}$$
     
  25. 1.25.

    Prove that {a; b} is an orthogonal wavelet filter bank if and only if \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b\})\) is an orthonormal basis for \(l_{2}(\mathbb{Z})\) for every \(J \in \mathbb{N}\).

     
  26. 1.26.
    Prove that \((\{\tilde{a};\tilde{b}\},\{a;b\})\) is a biorthogonal wavelet filter bank if and only if \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{\tilde{a};\tilde{b}\})\) and \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b\})\) are biorthogonal to each other in the space \(l_{2}(\mathbb{Z})\) for every \(J \in \mathbb{N}\):
    $$\displaystyle{\langle u,v\rangle = \left \{\begin{array}{@{}l@{\quad }l@{}} 1,\quad &\text{if }\tilde{u} = v\\ 0,\quad &\text{if } \tilde{u}\neq v,\end{array} \right.\qquad \forall \;u \in \mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b\}),\quad v \in \mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{\tilde{a};\tilde{b}\}).}$$
     
  27. 1.27.
    Prove that \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})\) is a dual framelet filter bank if and only if \((\mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\})\), \(\mathop{\mathrm{\mathsf{DAS}}}\nolimits (\{a;b_{1},\ldots,b_{s}\}))\) is a dual frame in \(l_{0}(\mathbb{Z})\), that is,
    $$\displaystyle{\langle v,w\rangle =\sum _{u\in \mathop{\mathrm{\mathsf{DAS}}}\nolimits _{J}(\{a;b_{1},\ldots,b_{s}\})}\langle v,\tilde{u}\rangle \langle u,w\rangle,\qquad \forall \;v,w \in l_{0}(\mathbb{Z}).}$$
    Note that the above summation is in fact finite since \(v,w \in l_{0}(\mathbb{Z})\).
     
  28. 1.28.

    Let \(m,\tilde{m},n,\tilde{n} \in \mathbb{Z}\). Define \(\widehat{a}(\xi ):= e^{-in\xi }\widehat{a_{m}^{B}}(\xi )\) and \(\widehat{\tilde{a}}(\xi ):= e^{-i\tilde{n}\xi }\widehat{a_{\tilde{m}}^{B}}(\xi )\). Show that \(\widehat{\tilde{a}}(\xi )\overline{\widehat{a}(\xi )} = 1 + O(\vert \xi \vert ^{3}),\xi \rightarrow 0\) cannot hold.

     
  29. 1.29.
    Suppose that {a; b 1, , b s } Θ is a tight framelet filter bank. Show that all the high-pass filters b 1, , b s have m vanishing moments if and only if
    $$\displaystyle{\widehat{\varTheta }(2\xi ) -\widehat{\varTheta } (\xi )\vert \widehat{a}(\xi )\vert ^{2} =\mathbb{ O}(\vert \xi \vert ^{2m}),\qquad \xi \rightarrow 0.}$$
    If in addition \(\widehat{\varTheta }(0)\neq 0\), then a must have m sum rules.
     
  30. 1.30.

    Define \(A(\xi ):=\prod _{ j=1}^{N}\frac{e^{-i\xi }-e^{it_{j}}} {1-e^{it_{j}}},\xi \in \mathbb{R}\) with \(t_{1},\ldots,t_{N} \in \mathbb{R}\setminus [2\pi \mathbb{Z}]\). Prove that \(A''(0) - [A'(0)]^{2} = -\sum _{j=1}^{N} \frac{1} {2(1-\cos (t_{j}))}\leqslant 0\) and the equality holds if and only if A(ξ) = 1.

     
  31. 1.31.

    Let \(a,b_{1},\ldots,b_{s},\varTheta \in l_{0}(\mathbb{Z})\) with \(\widehat{a}(0) = 1\) and \(\widehat{\varTheta }(0) = 1\). Suppose {a; b 1, , b s } Θ is a tight framelet filter bank. If all the roots of the Laurent polynomial \(\sum _{k\in \mathbb{Z}}a(k)z^{k}\) lie on the unit circle, prove that one of the high-pass filters b 1, , b s must have at most one vanishing moment.

     
  32. 1.32.

    Let \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\) be a dual framelet filter bank. For \(\lambda _{0},\ldots,\lambda _{s} \in \mathbb{T}\) and \(n_{\varTheta },n_{0},\ldots,n_{s} \in \mathbb{Z}\) such that \(\tilde{n}_{0} = n_{0} - n_{\varTheta },\tilde{n}_{\ell} = n_{\ell} + n_{\varTheta },n_{\ell} - n_{0} \in 2\mathbb{Z}\) for all = 1, , s, show that \((\{\lambda _{0}\tilde{a}(\cdot - n_{0});\lambda _{1}\tilde{b}_{1}(\cdot - n_{1}),\ldots,\lambda _{s}\tilde{b}_{s}(\cdot - n_{s})\},\{\lambda _{0}a(\cdot - n_{0});\lambda _{1}b_{1}(\cdot - n_{1}),\ldots,\lambda _{s}b_{s}(\cdot - n_{s})\})_{\varTheta (\cdot -n_{\varTheta })}\) is also a dual framelet filter bank.

     
  33. 1.33.
    Prove that the perfect reconstruction condition in (1.4.3) and (1.4.4) for a dual framelet filter bank \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\) is equivalent to
    $$\displaystyle\begin{array}{rcl} & & \left [\begin{array}{*{10}c} \widehat{\tilde{a}^{[0]}}(\xi )&\widehat{\tilde{b}_{ 1}^{[0]}}(\xi )&\cdots &\widehat{\tilde{b}_{ s}^{[0]}}(\xi ) \\ \widehat{\tilde{a}^{[1]}}(\xi )&\widehat{\tilde{b}_{1}^{[1]}}(\xi )&\cdots &\widehat{\tilde{b}_{s}^{[1]}}(\xi )\end{array} \right ]\left [\begin{array}{*{10}c} \widehat{\varTheta }(2\xi )&0\\ 0 &1\end{array} \right ] {}\\ & & \quad \times \left [\begin{array}{*{10}c} \widehat{a^{[0]}}(\xi )&\widehat{b_{ 1}^{[0]}}(\xi )&\cdots &\widehat{b_{ s}^{[0]}}(\xi ) \\ \widehat{a^{[1]}}(\xi )&\widehat{b_{1}^{[1]}}(\xi )&\cdots &\widehat{b_{s}^{[1]}}(\xi )\end{array} \right ]^{\star } = \frac{1} {2}\left [\begin{array}{*{10}c} \widehat{\varTheta ^{[0]}}(\xi )& 0 \\ 0 &\widehat{\varTheta ^{[1]}}(\xi )\end{array} \right ].{}\\ \end{array}$$
     
  34. 1.34.

    Prove item (iv) of Proposition 1.5.2.

     
  35. 1.35.

    Let \(u_{0},\ldots,u_{s} \in l_{0}(\mathbb{Z})\). Let \(\mathcal{W}^{per}\) be the coefficient matrix of the periodic discrete framelet analysis operator defined in (1.5.6) and let \(\mathcal{V}^{per}\) be the coefficient matrix of the periodic discrete framelet synthesis operator defined in (1.5.7) but with \(\tilde{u}_{0},\ldots,\tilde{u}_{s}\) being replaced by u 0, , u s . Show that \([\mathcal{W}^{per}]^{\star } = \mathcal{V}^{per}\), that is, \(\mathcal{V}^{per}\) is the complex conjugate of the transpose of \(\mathcal{W}^{per}\).

     
  36. 1.36.

    Let {a; b 1, , b s } be a filter bank. Suppose that there exists a positive constant C such that \(\|\mathcal{W}_{J}v\|_{(l_{2}(\mathbb{Z}))^{1\times (sJ+1)}}^{2}\leqslant C\|v\|_{l_{2}(\mathbb{Z})}^{2}\) for all \(v \in l_{2}(\mathbb{Z})\) and \(J \in \mathbb{N}\). Prove \(\vert \widehat{a}(0)\vert \leqslant 1\). If in addition \(\vert \widehat{a}(0)\vert = 1\), then \(\widehat{b_{1}}(0) = \cdots \widehat{b_{s}}(0) = 0\).

     
  37. 1.37.

    Let \(\zeta \in \mathbb{C}\) and λ > 0. Prove (i) \(\eta _{\lambda }^{soft}(\zeta ) = \mbox{ argmin}_{z\in \mathbb{C}} \frac{1} {2}\vert z -\zeta \vert ^{2} +\lambda \vert z\vert\), where the soft thresholding function η λ soft is defined in (1.3.2); (ii) \(\eta _{\lambda }^{Hard}(\zeta ) = \mbox{ argmin}_{z\in \mathbb{C}} \frac{1} {2}\vert z -\zeta \vert ^{2} +\lambda \vert z\vert _{ 0}\), where | 0 |0: = 0 and | z |0: = 1 for \(z \in \mathbb{C}\setminus \{0\}\), and η λ Hard (ζ) = { ζ} for | ζ | > λ, η λ Hard (ζ) = {0, ζ} if | ζ | = λ, and η λ Hard (ζ): = 0 for | ζ | < λ.

    The sampling factor used in this Chapter is 2. In fact, there are more general discrete framelet transforms and filter banks using a general sampling factor d, where d is a nonzero integer. For simplicity, here we only consider a positive sampling factor d with \(\mathsf{d}\geqslant 1\). Define \(\mathcal{S}_{u,\mathsf{d}}\) and \(\mathcal{T}_{u,\mathsf{d}}\) as in (1.3.12) and (1.3.13). For a filter bank \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\), a one-level discrete d-framelet decomposition is
    $$\displaystyle{\mathcal{W}v:= (w_{0},\ldots,w_{s})\quad \mbox{ with}\quad w_{\ell}:= \mathsf{d}^{-1/2}\mathcal{T}_{\tilde{ u}_{\ell},\mathsf{d}}v,\quad \ell = 0,\ldots,s,\;v \in l(\mathbb{Z})}$$
    and a one-level discrete d-framelet reconstruction is
    $$\displaystyle{\mathring{v} = \mathcal{V}(\mathring{w}_{0},\ldots,\mathring{w}_{s}) = \mathsf{d}^{-1/2}\sum _{ \ell=1}^{s}\mathcal{S}_{ u_{\ell},\mathsf{d}}\mathring{w}_{\ell},\qquad\mathring{w}_{0},\ldots,\mathring{w}_{s} \in l(\mathbb{Z}).}$$
    \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\) is called a dual d-framelet filter bank if it has the perfect reconstruction property, that is, \(\sum _{\ell=0}^{s}\mathcal{S}_{u_{\ell},\mathsf{d}}\mathcal{T}_{\tilde{u}_{\ell},\mathsf{d}}v = \mathsf{d}v\) for all \(v \in l(\mathbb{Z})\). {u 0, , u s } is called a tight d-framelet filter bank if ({u 0, , u s }, {u 0, , u s }) is a dual d-framelet filter bank. A dual d-framelet filter bank with s = | d | − 1 is called a biorthogonal d-wavelet filter bank and a tight d-framelet filter bank with s = | d | − 1 is called an orthogonal d-wavelet filter bank. The coset sequences u[γ: d] are
    $$\displaystyle{\widehat{u^{[\gamma:\mathsf{d}]}}(\xi ):=\sum _{ k\in \mathbb{Z}}u(\gamma +\mathsf{d}k)e^{-ik\xi },\quad \mbox{ that is},\quad u^{[\gamma:\mathsf{d}]} = u(\gamma +\cdot )\! \downarrow \!\mathsf{d} =\{ u(\gamma +\mathsf{d}k)\}_{ k\in \mathbb{Z}}.}$$
     
  38. 1.38.
    Prove that \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\) is a dual d-framelet filter bank if and only if
    $$\displaystyle{\widehat{\tilde{u}_{0}}(\xi )\overline{\widehat{u_{0}}(\xi +2\pi \gamma /\mathsf{d})} + \cdots +\widehat{\tilde{ u}_{s}}(\xi )\overline{\widehat{u_{s}}(\xi +2\pi \gamma /\mathsf{d})} =\boldsymbol{\delta } (\gamma ),\qquad \gamma = 0,\ldots,\mathsf{d} - 1.}$$
     
  39. 1.39.
    Prove \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\) is a dual d-framelet filter bank if and only if
    $$\displaystyle{\left [\begin{array}{*{10}c} \widehat{\tilde{u}_{0}^{[0:\mathsf{d}]}}(\xi ) & \widehat{\tilde{u}_{1}^{[0:\mathsf{d}]}}(\xi ) &\cdots & \widehat{\tilde{u}_{s}^{[0:\mathsf{d}]}}(\xi ) \\ \widehat{\tilde{u}_{0}^{[1:\mathsf{d}]}}(\xi ) & \widehat{\tilde{u}_{1}^{[1:\mathsf{d}]}}(\xi ) &\cdots & \widehat{\tilde{u}_{s}^{[1:\mathsf{d}]}}(\xi )\\ \vdots & \vdots & \ddots & \vdots \\ \widehat{\tilde{u}_{0}^{[\mathsf{d}-1:\mathsf{d}]}}(\xi )&\widehat{\tilde{u}_{1}^{[\mathsf{d}-1:\mathsf{d}]}}(\xi )&\cdots &\widehat{\tilde{u}_{s}^{[\mathsf{d}-1:\mathsf{d}]}}(\xi )\end{array} \right ]\left [\begin{array}{*{10}c} \widehat{u_{0}^{[0:\mathsf{d}]}}(\xi ) & \widehat{u_{1}^{[0:\mathsf{d}]}}(\xi ) &\cdots & \widehat{u_{s}^{[0:\mathsf{d}]}}(\xi ) \\ \widehat{u_{0}^{[1:\mathsf{d}]}}(\xi ) & \widehat{u_{1}^{[1:\mathsf{d}]}}(\xi ) &\cdots & \widehat{u_{s}^{[1:\mathsf{d}]}}(\xi )\\ \vdots & \vdots & \ddots & \vdots \\ \widehat{u_{0}^{[\mathsf{d}-1:\mathsf{d}]}}(\xi )&\widehat{u_{1}^{[\mathsf{d}-1:\mathsf{d}]}}(\xi )&\cdots &\widehat{u_{s}^{[\mathsf{d}-1:\mathsf{d}]}}(\xi )\end{array} \right ]^{\star } = \mathsf{d}^{-1}I_{\mathsf{ d}}.}$$
     
  40. 1.40.
    Let \((\{\tilde{u}_{0},\ldots,\tilde{u}_{s}\},\{u_{0},\ldots,u_{s}\})\) be a dual d-framelet filter bank. Prove that the following statements are equivalent:
    1. a.

      \(\widetilde{\mathcal{W}}\) is onto or \(\mathcal{V}\) is one-to-one.

       
    2. b.

      \(\mathcal{V}\widetilde{\mathcal{W}} =\mathrm{ Id}_{\,l(\mathbb{Z})}\) and \(\widetilde{\mathcal{W}}\mathcal{V} =\mathrm{ Id}_{(l(\mathbb{Z}))^{1\times (s+1)}}\), that is, \(\mathcal{V}^{-1} =\widetilde{ \mathcal{W}}\) and \(\widetilde{\mathcal{W}}^{-1} = \mathcal{V}\).

       
    3. c.

      s = | d | − 1.

       
     
  41. 1.41.
    Prove that {u 0, , u s } is a tight d-framelet filter bank if and only if
    $$\displaystyle{\|\mathcal{T}_{u_{0},\mathsf{d}}v\|_{l_{2}(\mathbb{Z})}^{2} + \cdots +\| \mathcal{T}_{ u_{s},\mathsf{d}}v\|_{l_{2}(\mathbb{Z})}^{2} = \mathsf{d}\|v\|_{ l_{2}(\mathbb{Z})}^{2},\qquad \forall \;v \in l_{ 2}(\mathbb{Z}).}$$
     
  42. 1.42.

    Prove that \(\mathcal{T}_{u,\mathsf{d}}\mathsf{p} = \mathsf{d}[\mathsf{p} {\ast} u^{\star }](\mathsf{d}\cdot )\) for any polynomial \(\mathsf{p}\).

     
  43. 1.43.
    Let \(u =\{ u(k)\}_{k\in \mathbb{Z}} \in l_{0}(\mathbb{Z})\) be a finitely supported sequence on \(\mathbb{Z}\) and \(\mathsf{p} \in \mathbb{P}\) be a polynomial. Show that the following statements are equivalent:
    1. a.

      \(\mathcal{S}_{u,\mathsf{d}}\mathsf{p}\) is a polynomial sequence;

       
    2. b.

      \(\sum _{k\in \mathbb{Z}}\mathsf{p}^{(\,j)}(-\mathsf{d}^{-1}\gamma - k)u(\gamma +\mathsf{d}k) =\sum _{k\in \mathbb{Z}}\mathsf{p}^{(j)}(-k)u(\mathsf{d}k)\) for all \(j \in \mathbb{N}_{0}\) and γ = 0, , d − 1;

       
    3. c.

      For all \(j \in \mathbb{N}_{0}\), \([\mathsf{p}^{(\,j)}(-\frac{0} {\mathsf{d}} - i\frac{d} {d\xi })\widehat{u^{[0:\mathsf{d}]}}(\xi )]\vert _{\xi =0} = [\mathsf{p}^{(\,j)}(-\frac{1} {\mathsf{d}} - i\frac{d} {d\xi })\widehat{u^{[1:\mathsf{d}]}}(\xi )]\vert _{\xi =0}\)\(= \cdots = [\mathsf{p}^{(\,j)}(-\frac{\mathsf{d}-1} {\mathsf{d}} - i\frac{d} {d\xi })\widehat{u^{[\mathsf{d}-1:\mathsf{d}]}}(\xi )]\vert _{\xi =0}\);

       
    4. d.

      \([\mathsf{p}^{(\,j)}(-\frac{i} {\mathsf{d}} \frac{d} {d\xi })\widehat{u}(\xi )]\vert _{\xi =\pi } = \cdots\)\(= [\mathsf{p}^{(\,j)}(-i\frac{\mathsf{d}-1} {\mathsf{d}} \frac{d} {d\xi })\widehat{u}(\xi )]\vert _{\xi =\pi } = 0\) for all \(j \in \mathbb{N}_{0}\).

       
    Moreover, if any of the above items holds, then \(\deg (\mathcal{S}_{u,\mathsf{d}}\mathsf{p})\leqslant \deg (\mathsf{p})\),
    $$\displaystyle{\mathcal{S}_{u,\mathsf{d}}\mathsf{p} = \mathsf{p}(\mathsf{d}^{-1}\cdot ) {\ast} u =\sum _{ j=0}^{\infty }\frac{(-i)^{\,j}} {\mathsf{d}^{\,j}j!} \mathsf{p}^{(\,j)}(\mathsf{d}^{-1}\cdot )\widehat{u}^{\,(\,j)}(0).}$$
     
  44. 1.44.
    For any positive integer \(m \in \mathbb{N}\), the following statements are equivalent
    1. a.

      \(\mathcal{S}_{u,\mathsf{d}}\mathsf{q} \in \mathbb{P}\) for some polynomial \(\mathsf{q} \in \mathbb{P}\) with \(\deg (\mathsf{q}) = m - 1\);

       
    2. b.

      \(\mathcal{S}_{u,\mathsf{d}}\mathbb{P}_{m-1} \subseteq \mathbb{P}_{m-1}\);

       
    3. c.

      \(\widehat{u}^{\,(\,j)}(\pi \gamma /\mathsf{d}) = 0\) for all \(0\leqslant j <m\) and \(1\leqslant \gamma <\mathsf{d}\), i.e., \(\widehat{u}(\xi +\pi \gamma /\mathsf{d}) =\mathbb{ O}(\vert \xi \vert ^{m})\) as ξ → 0 for \(1\leqslant \gamma <\mathsf{d}\);

       
    4. d.

      \(\widehat{u}(\xi ) = (1 + e^{-i\xi } + \cdots + e^{-i(\mathsf{d}-1)\xi })^{m}\widehat{v}(\xi )\) for some \(v \in l_{0}(\mathbb{Z})\);

       
    5. e.

      \(\sum _{k\in \mathbb{Z}}u(\gamma +\mathsf{d}k)( \tfrac{\gamma }{\mathsf{d}} + k)^{j} =\sum _{k\in \mathbb{Z}}u(\mathsf{d}k)k^{j}\) for all \(0\leqslant j <m\) and \(0\leqslant \gamma <\mathsf{d}\).

       
     
  45. 1.45.
    Show that \((\{\tilde{a};\tilde{b}_{1},\ldots,\tilde{b}_{s}\},\{a;b_{1},\ldots,b_{s}\})_{\varTheta }\) is a dual d-framelet filter bank, that is, it has the following perfect reconstruction property:
    $$\displaystyle{\varTheta ^{\star } {\ast} v = \frac{1} {\mathsf{d}}\mathcal{S}_{a,\mathsf{d}}(\varTheta ^{\star } {\ast}\mathcal{T}_{\tilde{ a},\mathsf{d}}v) + \frac{1} {\mathsf{d}}\sum _{\ell=1}^{s}\mathcal{S}_{ b_{\ell},\mathsf{d}}\mathcal{T}_{\tilde{b}_{\ell},\mathsf{d}}v,\qquad \forall \;v \in l(\mathbb{Z}),}$$
    if and only if for all γ = 0, , d − 1 and for all \(\xi \in \mathbb{R}\),
    $$\displaystyle{\widehat{\varTheta }(\mathsf{d}\xi )\widehat{\tilde{a}}(\xi )\overline{\widehat{a}(\xi +\tfrac{2\pi \gamma } {\mathsf{d}} )} +\widehat{\tilde{ b}_{1}}(\xi )\overline{\widehat{b_{1}}(\xi +\tfrac{2\pi \gamma } {\mathsf{d}} )} + \cdots +\widehat{\tilde{ b}_{s}}(\xi )\overline{\widehat{b_{s}}(\xi +\tfrac{2\pi \gamma } {\mathsf{d}} )} =\boldsymbol{\delta } (\gamma )\widehat{\varTheta }(\xi ).}$$
     
  46. 1.46.

    Suppose that {a; b 1, , b s } Θ is a tight d-framelet filter bank. Prove (i) \(\widehat{\varTheta }(\xi )\geqslant 0\) for all \(\xi \in \mathbb{R}\); (ii) All the high-pass filters b 1, , b s have m vanishing moments if and only if \(\widehat{\varTheta }(\mathsf{d}\xi ) -\widehat{\varTheta } (\xi )\vert \widehat{a}(\xi )\vert ^{2} =\mathbb{ O}(\vert \xi \vert ^{2m})\) as ξ → 0.

     

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Bin Han
    • 1
  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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