1 Introduction

Wavelet frames in \(L^{2}(\mathbb {R}^{d})\) have been widely investigated by many authors [1–8]. In particular, homogeneous wavelet dual frames in \(L^{2}(\mathbb {R}^{d})\) were first characterized by Han [9], and then studied by Bownik [3]. For homogeneous wavelet dual frames, regularity and vanishing moments have been both required. However, for nonhomogeneous wavelet dual frames in Sobolev space pairs \((H^{s}(\mathbb {R}^{d}), H^{-s}(\mathbb {R}^{d}))\), they can be separated. It makes it easy to construct dual frames (see [10–14] for details). This paper is devoted to characterizing nonhomogeneous wavelet dual frames in Sobolev spaces pairs \((H^{s}(\mathbb {R}^{d}), H^{-s}(\mathbb {R}^{d}))\) via a pair of equations.

Before proceeding, we introduce some notions and notations. We denote by \(\mathbb {Z}\) and \(\mathbb {N}\) the set of integers and the set of positive integers, respectively. Let \(d\in \mathbb {N}\). We denote by \(\mathbb {T}^{d}=[0,1)^{d}\) the d-dimensional torus. For a Lebesgue measurable set E in \(\mathbb {R}^{d}\), we denote by \(\vert E\vert \) its Lebesgue measure and \(\chi_{E}\) the characteristic function of E, respectively. And we write δ for the Dirac sequence, i.e., \(\delta_{0, 0}=1\) and \(\delta _{0, k}=0\) for \(0\neq k\in \mathbb {Z}^{d}\). The Fourier transform of a function \(f\in L^{1}(\mathbb {R}^{d}) \cap L^{2}(\mathbb {R}^{d})\) is defined by

$$\hat{f}(\cdot)= \int_{\mathbb {R}^{d}} f(x)e^{-2\pi i \langle x, \cdot\rangle} \,dx, $$

and extended to \(L^{2}(\mathbb {R}^{d})\) as usual, where \(\langle\cdot, \cdot\rangle\) denotes the Euclidean inner product in \(\mathbb {R}^{d}\).

For \(s\in \mathbb {R}\), we define Sobolev spaces \(H^{s}(\mathbb {R}^{d})\) as the space of all tempered distributions f such that

$$\Vert f\Vert ^{2}_{H^{s}(\mathbb {R}^{d})}= \int_{\mathbb {R}^{d}}\bigl\vert \hat{f}(\xi)\bigr\vert ^{2} \bigl(1+\Vert \xi \Vert ^{2}\bigr)^{s}\,d\xi< \infty, $$

where \(\Vert \cdot \Vert \) denotes the Euclidean norm on \(\mathbb {R}^{d}\). The inner product in \(H^{s}(\mathbb {R}^{d})\) is given by

$$\langle f, g\rangle_{H^{s}(\mathbb {R}^{d})}= \int_{\mathbb {R}^{d}}\hat{f}(\xi) \overline{\hat{g}(\xi)}\bigl(1+\Vert \xi \Vert ^{2}\bigr)^{s}\,d\xi, \quad f,\, g \in H^{s} \bigl(\mathbb {R}^{d}\bigr). $$

Moreover, for each \(g\in H^{-s}(\mathbb {R}^{d})\),

$$\langle f, g\rangle= \int_{\mathbb {R}^{d}}\hat{f}(\xi) \overline{\hat{g}(\xi)}\,d\xi, \quad f\in H^{s}\bigl(\mathbb {R}^{d}\bigr), $$

is a linear continuous functional in \(H^{s}(\mathbb {R}^{d})\). The \(H^{s}(\mathbb {R}^{d})\) and \(H^{-s}(\mathbb {R}^{d})\) form pairs of dual spaces.

For functions \(f, g: \mathbb {R}^{d} \mapsto \mathbb {C}\), define

$$[f, g]_{t}(\cdot)=\sum_{k\in \mathbb {Z}^{d}}f(\cdot+k) \overline{g(\cdot +k)}\bigl(1+\Vert \cdot+k\Vert ^{2} \bigr)^{t}, \quad t\in \mathbb {R}. $$

For convenience, we write

$$f_{j,k}(\cdot)=2^{\frac{jd}{2}}f\bigl(2^{j}\cdot-k\bigr)\quad \mbox{and}\quad f_{j,k}^{s}(\cdot)=2^{j(\frac{d}{2}-s)}f \bigl(2^{j}\cdot-k\bigr) $$

for a distribution f, \(j\in \mathbb {Z}\), \(k\in \mathbb {Z}^{d}\), and \(s\in \mathbb {R}\).

Let \(\mathbb {N}_{0}=\mathbb {N} \cup\{0\}\). Given \(L\in \mathbb {N}\) and \(s\in \mathbb {R}\), let \(\phi, \psi_{1}, \psi_{2}, \ldots, \psi_{L}\in H^{s}(\mathbb {R}^{d})\) and \(\tilde{\phi}, \tilde{\psi}_{1}, \tilde{\psi}_{2}, \ldots, \tilde {\psi}_{L}\in H^{-s}(\mathbb {R}^{d})\), we denote by \(X^{s}(\phi; \psi_{1}, \psi _{2}, \ldots, \psi_{L})\) and \(X^{-s}(\tilde{\phi}; \tilde{\psi}_{1}, \tilde{\psi}_{2}, \ldots, \tilde{\psi}_{L})\) the following two nonhomogeneous wavelet systems in \(H^{s}(\mathbb {R}^{d})\) and \(H^{-s}(\mathbb {R}^{d})\), respectively:

$$ X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L})= \bigl\{ \phi_{0,k}: k\in \mathbb {Z}^{d} \bigr\} \cup \bigl\{ \psi_{l,j,k}^{s}: j\in \mathbb {N}_{0}, k \in \mathbb {Z}^{d}, l=1, 2, \ldots, L \bigr\} $$
(1.1)

and

$$ X^{-s}(\tilde{\phi}; \tilde{\psi}_{1}, \tilde{ \psi}_{2}, \ldots, \tilde{\psi}_{L})= \bigl\{ \tilde{ \phi}_{0,k}: k\in \mathbb {Z}^{d} \bigr\} \cup \bigl\{ \tilde{ \psi}_{l,j,k}^{-s}: j\in \mathbb {N}_{0}, k\in \mathbb {Z}^{d}, l=1, 2, \ldots, L \bigr\} . $$
(1.2)

We say that \(X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L})\) is a nonhomogeneous wavelet frame in \(H^{s}(\mathbb {R}^{d})\) if there exist two positive constants A, B such that

$$\begin{aligned} A\Vert f\Vert _{H^{s}(\mathbb {R}^{d})}^{2}&\leq\sum _{k\in \mathbb {Z}^{d}}\bigl\vert \langle f, \phi_{0, k} \rangle_{H^{s}(\mathbb {R}^{d})}\bigr\vert ^{2}+\sum _{l=1}^{L}\sum_{j=0}^{\infty}\sum_{k\in \mathbb {Z}^{d}}\bigl\vert \bigl\langle f, \psi_{l, j, k}^{s}\bigr\rangle _{H^{s}(\mathbb {R}^{d})}\bigr\vert ^{2} \\ &\leq B\Vert f\Vert _{H^{s}(\mathbb {R}^{d})}^{2}, \quad \forall f\in H^{s}\bigl(\mathbb {R}^{d}\bigr), \end{aligned}$$
(1.3)

where A, B are called frame bounds; it is called a nonhomogeneous wavelet Bessel sequence in \(H^{s}(\mathbb {R}^{d})\) if the right-hand inequality in (1.3) holds, where B is called a Bessel bound. Furthermore, we say that \((X^{s}(\phi; \psi_{1}, \psi _{2}, \ldots, \psi_{L}), X^{-s}(\tilde{\phi}; \tilde{\psi}_{1}, \tilde{\psi}_{2}, \ldots, \tilde{\psi}_{L}))\) is a pair of nonhomogeneous wavelet dual frames in \((H^{s}(\mathbb {R}^{d}), H^{-s}(\mathbb {R}^{d}))\) if \(X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L})\) and \(X^{-s}(\tilde{\phi}; \tilde{\psi}_{1}, \tilde{\psi }_{2}, \ldots, \tilde{\psi}_{L})\) are Bessel sequences in \(H^{s}(\mathbb {R}^{d})\) and \(H^{-s}(\mathbb {R}^{d})\), respectively, and

$$ \langle f, g\rangle=\sum_{k\in \mathbb {Z}^{d}}\langle f, \tilde{\phi }_{0,k} \rangle \langle\phi_{0,k}, g\rangle+\sum _{l=1}^{L}\sum_{j=0}^{\infty}\sum_{k\in \mathbb {Z}^{d}}\bigl\langle f, \tilde{ \psi}_{l,j,k}^{-s} \bigr\rangle \bigl\langle \psi_{l,j,k}^{s}, g\bigr\rangle $$
(1.4)

holds for all \(f\in H^{s}(\mathbb {R}^{d})\) and \(g\in H^{-s}(\mathbb {R}^{d})\).

If \((X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L}), X^{-s}(\tilde {\phi}; \tilde{\psi}_{1}, \tilde{\psi}_{2}, \ldots, \tilde{\psi }_{L}))\) is a pair of dual frames in \((H^{s}(\mathbb {R}^{d}), H^{-s}(\mathbb {R}^{d}))\), then it follows from (1.4) that

$$f=\sum_{k\in \mathbb {Z}^{d}}\langle f, \tilde{\phi}_{0,k} \rangle\phi _{0,k}+\sum_{l=1}^{L} \sum_{j=0}^{\infty}\sum _{k\in \mathbb {Z}^{d}} \bigl\langle f, \tilde{\psi}_{l,j,k}^{-s} \bigr\rangle \psi_{l,j,k}^{s}, \quad f\in H^{s}\bigl(\mathbb {R}^{d}\bigr), $$

and

$$g=\sum_{k\in \mathbb {Z}^{d}}\langle g, \phi_{0,k} \rangle \tilde{\phi}_{0,k} +\sum_{l=1}^{L} \sum_{j=0}^{\infty}\sum _{k\in \mathbb {Z}^{d}} \bigl\langle g, \psi_{l,j,k}^{s} \bigr\rangle \tilde{\psi}_{l,j,k}^{-s},\quad g\in H^{-s}\bigl(\mathbb {R}^{d}\bigr), $$

with the series converging unconditionally in \(H^{s}(\mathbb {R}^{d})\) and \(H^{-s}(\mathbb {R}^{d})\), respectively.

The paper is organized as follows. Section 2 is devoted to some lemmas used latter. Section 3 is devoted to characterizing nonhomogeneous wavelet dual frames in \((H^{s}(\mathbb {R}^{d}), H^{-s}(\mathbb {R}^{d}))\) via a pair of equations.

2 Some lemmas

In this section, we give some auxiliary lemmas which are necessary in proving Theorem 3.1 below.

Definition 2.1

Define a function \(\kappa: \mathbb {Z}^{d}\to \mathbb {Z}\) by

$$\kappa(n)=\sup \bigl\{ j\geq0: 2^{-j}n\in \mathbb {Z}^{d} \bigr\} $$

for \(0\neq n\in \mathbb {Z}^{d}\), and set \(\kappa(0)=+\infty\).

Lemma 2.1

Let \(s\in \mathbb {R}\), \(j\in \mathbb {Z}\), and \(\psi\in H^{-s}(\mathbb {R}^{d})\). Then, for \(f\in H^{s}(\mathbb {R}^{d})\) and \(k\in \mathbb {Z}^{d}\), the kth Fourier coefficient of \([2^{\frac{jd}{2}}\hat{f}(2^{j}\cdot), \hat{\psi }(\cdot)]_{0}(\xi)\) is \(\langle f, \psi_{j, k}\rangle\). In particular,

$$ \bigl[2^{\frac{jd}{2}}\hat{f}\bigl(2^{j}\cdot\bigr), \hat{\psi}(\cdot)\bigr]_{0}(\xi )=\sum_{k\in \mathbb {Z}^{d}} \langle f, \psi_{j, k}\rangle e^{2\pi i\langle k, \xi\rangle} $$
(2.1)

if \(\{\psi_{j, k}: k\in \mathbb {Z}^{d}\}\) is a Bessel sequence in \(H^{-s}(\mathbb {R}^{d})\).

Proof

Since \(f\in H^{s}(\mathbb {R}^{d})\) and \(\psi\in H^{-s}(\mathbb {R}^{d})\), we have \(\hat {f}(2^{j}\cdot)\overline{\hat{\psi}(\cdot)}\in L^{1}(\mathbb {R}^{d})\), and thus

$$\begin{aligned} \int_{\mathbb {T}^{d}}\bigl[2^{\frac{jd}{2}}\hat{f}\bigl(2^{j} \cdot\bigr), \hat{\psi }(\cdot)\bigr]_{0}(\xi)e^{-2\pi i\langle k, \xi\rangle}\,d\xi &=2^{\frac{jd}{2}} \int_{\mathbb {T}^{d}}\sum_{l\in \mathbb {Z}^{d}} \hat{f} \bigl(2^{j}(\xi+l)\bigr)\overline{\hat{\psi}(\xi+l)} e^{-2\pi i\langle k, \xi\rangle} \,d\xi \\ &=2^{\frac{jd}{2}} \int_{\mathbb {R}^{d}}\hat{f}\bigl(2^{j}\xi\bigr) \overline{\hat{ \psi}(\xi)}e^{-2\pi i\langle k, \xi\rangle}\,d\xi \\ &=2^{-\frac{jd}{2}} \int_{\mathbb {R}^{d}}\hat{f}(\xi) \overline{\hat{\psi}\bigl(2^{-j} \xi\bigr)}e^{-2\pi i\langle k, 2^{-j}\xi \rangle}\,d\xi \\ &= \int_{\mathbb {R}^{d}}\hat{f}(\xi) \overline{ \bigl[\psi_{j, k}( \cdot) \bigr]^{\wedge}(\xi)}\,d\xi, \end{aligned}$$

by the Plancherel theorem. So

$$ \int_{\mathbb {T}^{d}}\bigl[2^{\frac{jd}{2}}\hat{f}\bigl(2^{j} \cdot\bigr), \hat{\psi }(\cdot)\bigr]_{0}(\xi)e^{-2\pi i\langle k, \xi\rangle}\,d\xi = \langle f, \psi_{j, k}\rangle. $$
(2.2)

If \(\{\psi_{j, k}: k\in \mathbb {Z}^{d}\}\) is a Bessel sequence in \(H^{-s}(\mathbb {R}^{d})\), then \(\{\langle f, \psi_{j, k}\rangle\}_{k\in \mathbb {Z}^{d}}\in\ell^{2}(\mathbb {Z}^{d})\), and thus (2.1) follows by (2.2). □

By a careful observation of the proof of [13], Proposition 2.1, we have the following.

Lemma 2.2

Let \(s\in \mathbb {R}\), \(\phi, \psi_{1}, \psi_{2}, \ldots, \psi_{L}\in H^{s}(\mathbb {R}^{d})\). Then \(X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L})\) is a Bessel sequence in \(H^{s}(\mathbb {R}^{d})\) with Bessel bound B if and only if

$$ \sum_{k\in \mathbb {Z}^{d}}\bigl\vert \langle g, \phi_{0,k}\rangle\bigr\vert ^{2}+\sum _{l=1}^{L}\sum_{j=0}^{\infty}\sum_{k\in \mathbb {Z}^{d}}\bigl\vert \bigl\langle g, \psi_{l, j, k}^{s} \bigr\rangle \bigr\vert ^{2}\leq B \Vert g\Vert ^{2}_{H^{-s}(\mathbb {R}^{d})} \quad \textit{for } g\in H^{-s} \bigl(\mathbb {R}^{d}\bigr). $$
(2.3)

Lemma 2.3

Let \(s\in \mathbb {R}\), \(\phi, \psi_{1}, \psi_{2}, \ldots, \psi_{L}\in H^{s}(\mathbb {R}^{d})\). Suppose that \(X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L})\) is a Bessel sequence in \(H^{s}(\mathbb {R}^{d})\) with Bessel bound B, then

$$ \bigl\vert \hat{\phi}(\cdot)\bigr\vert ^{2}+ \sum _{l=1}^{L}\sum_{j=0}^{\infty}2^{-2js} \bigl\vert \hat{\psi}_{l}\bigl(2^{-j}\cdot\bigr)\bigr\vert ^{2}\leq B\bigl(1+\Vert \cdot \Vert ^{2} \bigr)^{-s} $$
(2.4)

holds a.e. on \(\mathbb {R}^{d}\).

Proof

Since \(X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L})\) is a Bessel sequence in \(H^{s}(\mathbb {R}^{d})\) with Bessel bound B, by Lemma 2.2, we have

$$ \sum_{k\in \mathbb {Z}^{d}}\bigl\vert \langle g, \phi_{0,k}\rangle\bigr\vert ^{2}+ \sum _{l=1}^{L}\sum_{j=0}^{\infty}\sum_{k\in \mathbb {Z}^{d}} \bigl\vert \bigl\langle g, \psi_{l, j, k}^{s}\bigr\rangle \bigr\vert ^{2}\leq B \Vert g\Vert _{H^{-s}(\mathbb {R}^{d})}^{2} \quad \mbox{for } g\in H^{-s} \bigl(\mathbb {R}^{d}\bigr). $$
(2.5)

By Lemma 2.1 and an argument similar to that of [6], Theorem 1, we get

$$\begin{aligned} &\sum_{k\in \mathbb {Z}^{d}}\bigl\vert \langle g, \phi_{0,k}\rangle\bigr\vert ^{2}+ \sum _{l=1}^{L}\sum_{j=0}^{\infty}\sum_{k\in \mathbb {Z}^{d}} \bigl\vert \bigl\langle g, \psi_{l, j, k}^{s}\bigr\rangle \bigr\vert ^{2} \\ &\quad = \int_{\mathbb {R}^{d}}\hat{\phi}(\xi)\overline{\hat{g}(\xi)}\sum _{k\in \mathbb {Z}^{d}} \hat{g}(\xi+k)\overline{\hat{\phi}(\xi+k)}\,d\xi \\ &\quad\quad{} + \sum_{l=1}^{L}\sum _{j=0}^{\infty}2^{-2js} \int_{\mathbb {R}^{d}} \hat{\psi}_{l}\bigl(2^{-j}\xi \bigr)\overline{\hat{g}(\xi)}\sum_{k\in \mathbb {Z}^{d}} \hat{g}\bigl( \xi+2^{j}k\bigr)\overline{\hat{\psi}_{l} \bigl(2^{-j}\xi+k\bigr)}\,d\xi. \end{aligned}$$

It can be rewritten as

$$\begin{aligned} &\sum_{k\in \mathbb {Z}^{d}}\bigl\vert \langle g, \phi_{0,k}\rangle\bigr\vert ^{2}+ \sum _{l=1}^{L}\sum_{j=0}^{\infty}\sum_{k\in \mathbb {Z}^{d}} \bigl\vert \bigl\langle g, \psi_{l, j, k}^{s}\bigr\rangle \bigr\vert ^{2} \\ &\quad = \int_{\mathbb {R}^{d}}\bigl\vert \hat{g}(\xi)\bigr\vert ^{2} \Biggl(\bigl\vert \hat{\phi}(\xi)\bigr\vert ^{2}+ \sum _{l=1}^{L}\sum_{j=0}^{\infty}2^{-2js} \bigl\vert \hat{\psi}_{l}\bigl(2^{-j}\xi\bigr)\bigr\vert ^{2} \Biggr)\,d\xi \\ &\quad\quad{} + \int_{\mathbb {R}^{d}}\overline{\hat{g}(\xi)}\sum _{0\neq k\in \mathbb {Z}^{d}} \hat{g}(\xi+k) \\ &\quad\quad{}\times \Biggl(\hat{\phi}(\xi) \overline{\hat{ \phi}(\xi+k)}+ \sum_{l=1}^{L}\sum _{j=0}^{\kappa(k)}2^{-2js}\hat{\psi}_{l} \bigl(2^{-j}\xi\bigr) \overline{\hat{\psi}_{l} \bigl(2^{-j}(\xi+k)\bigr)} \Biggr)\,d\xi \end{aligned}$$
(2.6)

by the definition of κ.

Suppose (2.4) does not hold. Then there exists \(E\subset \mathbb {R}^{d}\) with \(\vert E\vert >0\) such that

$$\bigl\vert \hat{\phi}(\cdot)\bigr\vert ^{2}+ \sum _{l=1}^{L}\sum_{j=0}^{\infty}2^{-2js} \bigl\vert \hat{\psi}_{l}\bigl(2^{-j}\cdot\bigr)\bigr\vert ^{2}>B\bigl(1+\Vert \cdot \Vert ^{2} \bigr)^{-s} \quad \mbox{on } E, $$

and thus

$$\bigl\vert \hat{\phi}(\cdot)\bigr\vert ^{2}+ \sum _{l=1}^{L}\sum_{j=0}^{\infty}2^{-2js} \bigl\vert \hat{\psi}_{l}\bigl(2^{-j}\cdot\bigr)\bigr\vert ^{2}>B\bigl(1+\Vert \cdot \Vert ^{2} \bigr)^{-s} $$

on some \(E'=E\cap ( [0, 1)^{d}+k_{0} )\) with \(\vert E'\vert >0\) and \(k_{0} \in \mathbb {Z}^{d}\). Take g such that \(\hat{g}(\cdot)=(1+ \Vert \cdot \Vert ^{2})^{s/2}\chi_{E'}\) in (2.6), then we obtain

$$\sum_{k\in \mathbb {Z}^{d}}\bigl\vert \langle g, \phi_{0,k}\rangle\bigr\vert ^{2}+ \sum _{l=1}^{L}\sum_{j=0}^{\infty}\sum_{k\in \mathbb {Z}^{d}} \bigl\vert \bigl\langle g, \psi_{l, j, k}^{s}\bigr\rangle \bigr\vert ^{2}>B\bigl\vert E'\bigr\vert =B\Vert g\Vert _{H^{-s}(\mathbb {R}^{d})}^{2}, $$

contradicting (2.5). □

3 The characterization of nonhomogeneous wavelet dual frames in Sobolev spaces

This section is devoted to characterizing nonhomogeneous wavelet dual frames in \((H^{s}(\mathbb {R}^{d}), H^{-s}(\mathbb {R}^{d}))\). The following theorem provides us with a characterization via a pair of equations.

Theorem 3.1

Let \(s\in \mathbb {R}\), \(\phi, \psi_{1}, \psi_{2}, \ldots, \psi_{L}\in H^{s}(\mathbb {R}^{d})\) and \(\tilde{\phi}, \tilde{\psi}_{1}, \tilde{\psi}_{2}, \ldots, \tilde{\psi}_{L}\in H^{-s}(\mathbb {R}^{d})\). Define wavelet systems \(X^{s}(\phi ; \psi_{1}, \psi_{2}, \ldots, \psi_{L})\) and \(X^{-s}(\tilde{\phi}; \tilde{\psi}_{1}, \tilde{\psi}_{2}, \ldots, \tilde{\psi}_{L})\) as in (1.1) and (1.2), respectively. Suppose that \(X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L})\) is a Bessel sequence in \(H^{s}(\mathbb {R}^{d})\), and \(X^{-s}(\tilde{\phi}; \tilde{\psi}_{1}, \ldots , \tilde{\psi}_{L})\) is a Bessel sequence in \(H^{-s}(\mathbb {R}^{d})\). Then \((X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L}), X^{-s}(\tilde {\phi}; \tilde{\psi}_{1}, \tilde{\psi}_{2}, \ldots, \tilde{\psi }_{L}))\) is a pair of dual frames in \((H^{s}(\mathbb {R}^{d}), H^{-s}(\mathbb {R}^{d}))\) if and only if, for every \(k\in \mathbb {Z}^{d}\),

$$ \hat{\phi}(\cdot)\overline{\hat{\tilde{\phi}}(\cdot+k)}+ \sum _{l=1}^{L}\sum_{j=0}^{\kappa(k)} \hat{\psi}_{l}\bigl(2^{-j}\cdot\bigr) \overline{\hat{\tilde{ \psi}}_{l}\bigl(2^{-j}(\cdot+k)\bigr)} =\delta_{0,k} \quad \textit{a.e. on } \mathbb {R}^{d}. $$
(3.1)

Proof

By the definition, \((X^{s}(\phi; \psi_{1}, \psi_{2}, \ldots, \psi_{L}), X^{-s}(\tilde{\phi}; \tilde{\psi}_{1}, \tilde{\psi}_{2}, \ldots, \tilde{\psi}_{L}))\) is a pair of dual frames for \((H^{s}(\mathbb {R}^{d}), H^{-s}(\mathbb {R}^{d}))\) if and only if

$$\begin{aligned} &\sum_{k\in \mathbb {Z}^{d}}\langle f, \tilde{ \phi}_{0,k} \rangle \langle \phi_{0,k}, g\rangle+\sum _{l=1}^{L}\sum_{j=0}^{\infty}\sum_{k\in \mathbb {Z}^{d}} \bigl\langle f, \tilde{ \psi}_{l,j,k}^{-s} \bigr\rangle \bigl\langle \psi _{l,j,k}^{ s}, g\bigr\rangle \\ &\quad =\langle f, g\rangle,\quad f\in H^{s}\bigl(\mathbb {R}^{d}\bigr),\, g\in H^{-s}\bigl(\mathbb {R}^{d}\bigr). \end{aligned}$$
(3.2)

By the Plancherel theorem and Lemma 2.1, we deduce that

$$\begin{aligned} &\sum_{k\in \mathbb {Z}^{d}}\bigl\langle f, \tilde{\phi}(\cdot-k) \bigr\rangle \bigl\langle \phi(\cdot-k), g\bigr\rangle +\sum _{l=1}^{L}\sum_{j=0}^{\infty}\sum_{k\in \mathbb {Z}^{d}}\bigl\langle f, \tilde{ \psi}_{l,j,k}^{-s}\bigr\rangle \bigl\langle \psi_{l,j,k}^{s}, g\bigr\rangle \\ &\quad = \int_{\mathbb {T}^{d}} \biggl(\sum_{k\in \mathbb {Z}^{d}}\hat{f}( \xi+k)\overline {\hat{\tilde{\phi}}(\xi+k)} \biggr) \biggl(\sum _{k\in \mathbb {Z}^{d}}\hat{\phi}(\xi+k)\overline{\hat{g}(\xi +k)} \biggr)\,d\xi \\ &\quad\quad{} + \sum_{l=1}^{L}\sum _{j=0}^{\infty}2^{jd} \int_{\mathbb {T}^{d}} \biggl(\sum_{k\in \mathbb {Z}^{d}}\hat{f} \bigl(2^{j}(\xi+k)\bigr) \overline{\hat{\tilde{\psi}}_{l}( \xi+k)} \biggr) \biggl(\sum_{k\in \mathbb {Z}^{d}}\hat{ \psi}_{l}(\xi+k) \overline{\hat{g}\bigl(2^{j}(\xi+k)\bigr)} \biggr)\,d\xi \\ &\quad = \int_{\mathbb {R}^{d}}\sum_{k\in \mathbb {Z}^{d}} \hat{f}(\xi+k) \overline{\hat{\tilde{\phi}}(\xi+k)} \hat{\phi}(\xi)\overline{\hat{g}(\xi)}\,d\xi \\ &\quad\quad{} + \sum_{l=1}^{L}\sum _{j=0}^{\infty}2^{jd} \int_{\mathbb {R}^{d}} \sum_{k\in \mathbb {Z}^{d}}\hat{f} \bigl(2^{j}(\xi+k)\bigr) \overline{\hat{\tilde{\psi}}_{l}( \xi+k)}\hat{\psi}_{l}(\xi) \overline{\hat{g}\bigl(2^{j}\xi \bigr)}\,d\xi \\ &\quad = \int_{\mathbb {R}^{d}}\hat{f}(\xi)\overline{\hat{g}(\xi)} \Biggl( \hat{ \phi}(\xi)\overline{\hat{\tilde{\phi}}(\xi)} +\sum_{l=1}^{L} \sum_{j=0}^{\infty}\hat{\psi}_{l} \bigl(2^{-j}\xi\bigr) \overline{\hat{\tilde{\psi}}_{l} \bigl(2^{-j}\xi\bigr)} \Biggr)\,d\xi \\ &\quad\quad{} + \int_{\mathbb {R}^{d}}\overline{\hat{g}(\xi)} \Biggl(\sum _{0\neq k\in \mathbb {Z}^{d}}\hat{f}(\xi+k)\hat{\phi}(\xi) \overline{\hat{\tilde{ \phi}}(\xi+k)} \\ &\quad\quad{} +\sum_{l=1}^{L}\sum _{j=0}^{\infty}\sum_{0\neq k\in \mathbb {Z}^{d}} \hat{f}\bigl(\xi+2^{j}k\bigr) \hat{\psi}_{l} \bigl(2^{-j}\xi\bigr) \overline{\hat{\tilde{\psi}}_{l} \bigl(2^{-j}\xi+k\bigr)} \Biggr)\,d\xi \\ &\quad = \int_{\mathbb {R}^{d}}\hat{f}(\xi)\overline{\hat{g}(\xi)} \Biggl( \hat{ \phi}(\xi)\overline{\hat{\tilde{\phi}}(\xi)} +\sum_{l=1}^{L} \sum_{j=0}^{\infty}\hat{\psi}_{l} \bigl(2^{-j}\xi\bigr) \overline{\hat{\tilde{\psi}}_{l} \bigl(2^{-j}\xi\bigr)} \Biggr)\,d\xi \\ &\quad\quad{} + \int_{\mathbb {R}^{d}}\overline{\hat{g}(\xi)} \sum _{0\neq k\in \mathbb {Z}^{d}}\hat{f}(\xi+k) \Biggl( \hat{\phi}(\xi)\overline{\hat{ \tilde{\phi}}(\xi+k)} +\sum_{l=1}^{L}\sum _{j=0}^{\kappa(k)} \hat{\psi}_{l} \bigl(2^{-j}\xi\bigr) \overline{\hat{\tilde{\psi}}_{l} \bigl(2^{-j}(\xi+k)\bigr)} \Biggr)\,d\xi. \end{aligned}$$

And thus (3.2) can be rewritten as

$$\begin{aligned} & \int_{\mathbb {R}^{d}}\hat{f}(\xi)\overline{\hat{g}(\xi)} \Biggl( \hat{ \phi}(\xi)\overline{\hat{\tilde{\phi}}(\xi)} +\sum_{l=1}^{L} \sum_{j=0}^{\infty}\hat{\psi}_{l} \bigl(2^{-j}\xi\bigr) \overline{\hat{\tilde{\psi}}_{l} \bigl(2^{-j}\xi\bigr)} \Biggr)\,d\xi \\ &\quad \quad{} + \int_{\mathbb {R}^{d}}\overline{\hat{g}(\xi)} \sum _{0\neq k\in \mathbb {Z}^{d}}\hat{f}(\xi+k) \Biggl( \hat{\phi}(\xi)\overline{\hat{ \tilde{\phi}}(\xi+k)} +\sum_{l=1}^{L}\sum _{j=0}^{\kappa(k)} \hat{\psi}_{l} \bigl(2^{-j}\xi\bigr) \overline{\hat{\tilde{\psi}}_{l} \bigl(2^{-j}(\xi+k)\bigr)} \Biggr)\,d\xi \\ &\quad = \int_{\mathbb {R}^{d}}\hat{f}(\xi)\overline{\hat{g}(\xi)}\,d\xi. \end{aligned}$$
(3.3)

Obviously, (3.1) implies (3.3). To finish the proof, next we prove the converse implication.

Suppose (3.3) holds. By Lemma 2.3 and the Cauchy-Schwarz inequality, the series

$$\hat{\phi}(\cdot)\overline{\hat{\tilde{\phi}}(\cdot+k)}+ \sum _{l=1}^{L}\sum_{j=0}^{\kappa(k)} \hat{\psi}_{l}\bigl(2^{-j}\cdot\bigr) \overline{\hat{\tilde{ \psi}}_{l}\bigl(2^{-j}(\cdot+k)\bigr)} $$

with \(k\in \mathbb {Z}^{d}\) converges absolutely a.e. on \(\mathbb {R}^{d}\) and belongs to \(L^{\infty}(\mathbb {R}^{d})\), and almost all points in \(\mathbb {R}^{d}\) are Lebesgue points. Let \(\xi_{0}\in \mathbb {R}^{d}\) be such a point. For \(0<\epsilon<\frac{1}{2}\), take f and g such that

$$\hat{f}(\cdot)=\frac{ (1+\Vert \cdot \Vert ^{2} )^{-s/2}\chi _{B(\xi_{0},\epsilon)}}{ \sqrt{\vert B(\xi_{0},\epsilon)\vert }} \quad \mbox{and}\quad \hat{g}(\cdot)=\frac{ (1+\Vert \cdot \Vert ^{2} )^{s/2} \chi_{B(\xi_{0},\epsilon)}}{ \sqrt{\vert B(\xi_{0},\epsilon)\vert }} $$

in (3.3), where \(B(\xi_{0},\epsilon)=\{\xi\in \mathbb {R}^{d}: \vert \xi-\xi_{0}\vert <\epsilon\} \). Then

$$\frac{1}{\vert B(\xi_{0},\epsilon)\vert } \int_{B(\xi_{0},\epsilon)} \Biggl(\hat{\phi}(\xi)\overline{\hat{\tilde{\phi}}( \xi)}+ \sum_{l=1}^{L}\sum _{j=0}^{\infty}\hat{\psi}_{l} \bigl(2^{-j}\xi\bigr) \overline{\hat{\tilde{\psi}}_{l} \bigl(2^{-j}\xi\bigr)} \Biggr)\,d\xi =1, $$

letting \(\epsilon\to0\) and applying the Lebesgue differentiation theorem, we obtain

$$\hat{\phi}(\xi_{0})\overline{\hat{\tilde{\phi}}(\xi_{0})}+ \sum_{l=1}^{L}\sum _{j=0}^{\infty}\hat{\psi}_{l} \bigl(2^{-j}\xi_{0}\bigr) \overline{\hat{\tilde{ \psi}}_{l}\bigl(2^{-j}\xi_{0}\bigr)}=1. $$

For \(0\neq k_{0}\in \mathbb {Z}^{d}\), take f and g such that

$$\hat{f}(\cdot+k_{0})=\frac{ (1+\Vert \cdot \Vert ^{2} )^{-s/2}\chi _{B(\xi_{0},\epsilon)}}{ \sqrt{\vert B(\xi_{0},\epsilon)\vert }} \quad \mbox{and}\quad \hat{g}(\cdot)= \frac{ (1+\Vert \cdot \Vert ^{2} )^{s/2} \chi_{B(\xi_{0},\epsilon)}}{ \sqrt{\vert B(\xi_{0},\epsilon)\vert }} $$

in (3.3), where \(0<\epsilon<\frac{1}{2}\). Then

$$\frac{1}{\vert B(\xi_{0},\epsilon)\vert } \int_{B(\xi_{0},\epsilon)} \Biggl(\hat{\phi}(\xi)\overline{\hat{\tilde{\phi}}( \xi+k_{0})} +\sum_{l=1}^{L}\sum _{j=0}^{\kappa(k_{0})} \hat{\psi}_{l} \bigl(2^{-j}\xi\bigr) \overline{\hat{\tilde{\psi}}_{l} \bigl(2^{-j}(\xi+k_{0})\bigr)} \Biggr)\,d\xi=0, $$

letting \(\epsilon\to0\) and applying the Lebesgue differentiation theorem, we obtain

$$\hat{\phi}(\xi_{0})\overline{\hat{\tilde{\phi}}(\xi_{0}+k_{0})} +\sum_{l=1}^{L}\sum _{j=0}^{\kappa(k_{0})} \hat{\psi}_{l} \bigl(2^{-j}\xi_{0}\bigr) \overline{\hat{\tilde{ \psi}}_{l}\bigl(2^{-j}(\xi_{0}+k_{0}) \bigr)}=0. $$

By the arbitrariness of \(\xi_{0}\) and \(k_{0}\), we obtain (3.1). □