## 1 Introduction

We denote by ℤ and ℝ the set of integers and real numbers, respectively. Let ${L}_{p}\left(\mathbb{R}\right)$ stand for the classical Lebesgue space

${L}_{p}\left(\mathbb{R}\right):=\left\{f,{\int }_{\mathbb{R}}|f\left(t\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dt<+\mathrm{\infty }\right\}$

with the norm ${\parallel f\parallel }_{p}={\left({\int }_{\mathbb{R}}{|f\left(t\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{p}}$ and ${L}_{\mathrm{\infty }}\left(\mathbb{R}\right)$ consisting of all Lebesgue measurable and bounded functions on ℝ. Similarly, the discrete space ${\ell }_{p}\left(\mathbb{Z}\right):=\left\{\left\{{a}_{n}\right\},{\sum }_{n}{|{a}_{n}|}^{p}<+\mathrm{\infty },n\in \mathbb{Z}\right\}$ with ${\parallel \left\{{a}_{n}\right\}\parallel }_{p}={\left({\sum }_{n}{|{a}_{n}|}^{p}\right)}^{\frac{1}{p}}$. As usual, given $f\in {L}_{1}\left(\mathbb{R}\right)\cap {L}_{2}\left(\mathbb{R}\right)$, its Fourier transform is defined by

$\stackrel{ˆ}{f}\left(\omega \right):={\int }_{\mathbb{R}}f\left(x\right){e}^{-ix\omega }\phantom{\rule{0.2em}{0ex}}dx$

on ℝ. The Fourier transform of a function in ${L}_{2}\left(\mathbb{R}\right)$ is understood as the unitary extension. We write $h=f\ast g$ for the convolution $h\left(x\right)={\int }_{\mathbb{R}}f\left(x-t\right)g\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$, defined for any pair of functions f and g such that the integral exists almost everywhere. Clearly, $\stackrel{ˆ}{h}\left(\omega \right)=\stackrel{ˆ}{f}\left(\omega \right)\stackrel{ˆ}{g}\left(\omega \right)$ in the frequency domain, when all the Fourier transforms exist in that formula. Given $g\in {L}_{2}\left(\mathbb{R}\right)$, $\left\{g\left(x-k\right),k\in \mathbb{Z}\right\}$ is called a Riesz basis of its linearly generating space, if for each $\left\{{\lambda }_{k}\right\}\in {\ell }_{2}\left(\mathbb{Z}\right)$ there exist two positive constants A and B such that

$A\sum _{k\in \mathbb{Z}}{\lambda }_{k}^{2}\le {\parallel \sum _{k\in \mathbb{Z}}{\lambda }_{k}g\left(x-k\right)\parallel }_{2}^{2}\le B\sum _{k\in \mathbb{Z}}{\lambda }_{k}^{2}.$
(1.1)

The numbers A, B are called lower Riesz bound and upper Riesz bound, respectively.

Multiresolution analysis provides a classical method to construct wavelets.

Definition 1 A multiresolution analysis of ${L}_{2}\left(\mathbb{R}\right)$ means a sequence of closed linear subspaces ${V}_{j}$ of ${L}_{2}\left(\mathbb{R}\right)$ which satisfies

1. (i)

${V}_{j}\subset {V}_{j+1}$, $j\in \mathbb{Z}$,

2. (ii)

$f\left(x\right)\in {V}_{j}$ if and only if $f\left(2x\right)\in {V}_{j+1}$,

3. (iii)

$\overline{{\bigcup }_{j\in \mathbb{Z}}{V}_{j}}={L}_{2}\left(\mathbb{R}\right)$ and ${\bigcap }_{j\in \mathbb{Z}}{V}_{j}=\left\{0\right\}$,

4. (iv)

there exists a function $\varphi \in {L}_{2}\left(\mathbb{R}\right)$ such that $\left\{\varphi \left(x-k\right),k\in \mathbb{Z}\right\}$ forms a Riesz basis of ${V}_{0}$.

The function ϕ in Definition 1 is said to be a scaling function, if it satisfies

$\varphi \left(x\right)=\sum _{k}{a}_{k}\varphi \left(2x-k\right)$
(1.2)

for some sequence $\left\{{a}_{k}\right\}\in {\ell }_{2}\left(\mathbb{Z}\right)$. Define the Fourier series $\stackrel{ˆ}{c}$ of a sequence $\left\{{c}_{k}\right\}\in {\ell }_{2}\left(\mathbb{Z}\right)$ by

$\stackrel{ˆ}{c}\left(\xi \right):=\sum _{k\in \mathbb{Z}}{c}_{k}{e}^{-ik\xi },\phantom{\rule{1em}{0ex}}\xi \in \mathbb{R}.$

Then the refinement equation (1.2) becomes

$\stackrel{ˆ}{\varphi }\left(\xi \right)=\stackrel{ˆ}{a}\left(\xi /2\right)\stackrel{ˆ}{\varphi }\left(\xi /2\right),\phantom{\rule{1em}{0ex}}\xi \in \mathbb{R}.$

The function $\stackrel{ˆ}{a}$ is called the refinement mask of ϕ. The pseudo-spline of Type I was first introduced in [1] to construct tight framelets. The pseudo-spline of Type II was first studied by Dong and Shen in [2]. There have been many developments in the theory of pseudo-splines over the past ten years [3, 4]. Its applications in image denoising and image in-painting are also very extensive [5, 6]. The pseudo-spline is defined by its refinement mask. The refinement mask of a pseudo-spline of Type I with order $\left(m,\ell \right)$ is given by

$|{}_{1}\stackrel{ˆ}{a}\left(\xi \right){|}^{2}:=|{}_{1}{\stackrel{ˆ}{a}}_{m,\ell }\left(\xi \right){|}^{2}:={cos}^{2m}\left(\xi /2\right)\sum _{j=0}^{\ell }\left(\begin{array}{c}m+\ell \\ j\end{array}\right){sin}^{2j}\left(\xi /2\right){cos}^{2\left(\ell -j\right)}\left(\xi /2\right)$
(1.3)

and the refinement of a pseudo-spline of Type II with order $\left(m,\ell \right)$ is given by

${}_{2}\stackrel{ˆ}{a}\left(\xi \right):={}_{2}{\stackrel{ˆ}{a}}_{m,\ell }\left(\xi \right):={cos}^{2m}\left(\xi /2\right)\sum _{j=0}^{\ell }\left(\begin{array}{c}m+\ell \\ j\end{array}\right){sin}^{2j}\left(\xi /2\right){cos}^{2\left(\ell -j\right)}\left(\xi /2\right).$
(1.4)

The mask of Type I is obtained by taking the square root of the mask of Type II using the Fejér-Riesz lemma [7], i.e. ${}_{2}\stackrel{ˆ}{a}\left(\xi \right)={|{}_{1}\stackrel{ˆ}{a}\left(\xi \right)|}^{2}$. The corresponding pseudo-spline can be defined in terms of their Fourier transform, i.e.

${}_{k}\stackrel{ˆ}{\varphi }_{m,\ell }\left(\xi \right):=\prod _{j=1}^{\mathrm{\infty }}{}_{k}{\stackrel{ˆ}{a}}_{m,\ell }\left({2}^{-j}\xi \right),\phantom{\rule{1em}{0ex}}k=1,2.$

In order to smoothen the pseudo-spline, one can use the convolution method. Take the smoothed pseudo-spline

where ${\chi }_{\left[-\frac{1}{2},\frac{1}{2}\right]}$ denotes the characteristic function of interval $\left[-\frac{1}{2},\frac{1}{2}\right]$ and $n⩾m$. This is equivalent to

${\stackrel{ˆ}{\varphi }}_{n,m,\ell }\left(\xi \right)={\stackrel{ˆ}{\varphi }}_{m,\ell }\left(\xi \right){\left(\frac{sin\left(\xi /2\right)}{\xi /2}\right)}^{n-m}.$

Thus the symbol of ${\varphi }_{n,m,\ell }$ becomes

${\stackrel{ˆ}{a}}_{n,m,\ell }\left(\xi \right)={\stackrel{ˆ}{\varphi }}_{n,m,\ell }\left(2\xi \right)/{\stackrel{ˆ}{\varphi }}_{n,m,\ell }\left(\xi \right)={\stackrel{ˆ}{a}}_{m,\ell }\left(\xi \right){\left(cos\left(\xi /2\right)\right)}^{n-m}.$

Therefore, we define the smoothed pseudo-spline by its refinement mask for Type I:

$|{}_{1}{\stackrel{ˆ}{a}}_{n,m,\ell }{|}^{2}:={cos}^{2n}\left(\xi /2\right)\sum _{j=0}^{\ell }\left(\begin{array}{c}m+\ell \\ j\end{array}\right){sin}^{2j}\left(\xi /2\right){cos}^{2\left(\ell -j\right)}\left(\xi /2\right)$
(1.5)

and for Type II:

${}_{2}\stackrel{ˆ}{\varphi }_{r,m,\ell }:={cos}^{r}\left(\xi /2\right)\sum _{j=0}^{\ell }\left(\begin{array}{c}m+\ell \\ j\end{array}\right){sin}^{2j}\left(\xi /2\right){cos}^{2\left(\ell -j\right)}\left(\xi /2\right),$

where $r⩾2m$. When $r=2m$, it is the pseudo-spline. When $r\ne 2m$, it can be considered as an extension of pseudo-spline. Define the translated form of the Type II by

${}_{T}\stackrel{ˆ}{\varphi }_{r,m,\ell }\left(\xi \right):={{e}^{-ir\frac{\xi }{2}}}_{2}{\stackrel{ˆ}{\varphi }}_{r,m,\ell }\left(\xi \right).$

Then we get the differential relation

${}_{T}\varphi _{r+1,m,\ell }^{\prime }\left(x\right)={}_{T}{\varphi }_{r,m,\ell }\left(x\right){-}_{T}{\varphi }_{r,m,\ell }\left(x-1\right).$
(1.6)

This inherits the property of a B-spline.

Remark 1 One may think that smoothing the pseudo-splines by convolving them with B-splines seems unnecessary since one can simply increase m of the original pseudo-splines. However, by increasing m, we cannot get the differential relation (1.6), which is important for the construction of divergence free wavelets and curl free wavelets in the analysis of incompressible turbulent flows [8, 9].

Remark 2 Similar to the definition of (1.5), we can define a smoothed dual pseudo-spline by its refinement mask,

${\stackrel{ˆ}{b}}_{n,m,\ell }\left(\xi \right)={e}^{i\xi /2}{cos}^{2n+1}\left(\xi /2\right)\sum _{j=0}^{\ell }\left(\begin{array}{c}m+1/2+\ell \\ j\end{array}\right){sin}^{2j}\left(\xi /2\right){cos}^{2\left(\ell -j\right)}\left(\xi /2\right),$
(1.7)

as an extension of dual pseudo-splines in [3] and get the corresponding wavelets.

Remark 3 In addition, one can assume $n\in \mathbb{R}$ in (1.5) and (1.7), as a generalization of fractional splines in [10].

## 2 Some lemmas

This section gives some lemmas that will be used to prove several results of this paper. We start with some results from [2].

Lemma 1 [2]

For given nonnegative integers m, j, ,

This lemma will be used in Section 4 in order to prove the Riesz basis property of wavelets. Define ${P}_{m,\ell }\left(y\right):={\sum }_{j=0}^{\ell }\left(\begin{array}{c}m+\ell \\ j\end{array}\right){y}^{j}{\left(1-y\right)}^{\ell -j}$, ${R}_{m,\ell }\left(y\right):={\left(1-y\right)}^{m}{P}_{m,\ell }\left(y\right)$ and ${R}_{r,m,\ell }={\left(1-y\right)}^{\frac{r}{2}}{P}_{m,\ell }\left(y\right)$ where $y={sin}^{2}\left(\xi /2\right)$, r, m, are nonnegative integers and $r⩾2m$. Then one can find that

${R}_{m,\ell }\left({sin}^{2}\left(\xi /2\right)\right){=}_{2}{\stackrel{ˆ}{a}}_{m,\ell }\left(\xi \right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{R}_{r,m,\ell }\left({sin}^{2}\left(\xi /2\right)\right){=}_{2}{\stackrel{ˆ}{a}}_{r,m,\ell }\left(\xi \right)$

and the following lemma holds.

Lemma 2 [2]

For nonnegative integers m and with $\ell ⩽m-1$, let ${P}_{m,\ell }$ and ${R}_{m,\ell }$ be the polynomials defined above. Then

1. (1)

${P}_{m,\ell }\left(y\right)={\sum }_{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right){y}^{j}$;

2. (2)

${R}_{m,\ell }^{\prime }\left(y\right)=-\left(m+\ell \right)\left(\begin{array}{c}m+\ell -1\\ \ell \end{array}\right){y}^{\ell }{\left(1-y\right)}^{m-1}$.

With the two lemmas in hand, the basic property of the polynomial ${R}_{r,m,\ell }$, which will be used in Section 4, is given.

Lemma 3 For nonnegative integers r, m and ,

1. (1)

define $Q\left(y\right):={R}_{r,m,\ell }\left(y\right)+{R}_{r,m,\ell }\left(1-y\right)$; then

$\underset{y\in \left[0,1\right]}{min}Q\left(y\right)=Q\left(\frac{1}{2}\right)={2}^{1-\frac{r}{2}-\ell }\sum _{j=0}^{\ell }\left(\begin{array}{c}m+\ell \\ j\end{array}\right);$
2. (2)

define $S\left(y\right):={R}_{r,m,\ell }^{2}\left(y\right)+{R}_{r,m,\ell }^{2}\left(1-y\right)$; then

$\underset{y\in \left[0,1\right]}{min}S\left(y\right)=S\left(\frac{1}{2}\right)={2}^{1-r-2\ell }{\left(\sum _{j=0}^{\ell }\left(\begin{array}{c}m+\ell \\ j\end{array}\right)\right)}^{2}.$

Proof Since ${R}_{r,m,\ell }\left(y\right)={\left(1-y\right)}^{\frac{r}{2}-m}{R}_{m,\ell }\left(y\right)$, its derivative is

${R}_{r,m,\ell }^{\prime }\left(y\right)=-\left(\frac{r}{2}-m\right){\left(1-y\right)}^{\frac{r}{2}-m-1}{R}_{m,\ell }\left(y\right)+{\left(1-y\right)}^{\frac{r}{2}-m}{R}_{m,\ell }^{\prime }\left(y\right).$

So the derivative is ${Q}^{\prime }\left(y\right)={R}_{r,m,\ell }^{\prime }\left(y\right)+{R}_{r,m,\ell }^{\prime }\left(1-y\right)=I+\mathit{II}$, where

$I=\left(\frac{r}{2}-m\right){y}^{\frac{r}{2}-m-1}{R}_{m,\ell }\left(1-y\right)-\left(\frac{r}{2}-m\right){\left(1-y\right)}^{\frac{r}{2}-m-1}{R}_{m,\ell }\left(y\right)$

and

$\mathit{II}={\left(1-y\right)}^{\frac{r}{2}-m}{R}_{m,\ell }^{\prime }\left(y\right)-{y}^{\frac{r}{2}-m}{R}_{m,\ell }^{\prime }\left(1-y\right).$

Now, we compute them, respectively. For I, by using (1) of Lemma 2, one has

$I=\left(\frac{r}{2}-m\right)\sum _{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right)\left[{y}^{\frac{r}{2}-1}{\left(1-y\right)}^{j}-{y}^{j}{\left(1-y\right)}^{\frac{r}{2}-1}\right].$

For II, by using (2) of Lemma 2, one has

$\mathit{II}=\left(m+\ell \right)\left(\begin{array}{c}m+\ell -1\\ \ell \end{array}\right)\left[{y}^{\frac{r}{2}-1}{\left(1-y\right)}^{\ell }-{\left(1-y\right)}^{\frac{r}{2}-1}{y}^{\ell }\right].$

For $j=0,\dots ,\ell$, since ${y}^{\frac{r}{2}-1}{\left(1-y\right)}^{j}⩽{\left(1-y\right)}^{\frac{r}{2}-1}{y}^{j}$ for all $y\in \left[0,\frac{1}{2}\right]$ and ${y}^{\frac{r}{2}-1}{\left(1-y\right)}^{j}⩾{\left(1-y\right)}^{\frac{r}{2}-1}{y}^{j}$ for all $y\in \left[\frac{1}{2},1\right]$, one has

${Q}^{\prime }\left(y\right)=I+\mathit{II}\left\{\begin{array}{ll}⩽0,& y\in \left[0,\frac{1}{2}\right];\\ ⩾0,& y\in \left[\frac{1}{2},1\right].\end{array}$

This means $Q\left(y\right)$ reaches its minimum value at the point $y=1/2$. Furthermore,

$Q\left(1/2\right)=2{R}_{r,m,\ell }\left(1/2\right)={2}^{1-\frac{r}{2}}{P}_{m,\ell }\left(1/2\right)={2}^{1-\frac{r}{2}-\ell }\sum _{j=0}^{\ell }\left(\begin{array}{c}m+\ell \\ j\end{array}\right).$

This completes the proof of (1). For (2) of this lemma, since

$S\left(y\right)={R}_{r,m,\ell }^{2}\left(y\right)+{R}_{r,m,\ell }^{2}\left(1-y\right)={\left(1-y\right)}^{r-2m}{R}_{m,\ell }^{2}\left(y\right)+{y}^{r-2m}{R}_{m,\ell }^{2}\left(1-y\right),$

we have ${S}^{\prime }\left(y\right)=\mathit{III}+\mathit{IV}$, where

$\mathit{III}=\left(r-2m\right)\left[{y}^{r-2m-1}{R}_{m,\ell }^{2}\left(1-y\right)-{\left(1-y\right)}^{r-2m-1}{R}_{m,\ell }^{2}\left(y\right)\right]$

and

$\mathit{IV}=2{\left(1-y\right)}^{r-2m}{R}_{m,\ell }\left(y\right){R}_{m,\ell }^{\prime }\left(y\right)-2{y}^{r-2m}{R}_{m,\ell }\left(1-y\right){R}_{m,\ell }^{\prime }\left(1-y\right).$

For III, by (1) of Lemma 2, we have

$\begin{array}{rcl}\mathit{III}& =& \left(r-2m\right)\left[{y}^{r-1}{P}_{m,\ell }^{2}\left(1-y\right)-{\left(1-y\right)}^{r-1}{P}_{m,\ell }^{2}\left(y\right)\right]\\ =& \left(r-2m\right)\left({\left(\sum _{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right){y}^{\frac{1}{2}\left(r-1\right)}{\left(1-y\right)}^{j}\right)}^{2}\\ -{\left(\sum _{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right){\left(1-y\right)}^{\frac{1}{2}\left(r-1\right)}{y}^{j}\right)}^{2}\right)\\ =& \left(r-2m\right)\left(\sum _{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right)\left({y}^{\frac{1}{2}\left(r-1\right)}{\left(1-y\right)}^{j}+{\left(1-y\right)}^{\frac{1}{2}\left(r-1\right)}{y}^{j}\right)\right)\\ ×\left(\sum _{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right)\left({y}^{\frac{1}{2}\left(r-1\right)}{\left(1-y\right)}^{j}-{\left(1-y\right)}^{\frac{1}{2}\left(r-1\right)}{y}^{j}\right)\right).\end{array}$

For IV, by (2) of Lemma 2, we have

$\begin{array}{rcl}\frac{\mathit{IV}}{2}& =& {\left(1-y\right)}^{r-2m}{P}_{m,\ell }\left(y\right){R}_{m,\ell }^{\prime }\left(y\right)-{y}^{r-2m}{P}_{m,\ell }\left(1-y\right){R}_{m,\ell }^{\prime }\left(1-y\right)\\ =& {R}_{m,\ell }^{\prime }\left(y\right)\sum _{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right){y}^{j}{\left(1-y\right)}^{r-2m}-{R}_{m,\ell }^{\prime }\left(1-y\right)\sum _{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right){\left(1-y\right)}^{j}{y}^{r-2m}\\ =& \left(m+\ell \right)\left(\begin{array}{c}m-1+j\\ j\end{array}\right)\left(\sum _{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right)\left({y}^{r-1}{\left(1-y\right)}^{\ell +j}-{y}^{\ell +j}{\left(1-y\right)}^{r-1}\right)\right).\end{array}$

Since

${y}^{\frac{1}{2}\left(r-1\right)}{\left(1-y\right)}^{j}-{\left(1-y\right)}^{\frac{1}{2}\left(r-1\right)}{y}^{j}\left\{\begin{array}{ll}⩽0,& y\in \left[0,\frac{1}{2}\right];\\ ⩾0,& y\in \left[\frac{1}{2},1\right],\end{array}$

and for every j,

${y}^{r-1}{\left(1-y\right)}^{\ell +j}-{y}^{\ell +j}{\left(1-y\right)}^{r-1}\left\{\begin{array}{ll}⩽0,& y\in \left[0,\frac{1}{2}\right];\\ ⩾0,& y\in \left[\frac{1}{2},1\right],\end{array}$

we have

${S}^{\prime }\left(y\right)=\mathit{III}+\mathit{IV}\left\{\begin{array}{ll}⩽0,& y\in \left[0,\frac{1}{2}\right];\\ ⩾0,& y\in \left[\frac{1}{2},1\right].\end{array}$

This means $S\left(y\right)$ reaches its minimum at point $y=1/2$. Furthermore, we have

$S\left(1/2\right)=2{\left({2}^{-\frac{r}{2}}{P}_{m,\ell }\left(y\right)\right)}^{2}={2}^{1-r-2\ell }{\left(\sum _{j=0}^{\ell }\left(\begin{array}{c}m-1+j\\ j\end{array}\right)\right)}^{2}.$

This completes the lemma. □

## 3 Regularity and stability of scaling function

In this section, we discuss the regularity and stability of a scaling function generated by the refinement mask of a smoothed pseudo-spline. Let

$\stackrel{ˆ}{\varphi }\left(\xi \right):=\prod _{j=1}^{\mathrm{\infty }}\stackrel{ˆ}{a}\left({2}^{-j}\xi \right).$

Then the decay of $|\stackrel{ˆ}{\varphi }|$ can be characterized by $|\stackrel{ˆ}{a}|$ as stated in the following theorem.

Theorem 1 [2]

Let $\stackrel{ˆ}{a}\left(\xi \right)$ be a refinement mask of the refinable function ϕ of the form

$|\stackrel{ˆ}{a}\left(\xi \right)|={cos}^{n}\left(\xi /2\right)|\mathcal{L}\left(\xi \right)|,\phantom{\rule{1em}{0ex}}\xi \in \left[-\pi ,\pi \right].$

Suppose that

Then $|\stackrel{ˆ}{\varphi }\left(\xi \right)|⩽C{\left(1+|\xi |\right)}^{-n+\kappa }$, with $\kappa =log\left(\mathcal{L}\left(|\frac{2\pi }{3}\right)|\right)/log2$, and this decay is optimal.

In order to use this lemma, one needs to consider the polynomial corresponding to $\mathcal{L}\left(\xi \right)$. In fact, Dong and Shen give an important proposition to estimate it in the following proposition.

Proposition 1 [2]

Let ${P}_{m,\ell }\left(y\right)$ be defined as in Section  2, where m, are nonnegative integers with $\ell ⩽m-1$. Then

Combing Theorem 1 and Proposition 1, we have the following theorem, which characterizes the regularity of a smoothed pseudo-spline.

Theorem 2 Let 2ϕ be the smoothed pseudo-spline of Type II with order r, m, , then

$|{}_{2}\stackrel{ˆ}{\varphi }\left(\xi \right)|⩽C{\left(1+|\xi |\right)}^{-r+\kappa },$

where $\kappa =log\left({P}_{m,\ell }\left(\frac{3}{4}\right)\right)/log2$. Consequently, ${}_{2}\varphi \in {C}^{{\alpha }_{2}-ϵ}$ where ${\alpha }_{2}=r-\kappa -1$. Furthermore, let 1ϕ be the smoothed pseudo-spline of Type I with order n, m, . Then

$|{}_{1}\stackrel{ˆ}{\varphi }\left(\xi \right)|⩽C{\left(1+|\xi |\right)}^{-n+\frac{\kappa }{2}}.$

Consequently, ${}_{1}\varphi \in {C}^{{\alpha }_{1}-ϵ}$ with ${\alpha }_{1}=n-\frac{\kappa }{2}-1$.

Proof Notice that $|\mathcal{L}\left(\xi \right)|$ in Theorem 1 is exactly ${P}_{m,\ell }\left({sin}^{2}\left(\frac{\xi }{2}\right)\right)$ and $4y\left(1-y\right)={sin}^{2}\left(\xi \right)$; one can easily prove this theorem by Theorem 1 and Proposition 1. □

This theorem shows ${}_{k}\varphi \in {L}_{2}\left(\mathbb{R}\right)$ for $k=1,2$. Since $r⩾2m$, the regularity of ϕ is better than a pseudo-spline but the support is longer. For $r=6$, $m=2$, $\ell =1$ the smoothed pseudo-spline ${}_{2}\varphi _{r,m,\ell }$ is shown in Figure 1.

Now, we consider the stability of the smoothed pseudo-spline. When ϕ is compactly supported in ${L}_{2}\left(\mathbb{R}\right)$, it was shown by Jia and Micchelli [11] that the upper Riesz bound in (1.1) always exists. Furthermore, they assert that the existence of a lower Riesz bound is equivalent to

(3.1)

where 0 denotes the zero sequence in ${\ell }_{2}\left(\mathbb{Z}\right)$. Since a smoothed pseudo-spline is compactly supported and belongs to ${L}_{2}\left(\mathbb{R}\right)$ for $k=1,2$, the stability is equivalent to (3.1).

Theorem 3 Smoothed pseudo-splines are stable.

Proof By the definition of refinement mask, for each fixed $\frac{r}{2}⩾m⩾1$ and for any $0⩽\ell ⩽m-1$, ${cos}^{2m}\left(\xi /2\right)⩽{}_{2}{\stackrel{ˆ}{a}}_{r,m,\ell }\left(\xi \right)$ holds for all $\xi \in \mathbb{R}$. Therefore, we have

$|{\stackrel{ˆ}{B}}_{r}\left(\xi \right)|⩽|{\stackrel{ˆ}{\varphi }}_{r,m,\ell }\left(\xi \right)|,$

where ${B}_{r}$ stands for the B-spline with order r. Since ${B}_{r}$ is stable, the vector ${\left({\stackrel{ˆ}{B}}_{r}\left(\xi +2k\pi \right)\right)}_{k\in \mathbb{Z}}\ne \mathbf{0}$. Hence ${\left({\stackrel{ˆ}{\varphi }}_{r,m,\ell }\left(\xi +2k\pi \right)\right)}_{k\in \mathbb{Z}}\ne \mathbf{0}$.

For a smoothed pseudo-spline of Type I, since ${}_{2}\stackrel{ˆ}{a}_{2n,m,\ell }\left(\xi \right)={|{}_{1}{\stackrel{ˆ}{a}}_{n,m,\ell }|}^{2}={}_{1}{\stackrel{ˆ}{a}}_{n,m,\ell }\left(\xi \right){\cdot }_{1}{\stackrel{ˆ}{a}}_{n,m,\ell }\left(-\xi \right)$, one has

${}_{2}\stackrel{ˆ}{\varphi }_{2n,m,\ell }\left(\xi \right)={}_{1}{\stackrel{ˆ}{\varphi }}_{n,m,\ell }\left(\xi \right)\cdot {}_{1}{\stackrel{ˆ}{\varphi }}_{n,m,\ell }\left(-\xi \right).$

Therefore, the set of zeros of ${}_{1}\stackrel{ˆ}{\varphi }_{n,m,\ell }\left(\xi \right)$ is contained in that of ${}_{2}\stackrel{ˆ}{\varphi }_{2n,m,\ell }\left(\xi \right)$ and this guarantees the stability of ${}_{1}\varphi \left(\xi \right)$. □

This theorem shows the stability of a smoothed pseudo-spline. From the definition of a Riesz basis, one can find that the translate of a smoothed pseudo-spline is also linearly independent.

## 4 Riesz wavelets

Since all smoothed pseudo-splines are compactly supported, refinable, stable in ${L}_{2}\left(\mathbb{R}\right)$, the sequence of spaces ${\left({V}_{n}\right)}_{n\in \mathbb{Z}}$ defined via Definition 1 forms an MRA. The corresponding wavelets can be constructed by the classical method. Define

and $X\left(\psi \right):=\left\{{\psi }_{n,k}={2}^{n/2}\psi \left({2}^{n}-k\right),n,k\in \mathbb{Z}\right\}$. Then $X\left(\psi \right)$ is a Riesz basis. To prove this, the following theorem is needed.

Theorem 4 [2]

Let $\stackrel{ˆ}{a}\left(\xi \right)$ be a finitely supported refinement mask of a refinable function $\varphi \in {L}_{2}\left(\mathbb{R}\right)$ with $\stackrel{ˆ}{a}\left(0\right)=1$ and $\stackrel{ˆ}{a}\left(\pi \right)=0$, such that $\stackrel{ˆ}{a}$ can be factorized into the form

$|\stackrel{ˆ}{a}\left(\xi \right)|={cos}^{n}\left(\xi /2\right)|\mathcal{L}\left(\xi \right)|,\phantom{\rule{1em}{0ex}}\xi \in \left[-\pi ,\pi \right],$

whereis the Fourier series of a finitely supported sequence with $\mathcal{L}\left(\pi \right)\ne 0$. Suppose that

$|\stackrel{ˆ}{a}\left(\xi \right){|}^{2}+|\stackrel{ˆ}{a}\left(\xi +\pi \right){|}^{2}\ne 0,\phantom{\rule{1em}{0ex}}\xi \in \left[-\pi ,\pi \right].$

Define $\stackrel{ˆ}{\psi }\left(2\xi \right):={e}^{-i\xi }\overline{\stackrel{ˆ}{a}\left(\xi +\pi \right)}\stackrel{ˆ}{\varphi }\left(\xi \right)$ and $\stackrel{˜}{\mathcal{L}}:=\frac{\mathcal{L}\left(\xi \right)}{{|\stackrel{ˆ}{a}\left(\xi \right)|}^{2}+{|\stackrel{ˆ}{a}\left(\xi +\pi \right)|}^{2}}$. Assume that ${\parallel \mathcal{L}\left(\xi \right)\parallel }_{{L}_{\mathrm{\infty }}\left(\mathbb{R}\right)}<{2}^{n-1}$ and ${\parallel \stackrel{˜}{\mathcal{L}}\left(\xi \right)\parallel }_{{L}_{\mathrm{\infty }}\left(\mathbb{R}\right)}<{2}^{n-1}$. Then $X\left(\psi \right)$ is a Riesz basis for ${L}_{2}\left(\mathbb{R}\right)$.

From the above theorem, the key step is to estimate the upper Riesz bound of $|\mathcal{L}\left(\xi \right)|$ and $|\stackrel{˜}{\mathcal{L}}\left(\xi \right)|$. Notice that

$|{}_{1}{\stackrel{ˆ}{a}}_{n,m,\ell }\left(\xi \right){|}^{2}={}_{2}{\stackrel{ˆ}{a}}_{2n,m,\ell }\left(\xi \right)={cos}^{2n}\left(\xi /2\right){P}_{m,\ell }\left({sin}^{2}\left(\xi /2\right)\right).$

One has $|{}_{1}\mathcal{L}\left(\xi \right)|={\left({P}_{m,\ell }\left({sin}^{2}\left(\xi /2\right)\right)\right)}^{\frac{1}{2}}$, $|{\mathcal{L}}_{2}\left(\xi \right)|={P}_{m,\ell }\left({sin}^{2}\left(\xi /2\right)\right)$ and

$|{\stackrel{˜}{\mathcal{L}}}_{1}|=\frac{{\left({P}_{m,\ell }\left(y\right)\right)}^{\frac{1}{2}}}{{R}_{2n,m,\ell }\left(y\right)+{R}_{2n,m,\ell }\left(1-y\right)},\phantom{\rule{2em}{0ex}}|{\stackrel{˜}{\mathcal{L}}}_{2}|=\frac{{P}_{m,\ell }\left(y\right)}{{R}_{r,m,\ell }^{2}\left(y\right)+{R}_{r,m,\ell }^{2}\left(1-y\right)}.$

Thus, we have the following theorem.

Theorem 5 Let k ϕ, $k=1,2$ be the smoothed pseudo-spline of Types I and II with order $\left(r,n,m,\ell \right)$. The refinement masks k a are given in (1.3) and (1.4). Define

${}_{k}\stackrel{ˆ}{\psi }\left(2\xi \right):={e}^{-i\xi }{\overline{{}_{k}\stackrel{ˆ}{a}\left(\xi +\pi \right)}}_{k}\stackrel{ˆ}{\varphi }\left(\xi \right).$

Then $X\left(\psi \right)$ forms a Riesz basis for ${L}_{2}\left(\mathbb{R}\right)$.

Proof By (1) of Lemma 3, one obtains

$\begin{array}{rcl}{\parallel {}_{1}\stackrel{˜}{\mathcal{L}}\parallel }_{{L}_{\mathrm{\infty }}\left(\mathbb{R}\right)}& =& \underset{y\in \left[0,1\right]}{sup}\frac{{\left({P}_{m,\ell }\left(y\right)\right)}^{\frac{1}{2}}}{{R}_{2n,m,\ell }\left(y\right)+{R}_{2n,m,\ell }\left(1-y\right)}\\ ⩽& \frac{{\left(\begin{array}{c}m+\ell \\ \ell \end{array}\right)}^{\frac{1}{2}}}{{min}_{y\in \left[0,1\right]}\left({R}_{2n,m,\ell }\left(y\right)+{R}_{2n,m,\ell }\left(1-y\right)\right)}\\ ⩽& \frac{{2}^{n+\ell -1}{\left(\begin{array}{c}m+\ell \\ \ell \end{array}\right)}^{\frac{1}{2}}}{{\sum }_{j=0}^{\ell }\left(\begin{array}{c}m+\ell \\ j\end{array}\right)}.\end{array}$

Applying Lemma 1, one obtains $\parallel {}_{1}{\stackrel{˜}{\mathcal{L}}}_{\mathrm{\infty }}\parallel ⩽{2}^{n-1}<{2}^{n-\frac{1}{2}}$. Similarly, one can get

${\parallel {}_{2}\stackrel{˜}{\mathcal{L}}\parallel }_{{L}_{\mathrm{\infty }}\left(\mathbb{R}\right)}⩽{2}^{r-1}<{2}^{r-\frac{1}{2}}.$

Notice that

Hence, $|{}_{k}\mathcal{L}\left(\xi \right)|⩽|{}_{k}\stackrel{˜}{\mathcal{L}}\left(\xi \right)|$ for $k=1,2$. By using Theorem 4, one gets the desired result. □

By definition, the wavelets are also in ${L}_{2}\left(\mathbb{R}\right)$ and have the same regularity as the scaling function. Still, the support is longer than for pseudo-spline wavelets. For $r=6$, $m=2$, $\ell =1$, the smoothed pseudo-spline ${}_{2}\psi _{6,2,1}$ is shown in Figure 2.