A class of reproducing systems generated by a finite family in $L^2({\mathbb {R}}_+)$

Abstract

Reproducing systems in $L^2({\mathbb {R}})$ such as wavelet and Gabor dual frames have been extensively studied, but reducing systems in $L^2({\mathbb {R}}_+)$ with ${{\mathbb {R}}_+}=(0,\,\infty )$ have not. In practice, $L^2({\mathbb {R}}_+)$ models the causal space since the time variable cannot be negative. Due to ${\mathbb {R}}_+$ not being a group under addition, $L^2({\mathbb {R}}_+)$ admits no nontrivial shift invariant system and thus admits no traditional wavelet or Gabor analysis. However, $L^2({\mathbb {R}}_+)$ admits nontrivial dilation systems due to ${\mathbb {R}}_+$ being a group under multiplication. This paper addresses the frame theory of a class of dilation-and-modulation (${\mathcal{M}\mathcal{D}}$) systems generated by a finite family in $L^2({\mathbb {R}}_+)$. We obtain a parametric expression of ${\mathcal{M}\mathcal{D}}$-frames, and a density theorem for such ${\mathcal{M}\mathcal{D}}$-systems which is parallel to that of traditional Gabor systems in $L^2({\mathbb {R}})$. It is well known that an arbitrary Gabor frame must admit dual frames with the same structure. Interestingly, it is not the case for ${\mathcal{M}\mathcal{D}}$-frames. We prove that an ${\mathcal{M}\mathcal{D}}$-frame admits ${\mathcal{M}\mathcal{D}}$-dual frames if and only if $\log _ba$ is an integer, where a and b are dilation and modulation parameters, respectively. And in this case, we characterize and express all ${\mathcal{M}\mathcal{D}}$-dual generators for an arbitrarily given ${\mathcal{M}\mathcal{D}}$-frame. Some examples are also provided.

1 Introduction

Recall that a countable family $\{f_{j}\}_{j\in J}$ in a separable Hilbert space ${{\mathcal {H}}}$ is called a frame for ${{\mathcal {H}}}$ if there exist constants $0<A\le B<\infty$ satisfying

$$\begin{aligned} A\Vert f\Vert ^{2}\le \sum _{j\in J}|\langle f,\,f_{j}\rangle |^{2}\le B\Vert f\Vert ^{2}{ \text{ for } }f\in {{\mathcal {H}}}, \end{aligned}$$

(1.1)

where A and B are called frame bounds. If only the right-hand side inequality holds, we say that $\{f_{j}\}_{j\in J}$ is a Bessel sequence with Bessel bound B. And we say that it is a tight frame (Parseval frame) for ${{\mathcal {H}}}$ if $A=B$ ($A=B=1$) in (1.1). In addition, $\{f_{j}\}_{j\in J}$ is said to be a frame sequence in ${{\mathcal {H}}}$ if it is a frame for its closed linear span $\overline{\textrm{span}}(\{f_{j}\}_{j\in J})$. And it is called a Riesz sequence in ${\mathcal {H}}$ if there exist constants $0<A\le B<\infty$ satisfying

$$\begin{aligned} A\sum _{j\in J}|c_{j}|^{2}\le \Vert \sum _{j\in J}c_{j}f_{j}\Vert ^{2}\le B\sum _{j\in J}|c_{j}|^{2}{ \text{ for } }c\in l_{0}(J), \end{aligned}$$

(1.2)

where $l_{0}(J)$ denotes the set of all finitely supported sequences on J, A and B are called Riesz bounds. And it is called a Riesz basis for ${\mathcal {H}}$ if it is a Riesz sequence and complete. Given a frame $\{f_{j}\}_{j\in J}$ for ${{\mathcal {H}}}$, a sequence $\{g_{j}\}_{j\in J}$ in ${{\mathcal {H}}}$ is called a dual frame of $\{f_{j}\}_{j\in J}$ if it is a frame for ${{\mathcal {H}}}$ satisfying

$$\begin{aligned} f=\sum \limits _{j\in J} \langle f,\,g_{j}\rangle f_{j} { \text{ for } } f\in {{\mathcal {H}}}. \end{aligned}$$

(1.3)

By a standard argument, we have that $\{f_{j}\}_{j\in J}$ is also a dual of $\{g_{j}\}_{j\in J}$ if $\{g_{j}\}_{j\in J}$ is a dual of $\{f_{j}\}_{j\in J}$ (so, in this case, we say $\{f_{j}\}_{j\in J}$ and $\{g_{j}\}_{j\in J}$ form a pair of dual frames for ${\mathcal {H}}$), and that $\{f_{j}\}_{j\in J}$ and $\{g_{j}\}_{j\in J}$ form a pair of dual frames for ${\mathcal {H}}$ if they are Bessel sequences satisfying (1.3). We refer to [3, 12, 16, 32] for basics on frames. Equation (1.3) gives a reproducing formula for vectors in ${{\mathcal {H}}}$, where $\{g_{j}\}_{j\in J}$ is used for analyzing f and $\{f_{j}\}_{j\in J}$ for reconstructing f. So $\{f_{j}\}_{j\in J}$ and $\{g_{j}\}_{j\in J}$ satisfying (1.3) are also said to form a reproducing system. Dual frame pairs are stable reproducing systems. During the past more than 30 years, reproducing systems such as wavelet and Gabor dual frames for $L^2({\mathbb {R}}_+)$ have been extensively studied ( [2, 4,5,6,7, 9,10,11, 13, 17, 19,20,22, 29, 31]). But the theory of structured frames for $L^2({\mathbb {R}}_+)$ has not. It is because ${\mathbb {R}}$ is a group while ${\mathbb {R}}_+$ is not under addition. This results in $L^2({\mathbb {R}}_+)$ with ${{\mathbb {R}}_+}=(0,\,\infty )$ admitting no nontrivial shift invariant system and thus admitting no traditional wavelet or Gabor analysis. Observe that ${\mathbb {R}}_+$ is a group under multiplication. This makes $L^2({\mathbb {R}}_+)$ to be closed under dilation. This paper addresses a class of dilation-and-modulation reproducing systems generated by a finite family in $L^2({\mathbb {R}}_+)$.

For the moment, in order to explain the scope of this paper, we recall some notions and notations. ${\mathbb {Z}}$ and ${\mathbb {N}}$ denote the set of integers and the set of positive integers, respectively. Given a, $b>1$, a measurable function h defined on ${\mathbb {R}}_+$ is said to be b-dilation periodic if $h(b\cdot )=h(\cdot )$ a.e. on ${\mathbb {R}}_+$. Obviously, a b-dilation periodic function on ${\mathbb {R}}_+$ is determined by its values on $[1,\,b)$. $\{\Lambda _m\}_{m\in {\mathbb {Z}}}$ denotes the sequence of b-dilation periodic functions on ${\mathbb {R}}_+$ defined by

$$\begin{aligned} \Lambda _m(\cdot )=\frac{1}{\sqrt{b-1}}e^{\frac{2\pi im\cdot }{b-1}}{ \text{ on } }[1,\,b){ \text{ for } \text{ each }}\,\,\,m\in {\mathbb {Z}}. \end{aligned}$$

(1.4)

Given a finite family $\Psi =\{\psi _1,\,\psi _2,\,\ldots ,\, \psi _L\}$ in $L^2({\mathbb {R}}_+)$, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ denotes the dilation-and-modulation system generated by $\Psi$:

$$\begin{aligned} {{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)=\{\Lambda _m D_{a^j}\psi _l: m,\,j\in {\mathbb {Z}},\, 1\le l\le L\}, \end{aligned}$$

(1.5)

where

$$\begin{aligned} D_{c}f(\cdot )=\sqrt{c}f(c\cdot ) { \text{ for } }f\in L^2({\mathbb {R}}_+){ \text{ and } }c>0. \end{aligned}$$

For simplicity, we write ${{\mathcal{M}\mathcal{D}}}(\psi ,a,b)$ for ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ if $\Psi$ is a singleton $\{\psi \}$, and write ${{\mathcal{M}\mathcal{D}}}(\Psi ,\, a)$ for ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ if $a=b$. In this paper, we work under the following general setup:

General setup. (i) $a,\,b$ are two constants greater than 1 such that

$$\begin{aligned} \log _b a=\frac{p}{q} \end{aligned}$$

(1.6)

for some coprime positive integers p and q.

(ii) $\Psi =\{\psi _1,\,\psi _2,\,\ldots ,\, \psi _L\}$ is a finite family in $L^2({\mathbb {R}}_+)$.

From (1.6), we have $b^p=a^q$. Throughout this paper, we always write

$$\begin{aligned} \beta =b^p({ \text{ or } }a^q). \end{aligned}$$

(1.7)

In practice, the time variable cannot be negative, and $L^2({\mathbb {R}}_+)$ models the causal space. ${\mathcal{M}\mathcal{D}}$-systems of the form (1.5) have potential applications in analyzing causal signals. As pointed out in [24, 25], an ${\mathcal{M}\mathcal{D}}$-system is different from the Fourier transform version of a wavelet system in the Hardy space $H^2({\mathbb {R}})$, and $L^2({\mathbb {R}}_+)$ is not closed under the Fourier transform. So the Fourier transform is not applicable to analysis for $L^2({\mathbb {R}}_+)$ although it is a powerful tool for wavelet and Gabor analysis in $L^2({\mathbb {R}})$. Some special cases of (1.5) were studied in [23,24,25,26,27, 30]. [23, 25] investigated ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a)$-frames with $\Psi$ being a finite family of characteristic functions, where [25] is for $\textrm{card}(\Psi )=1$ and [23] is for general $\textrm{card}(\Psi )$. [27] characterized ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a)$-frames and dual frames, and [30] is for the vector-valued case. [24] presented a density theorem for singly generated ${{\mathcal{M}\mathcal{D}}}$-system of the form ${{\mathcal{M}\mathcal{D}}}(\psi ,a,b)$. This paper focuses on ${\mathcal{M}\mathcal{D}}$-dual pairs of the form (1.5) under the general setup. Two frames ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ and ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ for $L^2({\mathbb {R}}_+)$ form a pair of dual frames if

$$\begin{aligned} f=\sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\,\Lambda _m D_{a^j}\psi _l\rangle \Lambda _m D_{a^j}\phi _l{ \text{ for } }f\in L^2({\mathbb {R}}_+), \end{aligned}$$

(1.8)

equivalently,

$$\begin{aligned} f=\sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\,\Lambda _m D_{a^j}\phi _l\rangle \Lambda _m D_{a^j}\psi _l{ \text{ for } }f\in L^2({\mathbb {R}}_+). \end{aligned}$$

(1.9)

Given Bessel sequences ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ and ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ in $L^{2}({\mathbb {R}}_+)$, define the mixed frame operator $S_{\Psi ,\,\Phi }$ by

$$\begin{aligned} S_{\Psi ,\,\Phi }f=\sum \limits _{l=1}^{L}\sum \limits _{m,\,j\in {\mathbb {Z}}}\langle f,\,\Lambda _mD_{a^j}\phi _l\rangle \Lambda _mD_{a^j}\psi _l{ \text{ for } }f\in L^{2}({\mathbb {R}}_+). \end{aligned}$$

Then, it is a bounded operator on $L^{2}({\mathbb {R}}_+)$. And (1.8) and (1.9) may be written as $S_{\Phi ,\,\Psi }=I$ and $S_{\Psi ,\,\Phi }=I$, respectively, where I is the identity operator on $L^2({\mathbb {R}}_+)$. Write $S_{\Psi ,\Psi }=S_{\Psi }$.

Recall that an arbitrary Gabor frame for $L^2({\mathbb {R}})$ admits dual frames with Gabor structure. In contrast, this paper points out that not every ${\mathcal{M}\mathcal{D}}$-frame has the duals with the same structure. We prove that $\log _ba\in {\mathbb {N}}$ is the necessary and sufficient condition for an ${\mathcal{M}\mathcal{D}}$-frame ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ admitting ${\mathcal{M}\mathcal{D}}$-dual frames (see Theorem 3.1), and present a parametric expression of ${\mathcal{M}\mathcal{D}}$-frames and their ${\mathcal{M}\mathcal{D}}$-duals (see Theorems 2.3, 3.2 and 3.4). It is one of the novelties of this paper. Another feature is our method of proof. Recall that the traditional Fourier transform is not applicable to analysis for $L^2({\mathbb {R}}_+)$. We introduce the $\Theta _\beta$-transform matrix methods, and, with the help of dilation congruence and disjointness, reduce all problems to designing suitable $\Theta _\beta$-transform matrix-valued functions on $[1,\,a^{\frac{1}{p}})\times [0,\,1)$ of finite order. All entries of such matrices are mutually independent. This gives us much freedom in designing the $\Theta _\beta$-transform matrix-valued functions and makes all conditions of theorems easily realized. Of course, our general setup is a technical requirement. How to treat the case of $\log _ba$ being irrational is unresolved.

The rest of this paper is organized as follows. Section 2 is devoted to parametric expression of ${\mathcal{M}\mathcal{D}}$-Bessel sequences (frames and Riesz bases) and the density theorem of ${\mathcal{M}\mathcal{D}}$-systems. Section 3 focuses on proving $\log _ba\in {\mathbb {N}}$ being the necessary and sufficient condition for an ${\mathcal{M}\mathcal{D}}$-frame to admit ${\mathcal{M}\mathcal{D}}$-duals, and a parametric expression of ${\mathcal{M}\mathcal{D}}$-duals. Some examples are also provided.

Before proceeding, we introduce some notations and notions. Throughout this paper, inclusion or equality between two measurable sets in ${\mathbb {R}}_+$ (equality or inequality between two functions) means it holding up to a set of measure zero. Given M, $N\in {\mathbb {N}}$, we denote by $I_M$ the $M\times M$ identity matrix, by ${{\mathbb {N}}}_M$ the set $\{0,\,1,\,2,\,\ldots ,M-1\}$, by ${{\mathcal {M}}}_{M,\,N}$ the set of $M\times N$ complex matrices. For ${{\mathcal {A}}}=({{\mathcal {A}}}_{m,n})_{1\le m\le M,\,1\le n\le N}\in {{\mathcal {M}}}_{M,\,N}$, we denote by $\Vert {{\mathcal {A}}}\Vert _2$ its Frobenius norm, i.e.,

$$\begin{aligned} \Vert {{\mathcal {A}}}\Vert _2=\left( \sum \limits _{n=1}^{N}\sum \limits _{m=1}^{M}|{{\mathcal {A}}}_{m,n}|^2\right) ^{\frac{1}{2}}. \end{aligned}$$

Obviously, $\Vert {{\mathcal {A}}}\Vert _2=\Vert {{\mathcal {A}}}\Vert _{{{\mathbb {C}}}^M}$ if $N=1$. We denote by ${{\mathcal {A}}}^t$ and ${{\mathcal {A}}}^*$ its transpose and conjugate transpose, respectively, by $I_L\otimes {{\mathcal {A}}}$ the block matrix (with L blocks) of the form $I_L\otimes {{\mathcal {A}}}=\textrm{diag}({{\mathcal {A}}},\,{{\mathcal {A}}},\ldots ,\,{{\mathcal {A}}})$, by $\{e_{m,\,j}\}$ the function sequence:

$$\begin{aligned} e_{m,\,j}(x,\,\xi )=\Lambda _m(x)e^{2\pi ij\xi }{ \text{ for } }m,\,j\in {\mathbb {Z}}{ \text{ and } }(x,\,\xi )\in {{\mathbb {R}}}_+\times {{\mathbb {R}}}, \end{aligned}$$

(1.10)

where $\Lambda _m$ is as in (1.4); and by $\{e_k:\,k\in {{\mathbb {N}}}_M\}$ denotes the canonical orthonormal basis for ${{\mathbb {C}}}^M$, i.e., each $e_k$ is the vector with the kth component being 1 and the others being 0.

Definition 1.1

Let a, b and $\Psi$ be as in the general setup, and $\beta$ be as in (1.7). Define $\Theta _\beta :\,L^{2}({\mathbb {R}}_+)\rightarrow L^{2}_{loc}({{\mathbb {R}}_+}\times {{\mathbb {R}}})$, $\Gamma :\,L^{2}({\mathbb {R}}_+)\rightarrow L^{2}_{loc}({{\mathbb {R}}_+}\times {{\mathbb {R}}},\,{{\mathbb {C}}}^p)$ and $\Upsilon :\,L^{2}({\mathbb {R}}_+)\rightarrow L^{2}_{loc}({{\mathbb {R}}_+}\times {{\mathbb {R}}},\,{{\mathcal {M}}}_{q,\,p})$ by

$$\begin{aligned} \Theta _{\beta }f(x,\,\xi )= & {} \sum \limits _{j\in {\mathbb {Z}}}\beta ^{\frac{j}{2}}f(\beta ^{j}x)e^{-2\pi ij\xi }, \end{aligned}$$

(1.11)

$$\begin{aligned} \Gamma f(x,\xi )= & {} \left( b^{\frac{s}{2}}\Theta _{\beta }f(b^{s}x,\,\xi )\right) _{s\in {\mathbb {N}}_p}, \end{aligned}$$

(1.12)

$$\begin{aligned} \Upsilon f(x,\xi )= & {} \left( a^{\frac{r}{2}}b^{\frac{s}{2}}\Theta _{\beta }f(a^rb^{s}x,\,\xi )\right) _{r\in {{\mathbb {N}}}_q,\,s\in {\mathbb {N}}_p} \end{aligned}$$

(1.13)

for $f\in L^2({\mathbb {R}}_+)$ and a.e. $(x,\,\xi )\in {{\mathbb {R}}_+}\times {\mathbb {R}}$, respectively. We associate $\Psi$ with the matrix-valued function (so-called $\Theta _{\beta }$-transform matrix) from ${\mathbb {R}}_+\times {\mathbb {R}}$ into ${{\mathcal {M}}}_{Lq,\,p}$ defined by

$$\begin{aligned} {\varvec{\Psi }}(x,\,\xi )=\left( \begin{array}{c} {\Upsilon \psi _1}(x,\,\xi ) \\ {\Upsilon \psi _2}(x,\,\xi ) \\ \vdots \\ {\Upsilon \psi _L}(x,\,\xi )\\ \end{array} \right) \end{aligned}$$

(1.14)

for a.e. $(x,\xi )\in {{\mathbb {R}}_+}\times {{\mathbb {R}}}$.

Remark 1.1

(i) The $\Theta _{\beta }$-transform herein appeared before in [14] called multiplicative Zak transform and [18] called generalized Zak transform. A more general definition appeared in [1] to study reproducing systems on LCA groups.

(ii) By ([24], Remark 2.4) and a standard argument, $\Theta _{\beta }$, $\Gamma$ and $\Upsilon$ are all well defined. And by Lemma 2.2 (v), $\Psi$ is uniquely determined by the values of its $\Theta _{\beta }$-transform matrix ${\varvec{\Psi }}(x,\,\xi )$ for $(x,\,\xi )\in E_{\Upsilon }\times [0,\,1)$ where $E_{\Upsilon }$ is $({a^{\frac{1}{p}}})^{{\mathbb {Z}}}$-dilation congruent to $[1,\,a^{\frac{1}{p}})$. For an arbitrary $Lq\times p$ matrix-valued function defined on $E_{\Upsilon }\times [0,\,1)$ with entries in $L^{2}(E_{\Upsilon }\times [0,\,1))$ determines a unique $\Psi$ in $L^2({\mathbb {R}}_+)$ via (1.14) restricted to $E_{\Upsilon }\times [0,1)$. And the unique $\Psi =\{\psi _1,\,\psi _2,\,\ldots ,\,\psi _L\}$ is defined by

$$\begin{aligned} \psi _{l}(\beta ^ja^rb^sx)= & {} \beta ^{-\frac{j}{2}}a^{-\frac{r}{2}}b^{-\frac{s}{2}}\int _{[0,\,1)}{(\Upsilon \psi _l)}_{r,s}(x,\,\xi )e^{2\pi i j \xi }d\xi { \text{ for } }(r,\,s,\,j)\nonumber \\\in & {} {{\mathbb {N}}}_q\times {{\mathbb {N}}}_p\times {{\mathbb {Z}}}{\text{ and } \text{ a.e. } }x\in E_{\Upsilon } \end{aligned}$$

(1.15)

for $1\le l\le L$.

Definition 1.2

For a measurable set $S\subset {\mathbb {R}}_+$, a collection $\{S_i\}_{i\in {\mathcal {I}}}$ of measurable subsets of S is called a partition of S if $\chi _{_S}=\sum \limits _{i\in {\mathcal {I}}}\chi _{_{S_i}}$, where $\chi _{_E}$ denotes the characteristic function of E for a set E. And for $\alpha >0$, measurable subsets T, ${\widetilde{T}}$ of ${\mathbb {R}}_+$ and a collection $\{\Gamma _i:\,i\in {\mathcal {I}}\}$ of measurable subsets of ${\mathbb {R}}_+$ with ${\mathcal {I}}$ being at most countable, we say that T is ${\alpha }^{{\mathbb {Z}}}$-dilation congruent to ${\widetilde{T}}$ if there exists a partition $\{T_k:\,k\in {\mathbb {Z}}\}$ of T such that $\{\alpha ^kT_k:\,k\in {\mathbb {Z}}\}$ is a partition of ${\widetilde{T}}$, and that $\{\Gamma _i:\,i\in {{\mathcal {I}}}\}$ is ${\alpha ^{{\mathbb {Z}}}}$-dilation disjoint if

$$\begin{aligned} \Gamma _i\cap {\alpha }^k\Gamma _j=\emptyset \end{aligned}$$

for $i,\,j\in {{\mathcal {I}}}$ and $k\in {\mathbb {Z}}$ with $(i,\,k)\ne (j,\,0)$.

Remark 1.2

By Definition 1.2, ${\widetilde{T}}$ is $\alpha ^{{\mathbb {Z}}}$-dilation congruent to T if T is $\alpha ^{{\mathbb {Z}}}$-dilation congruent to $\widetilde{T}$. So we say that T and ${{\widetilde{T}}}$ are $\alpha ^{{\mathbb {Z}}}$-dilation congruent. Also observe that only finitely many $T_{k}$ among $\{T_{k}: \,k\in {\mathbb {Z}}\}$ are nonempty if T and ${\widetilde{T}}$ are contained in some bounded subinterval $[M,\,N]$ of ${\mathbb {R}}_+$.

Definition 1.3

Let a, b, $\Psi$ be as in the general setup, and ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ be a Bessel sequence in $L^2({\mathbb {R}}_+)$. We say ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ has Riesz property if $c=0$ is the unique solution to

$$\begin{aligned} \sum \limits _{l=1}^{L}\sum \limits _{m,\,j\in {\mathbb {Z}}}c_{l,m,j}\Lambda _mD_{a^j}\psi _l=0 \end{aligned}$$

in $l^2(\{1,2,\ldots , L\}\times {{\mathbb {Z}}^2})$.

2 Parametric expression and density theorem

This section is devoted to parametric expression of ${\mathcal{M}\mathcal{D}}$-Bessel sequences (frames and Riesz bases) and the density theorem of ${\mathcal{M}\mathcal{D}}$-systems (see Theorems 2.3 and 2.4). For this purpose, we need to establish some lemmas. The first one gives a dilation congruence-based partition of ${\mathbb {R}}_+$.

Lemma 2.1

Given $\alpha >1$, let E be $\alpha ^{{\mathbb {Z}}}$-dilation congruent to $[1,\,\alpha )$. Then, $\{\alpha ^jE:\,j\in {\mathbb {Z}}\}$ is a partition of ${{\mathbb {R}}}_+$.

Proof

Suppose $\{E_k:\,k\in {\mathbb {Z}}\}$ is a partition of E such that $\{{\alpha }^kE_k:\,k\in {\mathbb {Z}}\}$ is a partition of $[1,\,\alpha )$. Also observe that $\{{\alpha }^j[1,\,\alpha ):\,j\in {\mathbb {Z}}\}$ is a partition of ${{\mathbb {R}}}_+$. It follows that $\{{\alpha }^{k+j}E_k:\,j,\,k\in {\mathbb {Z}}\}$ is a partition of ${{\mathbb {R}}}_+$. Thus,

$$\begin{aligned} \chi _{_{{\mathbb {R}}_+}}(\cdot )=\sum \limits _{k\in {\mathbb {Z}}}\sum \limits _{j\in {\mathbb {Z}}}\chi _{_{\alpha ^{k+j}E_k}}(\cdot )=\sum \limits _{k\in {\mathbb {Z}}}\sum \limits _{j\in {\mathbb {Z}}}\chi _{_{\alpha ^{j}E_k}}(\cdot ). \end{aligned}$$

Interchanging the order of summation gives

$$\begin{aligned} \chi _{_{{\mathbb {R}}_+}}(\cdot )=\sum \limits _{j\in {\mathbb {Z}}}\sum \limits _{k\in {\mathbb {Z}}}\chi _{_{\alpha ^{j}E_k}}(\cdot )=\sum \limits _{j\in {\mathbb {Z}}}\chi _{_{\alpha ^{j}E}}(\cdot ), \end{aligned}$$

where we used the fact that $\{E_k:\,k\in {\mathbb {Z}}\}$ is a partition of E in the last equality. The proof is completed. $\square$

Lemma 2.2

Let $a,\,b$ be as in the general setup. Then,

(i) $\{e_{m,\,j}:\,m,\,j\in {\mathbb {Z}}\}$ is an orthonormal basis for $L^2([1,\,b)\times [0,\,1))$.

(ii)

$$\begin{aligned} \int _{[1,\,b)\times [0,\,1)}|f(x,\,\xi )|^2dxd\xi =\sum \limits _{m,\,j\in {\mathbb {Z}}}\left| \int _{[1,\,b)\times [0,\,1)}f(x,\,\xi )\overline{e_{m,j}(x,\,\xi )}dxd\xi \right| ^2 \end{aligned}$$

for $f\in L^1([1,\,b)\times [0,\,1))$.

(iii) $\Theta _{\beta }$ has the quasi-periodicity property:

$$\begin{aligned} \Theta _{\beta }f(\beta ^jx,\,\xi +m)=\beta ^{-\frac{j}{2}}e^{2\pi i j\xi }\Theta _{\beta }f(x,\,\xi ){ \text{ for } }m,\,j\in {{\mathbb {Z}}}{ \text{ and } }f\in L^2({{\mathbb {R}}}_+)\nonumber \\ \end{aligned}$$

(2.1)

for a.e. $(x,\,\xi )\in {{\mathbb {R}}}_+\times {{\mathbb {R}}}$.

(iv) For $f\in L^2({\mathbb {R}}_+)$ and $(j,\,m,\,r)\in {{\mathbb {Z}}}\times {{\mathbb {Z}}}\times {{\mathbb {N}}}_q$,

$$\begin{aligned} \Gamma \Lambda _mD_{a^{jq+r}}f(x,\,\xi )= & {} e_{m,\,j}(x,\,\xi )\left( a^{\frac{r}{2}}b^{\frac{s}{2}}\Theta _{\beta }f(a^rb^sx,\,\xi )\right) _{s\in {{\mathbb {N}}}_p}\nonumber \\= & {} a^{\frac{r}{2}}e_{m,\,j}(x,\,\xi )\Gamma f(a^rx,\,\xi ), \end{aligned}$$

(2.2)

for a.e. $(x,\,\xi )\in {{\mathbb {R}}}_+\times {{\mathbb {R}}}$.

(v) The mapping $\Theta _\beta$, $\Gamma$ and $\Upsilon$ are unitary operators from $L^2({\mathbb {R}}_+)$ onto $L^2(E_{\Theta }\times [0,\,1))$, $L^2(E_{\Gamma }\times [0,\,1),\,{{\mathbb {C}}}^p)$ and $L^2(E_{\Upsilon }\times [0,\,1),\,{{\mathcal {M}} }_{q,p})$, respectively, where $E_{\Theta }$ is $\beta ^{{\mathbb {Z}}}$-dilation congruent to $[1,\,\beta )$, $E_{\Gamma }$ is $b^{{\mathbb {Z}}}$-dilation congruent to $[1,\,b)$ and $E_{\Upsilon }$ is $({a^{\frac{1}{p}}})^{{\mathbb {Z}}}$-dilation congruent to $[1,\,a^{\frac{1}{p}})$.

Proof

(i)–(iv) are repeated from ([24], Lemmas 3.1,2.2). Next we prove (v). Obviously $\Theta _\beta$, $\Gamma$ and $\Upsilon$ are all linear. We only need to prove that they are norm-preserving and onto. Fix an arbitrary $f\in L^2({\mathbb {R}}_+)$. We have

$$\begin{aligned} \Vert \Theta _{\beta }f\Vert ^2_{E_{\Theta }\times [0,\,1)}= & {} \int _{E_{\Theta }\times [0,\,1)}|\Theta _{\beta }f(x,\,\xi )|^{2}dxd\xi \\= & {} \int _{E_{\Theta }}dx\int _{[0,\,1)}|\sum \limits _{k\in {\mathbb {Z}}}{\beta }^{\frac{k}{2}}f(\beta ^kx)e^{-2\pi ik\xi }|^2d\xi \\= & {} \int _{E_{\Theta }}\sum \limits _{k\in {\mathbb {Z}}}{\beta }^{k}|f(\beta ^kx)|^2dx, \end{aligned}$$

where the last equality follows from the fact that $\{e^{-2\pi ik \cdot }:\,k\in {\mathbb {Z}}\}$ is an orthonormal basis for $L^2([0,\,1))$. Since $E_{\Theta }$ is $\beta ^{{\mathbb {Z}}}$-dilation congruent to $[1,\,\beta )$, there exists a partition of $\{E_j:\,j\in {\mathbb {Z}}\}$ of $E_{\Theta }$ such that $\{\beta ^jE_j:\,j\in {\mathbb {Z}}\}$ is a partition of $[1,\,\beta ).$ It follows that

$$\begin{aligned} \Vert \Theta _{\beta }f\Vert ^2_{E_{\Theta }\times [0,\,1)}= & {} \sum \limits _{j\in {\mathbb {Z}}}\int _{E_j}\sum \limits _{k\in {\mathbb {Z}}}{\beta }^k|f(\beta ^kx)|^2dx\\= & {} \sum \limits _{j\in {\mathbb {Z}}}\int _{\beta ^jE_j}\sum \limits _{k\in {\mathbb {Z}}}{\beta }^{k-j}|f(\beta ^{k-j}x)|^2dx\\= & {} \int _{[1,\,\beta )}\sum \limits _{k\in {\mathbb {Z}}}{\beta ^k}|f(\beta ^kx)|^2dx\\= & {} \Vert f\Vert _{L^2({\mathbb {R}})}^2. \end{aligned}$$

This implies that $\Theta _\beta$ is norm-preserving. Next, we prove that it is onto. Let $F\in L^2(E_{\Theta }\times [0,\,1))$. Define f on ${\mathbb {R}}_+$ by

$$\begin{aligned} f(\beta ^jx)=\beta ^{-\frac{j}{2}}\int _{[0,\,1)}F(x,\,\xi )e^{2\pi ij\xi }d\xi { \text{ for } }j\in {\mathbb {Z}}{ \text{ and } }x\in E_{\Theta }. \end{aligned}$$

Since $E_{\Theta }$ is $\beta ^{{\mathbb {Z}}}$-dilation congruent to $[1,\,\beta )$, $\{\beta ^jE_{\Theta }:j\in {\mathbb {Z}}\}$ is a partition of ${\mathbb {R}}_+$ by Lemma 2.1. It follows that f is well defined on ${\mathbb {R}}_+$, and

$$\begin{aligned} \int _{{\mathbb {R}}_+}|f(x)|^2dx= & {} \int _{E_{\Theta }}\sum \limits _{j\in {\mathbb {Z}}}\beta ^j|f(\beta ^jx)|^2dx\\= & {} \int _{E_{\Theta }}\sum \limits _{j\in {\mathbb {Z}}}|\int _{[0,\,1)}F(x,\,\xi )e^{2\pi ij\xi }d\xi |^2dx\\= & {} \int _{E_{\Theta }\times [0,\,1)}|F(x,\,\xi )|^2 dxd\xi \\< & {} \infty . \end{aligned}$$

And by a standard argument, we have

$$\begin{aligned} \Theta _\beta f(x,\,\xi )= & {} \sum \limits _{j\in {\mathbb {Z}}}\beta ^{\frac{j}{2}}f(\beta ^jx)e^{-2\pi ij \xi }\\= & {} \sum \limits _{j\in {\mathbb {Z}}}\int _{[0,\,1)}F(x,\,\omega )e^{2\pi i jt}d\omega e^{-2\pi i j\xi }\\= & {} F(x,\,\xi ) \end{aligned}$$

for a.e. $(x,\,\xi )\in E_{\Theta }\times [0,\,1)$. Hence, $\Theta _\beta$ is onto.

Now we prove that $\Gamma$ is norm-preserving and onto. Fix an arbitrary $f\in L^2({\mathbb {R}}_+)$. Then,

$$\begin{aligned} \int _{E_{\Gamma }\times [0,\,1)}\Vert \Gamma f(x,\,\xi )\Vert ^2_{{{\mathbb {C}}}^p}dxd\xi= & {} \int _{E_{\Gamma }\times [0,\,1)}\sum \limits _{s\in {\mathbb {N}}_p}\bigg |\sum \limits _{k\in {\mathbb {Z}}}b^{\frac{s}{2}}\beta ^{\frac{k}{2}}f(b^{s}\beta ^kx)e^{-2\pi ik\xi }\bigg |^2 dxd\xi \\= & {} \int _{E_{\Gamma }}\sum \limits _{s\in {\mathbb {N}}_p}\sum \limits _{k\in {\mathbb {Z}}}b^{kp+s}|f(b^{kp+s}x)|^2dx\\= & {} \int _{E_{\Gamma }}\sum \limits _{k\in {\mathbb {Z}}}b^k|f(b^kx)|^2dx\\= & {} \sum \limits _{k\in {\mathbb {Z}}}\int _{b^kE_{\Gamma }}|f(x)|^2dx. \end{aligned}$$

Since $E_{\Gamma }$ is $b^{{\mathbb {Z}}}$-dilation congruent to $[1,\,b)$, $\{b^kE_{\Gamma }:\,k\in {\mathbb {Z}}\}$ is a partition of ${\mathbb {R}}_+$ by Lemma 2.1. Thus,

$$\begin{aligned} \int _{E_{\Gamma }\times [0,\,1)}\Vert \Gamma f(x,\,\xi )\Vert ^2_{{{\mathbb {C}}}^p}dxd\xi= & {} \Vert f\Vert ^2_{L^2({{\mathbb {R}}}_+)}. \end{aligned}$$

This implies that $\Gamma$ is norm-preserving. Next, we prove that it is onto. Let $F=\left( F_s\right) _{s\in {{\mathbb {N}}}_p}\in L^2(E_{\Gamma }\times [0,\,1),\,{{\mathbb {C}}}^p)$. Define f on ${\mathbb {R}}_+$ by

$$\begin{aligned} f(\beta ^kb^sx)= & {} b^{-\frac{s}{2}}{\beta }^{-\frac{k}{2}}\int _{[0,\,1)}F_s(x,\,\xi )e^{2\pi ik\xi }d\xi { \text{ for } }(s,\,k)\in {{\mathbb {N}}_p}\times {{\mathbb {Z}}}{ \text{ and } }x\in E_{\Gamma }. \end{aligned}$$

By the same procedure as above, we may prove $f\in L^2({\mathbb {R}}_+)$ and $\Gamma f=F$. Hence, $\Gamma$ is onto.

Finally, we prove that $\Upsilon$ is norm-preserving and onto. Fix an arbitrary $f\in L^2({\mathbb {R}}_+)$. Then,

$$\begin{aligned}{} & {} \int _{E_{\Upsilon }\times [0,\,1)}\Vert \Upsilon f(x,\,\xi )\Vert ^2_{2}dxd\xi \\{} & {} \quad =\int _{E_{\Upsilon }\times [0,\,1)}\sum \limits _{r\in {\mathbb {N}}_q}\sum \limits _{s\in {\mathbb {N}}_p}|\sum \limits _{k\in {\mathbb {Z}}}a^{\frac{r}{2}}b^{\frac{s}{2}}\beta ^{\frac{k}{2}}f(a^rb^{s}\beta ^kx)e^{-2\pi ik\xi }|^2 dxd\xi \\{} & {} \quad =\int _{E_{\Upsilon }}\sum \limits _{r\in {\mathbb {N}}_q}\sum \limits _{s\in {\mathbb {N}}_p}\sum \limits _{k\in {\mathbb {Z}}}(a^{\frac{1}{p}})^{rp+qs+kpq}|f((a^{\frac{1}{p}})^{pr+qs+pqk}x)|^2dx\\{} & {} \quad =\int _{E_{\Upsilon }}\sum \limits _{k\in {\mathbb {Z}}}a^{\frac{k}{p}}|f(a^{\frac{k}{p}}x)|^2dx\\{} & {} \quad =\sum \limits _{k\in {\mathbb {Z}}}\int _{a^{\frac{k}{p}}E_{\Upsilon }}|f(x)|^2dx \end{aligned}$$

where the third equality uses the fact that ${\mathbb {Z}}$ is the disjoint union of $pr+qs+pq{{\mathbb {Z}}}$ with $r\in {{\mathbb {N}}_q}$ and $s\in {{\mathbb {N}}_p}$. Since $E_{\Upsilon }$ is $(a^{\frac{1}{p}})^{{\mathbb {Z}}}$-dilation congruent to $[1,\,a^{\frac{1}{p}})$, $\big \{a^{\frac{k}{p}}E_{\Upsilon }:\,k\in {\mathbb {Z}}\big \}$ is a partition of ${\mathbb {R}}_+$ by Lemma 2.1. Thus,

$$\begin{aligned} \int _{E_{\Upsilon }\times [0,\,1)}\Vert \Upsilon f(x,\,\xi )\Vert ^2_{2}dxd\xi= & {} \Vert f\Vert ^2_{L^2({{\mathbb {R}}}_+)}. \end{aligned}$$

This implies that $\Upsilon$ is norm-preserving. Next, we prove that it is onto. Let $F=\left( F_{r,\,s}\right) _{r\in {{\mathbb {N}}}_q,\,s\in {{\mathbb {N}}}_p}\in L^2(E_{\Upsilon }\times [0,\,1),\,{{\mathcal {M}}}_{q,p})$. Define f on ${\mathbb {R}}_+$ by

$$\begin{aligned} f(\beta ^{k}a^{r}b^{s}x)= & {} a^{-\frac{r}{2}}b^{-\frac{s}{2}}{\beta }^{-\frac{k}{2}}\nonumber \\{} & {} \int _{[0,\,1)}F_{r,s}(x,\,\xi )e^{2\pi ik\xi }d\xi { \text{ for } }(r,\,s,\,k)\nonumber \\{} & {} \in {{\mathbb {N}}_q}\times {{\mathbb {N}}_p}\times {{\mathbb {Z}}}{\text{ and } }x\in E_{\Upsilon }. \end{aligned}$$

(2.3)

By the same procedure as above, we may prove $f\in L^2({\mathbb {R}}_+)$ and $\Upsilon f=F$. Hence, $\Upsilon$ is onto. The proof is completed. $\square$

By ([24], Lemma 3.5) and a simple computation, we have the following lemma.

Lemma 2.3

Let $a,\,b$ and $\Psi$ be as in the general setup. Then, we have

(i) For $(r,\,j)\in {{\mathbb {N}}}_q\times {\mathbb {Z}}$,

$$\begin{aligned} {\varvec{\Psi }}(a^{jq+r}x,\,\xi )=a^{-\frac{jq+r}{2}}e^{2\pi ij\xi }\left( I_L\otimes U_r(\xi )\right) {\varvec{\Psi }}(x,\,\xi ), \end{aligned}$$

(2.4)

for a.e. $(x,\,\xi )\in {{\mathbb {R}}_+}\times {\mathbb {R}}$, where

$$\begin{aligned} U_r=\left\{ \begin{array}{ll} I_q&{}\quad \hbox {if }~r=0;\\ \left( \begin{array}{ll} 0 &{} I_{q-r} \\ e^{2\pi i \xi }I_r &{} 0 \\ \end{array} \right)&\quad \hbox {if } r\in {{\mathbb {N}}}_q\setminus \{0\}. \end{array} \right. \end{aligned}$$

(ii)

$$\begin{aligned} {\varvec{\Psi }}(a^{\frac{1}{p}}x,\xi )=a^{-\frac{1}{2p}}\left( I_L\otimes {{\mathcal {L}}}_{p,q}(\xi )\right) {\varvec{\Psi }}(x,\,\xi ){{\mathcal {R}}}_{p,q}(\xi ), \end{aligned}$$

(2.5)

for a.e. $(x,\,\xi )\in {{\mathbb {R}}_+}\times {\mathbb {R}}$, where $(r',\,s')\in ({{\mathbb {N}}}_q\backslash \{0\})\times ({{\mathbb {N}}}_p\backslash \{0\})$ satisfying $pr'+qs'=pq+1$,

$$\begin{aligned} {{\mathcal {L}}}_{p,\,q}(\xi )=\left\{ \begin{array}{ll} \left( \begin{array}{ll} 0 &{} I_{q-r'} \\ e^{2\pi i\xi }I_{r'} &{} 0 \\ \end{array} \right) &{}\hbox { if } ~p,\,q>1;\\ I_q &{}\hbox { if }\,~ p=1 \hbox {or} q=1,\\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} {{\mathcal {R}}}_{p,\,q}(\xi )=\left\{ \begin{array}{ll} \left( \begin{array}{ll} 0 &{} I_{s'} \\ e^{2\pi i\xi }I_{p-s'} &{} 0 \\ \end{array} \right) &{}\hbox {if }~ p,\,q>1;\\ I_p &{}\hbox {if }~ p=1 \hbox {or} q=1.\\ \end{array} \right. \end{aligned}$$

(iii) For $(s,\,j)\in {{\mathbb {N}}}_p\times {\mathbb {Z}}$ and $f\in L^2({{\mathbb {R}}}_+)$,

$$\begin{aligned} {\varvec{\Psi }}(b^{jp+s}x,\xi )=b^{-\frac{jp+s}{2}}e^{2\pi i j\xi }{\varvec{\Psi }}(x,\xi ){{\mathcal {M}}}_s(\xi ) \end{aligned}$$

(2.6)

and

$$\begin{aligned} \Gamma f(b^{jp+s}x,\,\xi )=b^{-\frac{jp+s}{2}}e^{2\pi i j\xi }\overline{{{\mathcal {M}}^*}_s(\xi )}\Gamma f(x,\,\xi ) \end{aligned}$$

(2.7)

for a.e. $(x,\,\xi )\in {{\mathbb {R}}_+}\times {\mathbb {R}}$, where

$$\begin{aligned} {{\mathcal {M}}}_s(\xi )=\left\{ \begin{array}{ll} I_p&{}\hbox { if }\,\, s=0;\\ \left( \begin{array}{cc} 0 &{} e^{2\pi i \xi }I_{s} \\ I_{p-s} &{} 0 \\ \end{array} \right)&\hbox { if }\,\, s\in {{\mathbb {N}}}_p\setminus \{0\}. \end{array} \right. \end{aligned}$$

The following lemma is a generalization of ([24], Lemma 3.6).

Lemma 2.4

Let $a,\,b$ and $\Psi$ be as in the general setup. Then, for positive constants A and B with $A\le B$, we have

(i)

$$\begin{aligned} AI_p\le {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\le BI_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,b)\times [0,\,1) \end{aligned}$$

(2.8)

if and only if

$$\begin{aligned} a^{\frac{q-1}{p}}AI_p\le {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\le BI_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

(2.9)

(ii)

$$\begin{aligned} {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\le BI_p{ \text{ for } \text{ a.e. }}(x,\,\xi )\in [1,\,b)\times [0,\,1) \end{aligned}$$

(2.10)

if and only if

$$\begin{aligned} {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\le BI_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

(2.11)

Proof

We only prove (i), and (ii) can be proved similarly. Since $b=a^{\frac{q}{p}}$ and $[1,\,b)=\bigcup \limits _{r\in {{\mathbb {N}}}_q}a^{\frac{r}{p}}[1,\,a^\frac{1}{p})$, (2.8) holds if and only if

$$\begin{aligned} AI_p\le {\varvec{\Psi }}^*(a^{\frac{r}{p}}x,\,\xi ){\varvec{\Psi }}(a^{\frac{r}{p}}x,\,\xi )\le BI_p \end{aligned}$$

(2.12)

for $r\in {{\mathbb {N}}}_q$ and a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$. This can be rewritten as

$$\begin{aligned} AI_p\le a^{-\frac{r}{p}}{{\mathcal {R}}}_{p,q}^*(\xi ){\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi ){{\mathcal {R}}}_{p,q}(\xi )\le BI_p \end{aligned}$$

for $r\in {{\mathbb {N}}}_q$ and a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$ by Lemma 2.3 (ii). It is in turn equivalent to (2.9) by a standard argument. The proof is completed. $\square$

Remark 2.1

Lemma 2.4 implies that

$$\begin{aligned} a^{\frac{q-1}{p}}A\le B \end{aligned}$$

is necessary for (2.8) or (2.9) to hold. In particular, $q=1$ if $A=B$ in Lemma 2.4.

By ([24], Lemma 4.1), under the general setup, we have

$$\begin{aligned}{} & {} \sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}|\langle f,\,\Lambda _mD_{a^j}\psi _l\rangle _{L^2({{\mathbb {R}}_+})}|^2\nonumber \\{} & {} =\int _{[1,b)\times [0,1)}\left\| \overline{{\varvec{\Psi }}(x,\xi )}\Gamma f(x,\xi )\right\| _{{{\mathbb {C}}}^{Lq}}^{2}dxd\xi { \text{ for } }f\in L^2({\mathbb {R}}_+). \end{aligned}$$

(2.13)

Again by Lemma 2.2 (v), this implies that ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^2({\mathbb {R}}_+)$ with bound B if and only if

$$\begin{aligned}{} & {} \int _{[1,\,b)\times [0,\,1)}\left\| {\overline{{\varvec{\Psi }}(x,\,\xi )}}\Gamma f(x,\,\xi )\right\| _{{\mathbb {C}}^{Lq}}^2dxd\xi \nonumber \\{} & {} \le B\int _{[1,\,b)\times [0,\,1)}\Vert \Gamma f(x,\,\xi )\Vert _{{\mathbb {C}}^p}^2dxd\xi { \text{ for } }f\in L^2({\mathbb {R}}_+). \end{aligned}$$

(2.14)

The following lemma reduces (2.14) to the spectral condition of ${\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )$ on $[1,\,a^{\frac{1}{p}})\times [0,\,1)$.

Lemma 2.5

Let $a,\,b$ and $\Psi$ be as in the general setup. Then, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^{2}({\mathbb {R}}_+)$ with bound B if and only if

$$\begin{aligned} {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\le BI_p{ \text{ for } \text{ a.e. }}(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

(2.15)

Proof

By Lemma 2.4, (2.15) is equivalent to

$$\begin{aligned} {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\le BI_p{ \text{ for } \text{ a.e. }}(x,\,\xi )\in [1,\,b)\times [0,\,1). \end{aligned}$$

(2.16)

Next we prove that ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^{2}({\mathbb {R}}_+)$ with bound B if and only if (2.16) holds. Observe that

$$\begin{aligned}{} & {} \int _{[1,\,b)\times [0,\,1)}\left\| \overline{{\varvec{\Psi }}(x,\xi )}\Gamma f(x,\xi )\right\| _{{{\mathbb {C}}}^{Lq}}^{2}dxd\xi = \int _{[1,\,b)\times [0,\,1)}\left\| {{\varvec{\Psi }}(x,\xi )}\overline{\Gamma f(x,\xi )}\right\| _{{{\mathbb {C}}}^{Lq}}^{2}dxd\xi \\{} & {} \quad =\int _{[1,\,b)\times [0,\,1)}\left\langle {\varvec{\Psi }}^*(x,\xi ){\varvec{\Psi }}(x,\xi )\overline{\Gamma f(x,\xi )},\,\overline{\Gamma f(x,\xi )}\right\rangle _{{{\mathbb {C}}}^{p}} dxd\xi \end{aligned}$$

for $f\in L^2({\mathbb {R}}_+)$. We have that (2.14) holds, and thus ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^2({\mathbb {R}}_+)$ with bound B if (2.16) holds.

Now we prove the converse implication. Suppose ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^2({\mathbb {R}}_+)$ with bound B. Then, (2.14) holds. Fix an arbitrary $c\in {{\mathbb {C}}}^p$ and $E\subset [1,\,b)\times [0,\,1)$ with $|E|>0$. Take f in (2.14) by

$$\begin{aligned} \Gamma f(x,\,\xi )=\chi _{_E} c{ \text{ for } }(x,\,\xi )\in [1,\,b)\times [0,\,1). \end{aligned}$$

Then, f is well defined by Lemma 2.2 (v), and

$$\begin{aligned} \int _E\left\| \overline{{\varvec{\Psi }}(x,\xi )}c\right\| ^2_{{{\mathbb {C}}}^{Lq}}dxd\xi \le B|E|\Vert c\Vert ^2. \end{aligned}$$

This implies that

$$\begin{aligned} \left\| \overline{{\varvec{\Psi }}(x,\xi )}c\right\| ^2_{{{\mathbb {C}}}^{Lq}}\le B\Vert c\Vert ^2{ \text{ for } }c\in {{\mathbb {C}}}^p{ \text{ and } \text{ a.e. }}(x,\,\xi )\in [1,\,b)\times [0,\,1) \end{aligned}$$

by the arbitrariness of E and ([28], Theorem 1.40), equivalently, (2.16) holds. The proof is completed. $\square$

Remark 2.2

Observe that (2.15) is equivalent to

$$\begin{aligned} \left\| {{\varvec{\Psi }}(x,\xi )}c\right\| _{{{\mathbb {C}}}^{Lq}}\le \sqrt{B}\left\| c\right\| { \text{ for } }c\in {{\mathbb {C}}}^p{\text{ and } \text{ a.e. } }(x,\,\xi )\in [1,\,b)\times [0,\,1). \end{aligned}$$

Since all norms on an arbitrarily finite-dimensional normed linear spaces are mutually equivalent, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b )$ is a Bessel sequence in $L^2({\mathbb {R}}_+)$ if and only if all entries of ${\varvec{\Psi }}(x,\xi )$ belong to $L^{\infty }([1,\,a^{\frac{1}{p}})\times [0,\,1))$, equivalently, they belong to $L^{\infty }({{\mathbb {R}}}_+\times {{\mathbb {R}}})$ by ([24], Lemma 3.3) and quasi-periodicity of $\Theta _{\beta }$-transform.

The following lemma is a partial generalization of (2.13).

Lemma 2.6

Let $a,\,b$ and $\Psi$ be as in the general setup, and $\Phi =\{\phi _1,\,\phi _2,\,\ldots ,\,\phi _L\}$ be a sequence in $L^2({\mathbb {R}}_+)$. Then,

$$\begin{aligned}{} & {} \sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\,\Lambda _mD_{a^j}\phi _l\rangle _{L^2({{\mathbb {R}}_+})}\langle \Lambda _mD_{a^j}\psi _l,\,g \rangle _{L^2({{\mathbb {R}}_+})}\nonumber \\{} & {} \quad =\left\langle \overline{{\varvec{\Psi }}^{*}{\varvec{\Phi }}}\Gamma f,\,\Gamma g\right\rangle _{L^2([1,\,b)\times [0,\,1),\,{{\mathbb {C}}}^p)} \end{aligned}$$

(2.17)

for $f,\,g$ satisfying

$$\begin{aligned}{} & {} \int _{[1,\,b)\times [0,\,1)}\left\| \overline{{\varvec{\Phi }}(x,\,\xi )}\Gamma f(x,\,\xi )\right\| _{{{\mathbb {C}}}^{Lq}}^{2}dxd\xi ,\,\nonumber \\{} & {} \quad \int _{[1,\,b)\times [0,\,1)}\left\| \overline{{\varvec{\Psi }}(x,\,\xi )}\Gamma g(x,\,\xi )\right\| _{{{\mathbb {C}}}^{Lq}}^{2}dxd\xi <\infty . \end{aligned}$$

(2.18)

Proof

Arbitrarily fix $f,\,g$ satisfying (2.18). By (2.13) and (2.18),

$$\begin{aligned} \{\langle f,\,\Lambda _mD_{a^j}\phi _l\rangle \}_{m,j\in {\mathbb {Z}},\,1\le l\le L},\,\{\langle g,\,\Lambda _mD_{a^j}\psi _l\rangle \}_{m,j\in {\mathbb {Z}},\,1\le l\le L}\in l^2(\{1,\,2,\,\ldots ,\,L\}\times {{\mathbb {Z}}}^2).\nonumber \\ \end{aligned}$$

(2.19)

So the left-hand side of (2.17) is well defined, and

$$\begin{aligned}{} & {} \sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\,\Lambda _mD_{a^j}\phi _l\rangle _{L^2({{\mathbb {R}}_+})}\langle \Lambda _mD_{a^j}\psi _l,\,g \rangle _{L^2({\mathbb {R}}_+)}\nonumber \\= & {} \sum \limits _{l=1}^{L}\sum \limits _{{r\in {\mathbb {N}}}_q}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\,\Lambda _mD_{a^{jq+r}}\phi _l\rangle _{L^2({{\mathbb {R}}_+})}\langle \Lambda _mD_{a^{jq+r}}\psi _l,\,g \rangle _{L^2({\mathbb {R}}_+)}. \end{aligned}$$

(2.20)

Applying Lemma 2.2 to (2.20) gives

$$\begin{aligned}{} & {} \sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\,\Lambda _mD_{a^j}\phi _l\rangle _{L^2({{\mathbb {R}}_+})}\langle \Lambda _mD_{a^j}\psi _l,\,g \rangle _{L^2({\mathbb {R}}_+)}\nonumber \\= & {} \sum \limits _{l=1}^{L}\sum \limits _{r\in {{\mathbb {N}}}_q}\sum \limits _{m,j\in {\mathbb {Z}}}\left\langle \Gamma f(x,\,\xi ),\,e_{m,j}(x,\,\xi )\left( a^{\frac{r}{2}}b^{\frac{s}{2}} \Theta _{\beta }\phi _l(a^{r}b^{s}x,\,\xi )\right) _{s\in {{\mathbb {N}}}_p}\right\rangle _{L^2([1,\,b)\times [0,\,1))}\times \nonumber \\{} & {} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\langle e_{m,j}(x,\,\xi )\left( a^{\frac{r}{2}}b^{\frac{s}{2}}\Theta _{\beta }\psi _l(a^{r}b^{s}x,\,\xi )\right) _{s\in {{\mathbb {N}}}_p},\,\Gamma g(x,\,\xi )\right\rangle _{L^2([1,\,b)\times [0,\,1))}\nonumber \\= & {} \sum \limits _{l=1}^{L}\sum \limits _{r\in {{\mathbb {N}}}_q}\left\langle \left( \overline{\Upsilon {\phi _l}(x,\,\xi )}\Gamma f(x,\,\xi )\right) _{r},\,e_{m,j}(x,\,\xi )\right\rangle _{L^2([1,\,b)\times [0,1))}\times \nonumber \\{} & {} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\langle e_{m,j}(x,\,\xi ),\,\left( \overline{\Upsilon {\psi _l}(x,\,\xi )}\Gamma g(x,\,\xi )\right) _{r}\right\rangle _{L^2([1,\,b)\times [0,1))}. \end{aligned}$$

(2.21)

Also observe that

$$\begin{aligned} \left( {\overline{\Upsilon {\phi _l}(\cdot ,\,\cdot )}}\Gamma f(\cdot ,\,\cdot )\right) _{r},\,\left( {\overline{\Upsilon {\psi _l}(\cdot ,\,\cdot )}}\Gamma g(\cdot ,\,\cdot )\right) _{r}\in L^2([1,\,b)\times [0,\,1)){ \text{ for } \text{ each } }r\in {{\mathbb {N}}}_q \end{aligned}$$

by (2.18). It follows that

$$\begin{aligned}{} & {} \sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\,\Lambda _mD_{a^j}\phi _l\rangle _{L^2({{\mathbb {R}}_+})}\langle \Lambda _mD_{a^j}\psi _l,\,g \rangle _{L^2({\mathbb {R}}_+)}\\= & {} \sum \limits _{l=1}^{L}\sum \limits _{r\in {{\mathbb {N}}}_q}\left\langle \left( \overline{\Upsilon {\phi _l}}\Gamma f\right) _{r},\,\left( \overline{\Upsilon {\psi _l}}\Gamma g\right) _{r}\right\rangle _{L^2([1,\,b)\times [0,1))}\\= & {} \left\langle \overline{{\varvec{\Psi }}^*{\varvec{\Phi }}}\Gamma f,\,\Gamma g\right\rangle _{L^2([1,\,b)\times [0,1),\,{{\mathbb {C}}}^{p})}. \end{aligned}$$

Therefore, (2.17) holds. The proof is completed. $\square$

The following two lemmas are repeated from ([24], Lemmas 4.2, 4.3). The first one is a variation of ([15], Corollary 2.4), and the second one is obtained by an application of the spectral theorem of self-adjoint matrices (see [8], p.978).

Lemma 2.7

An arbitrary $\mu \times \nu$ matrix-valued measurable function ${{\mathcal {A}}}(\cdot )$ on a measurable set E in ${{\mathbb {R}}}^d$ must have the form

$$\begin{aligned} {{\mathcal {A}}}(\cdot )=U(\cdot )\left( \begin{array}{cc} \Lambda (\cdot ) &{} 0 \\ 0 &{} 0 \\ \end{array} \right) V(\cdot ){ \text{ a.e. } \text{ on } }E, \end{aligned}$$

where $U(\cdot )$ and $V(\cdot )$ are $\mu \times \mu$ and $\nu \times \nu$ unitary matrix-valued measurable functions on E, respectively, and $\Lambda (\cdot )$ is a $\min (\mu ,\,\nu )\times \min (\mu ,\,\nu )$ diagonal matrix-valued measurable function on E.

Lemma 2.8

Let ${{\mathcal {A}}}(\cdot )$ be an $m\times n$ matrix-valued measurable function defined on a measurable set E. Then, the orthogonal projection operator $P_{ker({\mathcal {A}}(\cdot ))}$ from ${\mathbb {C}}^n$ onto the kernel space $ker({\mathcal {A}}(\cdot ))$ of ${{\mathcal {A}}}(\cdot )$ is measurable on E, and

$$\begin{aligned} P_{ker({\mathcal {A}}(\cdot ))}=\lim \limits _{n\rightarrow \infty } exp(-n{{\mathcal {A}}^*(\cdot )A(\cdot )}). \end{aligned}$$

The following lemma characterizes the Riesz property of ${\mathcal{M}\mathcal{D}}$-Bessel sequences.

Lemma 2.9

Let $a,\,b$ and $\Psi$ be as in the general setup. Suppose ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^2({\mathbb {R}}_+)$. Then, the following are equivalent:

(i) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ has Riesz property.

(ii) $\textrm{rank}({\varvec{\Psi }})(x,\,\xi )=Lq{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^\frac{1}{p})\times [0,\,1).$

Proof

Since ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^2({\mathbb {R}}_+)$,

$$\begin{aligned} \sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}c_{l,m,j}\Lambda _mD_{a^j}\psi _l \end{aligned}$$

is well defined and converges unconditionally for $c\in l^2(\{1,\,2,\ldots ,\,L\}\times {{\mathbb {Z}}^2})$. Thus,

$$\begin{aligned} \sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}c_{l,m,j}\Lambda _mD_{a^j}\psi _l=0 \end{aligned}$$

(2.22)

if and only if

$$\begin{aligned} \sum \limits _{l=1}^{L}\sum \limits _{r\in {{\mathbb {N}}}_q}\sum \limits _{m,j\in {\mathbb {Z}}}c_{l,m,qj+r}\Lambda _mD_{a^{qj+r}}\psi _l=0 \end{aligned}$$

(2.23)

for some $c\in l^2(\{1,\,2,\ldots ,\,L\}\times {{\mathbb {Z}}^2})$. By Lemma 2.2 (v), (2.23) is equivalent to

$$\begin{aligned} \sum \limits _{l=1}^{L}\sum \limits _{r\in {{\mathbb {N}}}_q} \sum \limits _{m,j\in {\mathbb {Z}}}c_{l,m,qj+r}\Gamma \Lambda _mD_{a^{qj+r}}\psi _l=0, \end{aligned}$$

equivalently,

$$\begin{aligned} \sum \limits _{l=1}^{L}\sum \limits _{r\in {{\mathbb {N}}}_q} \left( \sum \limits _{m,j\in {\mathbb {Z}}}c_{l,m,qj+r}e_{m,j}(x,\,\xi )\right) \left( a^{\frac{r}{2}}b^{\frac{s}{2}}\Theta _{\beta }\psi _l(a^rb^sx,\xi )\right) _{s\in {\mathbb {N}}_p}=0, \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,b)\times [0,\,1)$, i.e.,

$$\begin{aligned} {\varvec{\Psi }}^*(x,\,\xi )C(x,\,\xi )=0{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,b)\times [0,\,1), \end{aligned}$$

(2.24)

where

$$\begin{aligned} C(x,\xi )=\left( C^l(x,\xi )\right) _{1\le l\le L}{ \text{ and } } C^l(x,\xi )=\left( {\sum \limits _{m,j\in {\mathbb {Z}}}\overline{c_{l,m,jq+r}e_{m,j}(x,\xi )}} \right) _{r\in {\mathbb {N}}_q}. \end{aligned}$$

(2.25)

Thus, (i) is equivalent to $C(x,\xi )=0$ is a unique solution to (2.24) in $L^2([1,b)\times [0,1),\,{{\mathbb {C}}}^{Lq})$. Observe

$$\begin{aligned} \textrm{rank}({\varvec{\Psi }})(x,\,\xi )=Lq{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^\frac{1}{p})\times [0,\,1) \end{aligned}$$

if and only if

$$\begin{aligned} \textrm{rank}({\varvec{\Psi }})(x,\,\xi )=Lq{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,b)\times [0,\,1) \end{aligned}$$

(2.26)

by Lemma 2.3 (ii) and the fact that $b=a^{\frac{q}{p}}$. So we only need to prove that $C(x,\xi )=0$ being a unique solution to (2.24) in $L^2([1,b)\times [0,1),\,{{\mathbb {C}}}^{Lq})$ if and only if (2.26) holds.

The sufficiency is obvious. Now, we prove the necessity by proving that (2.24) has a nonzero solution in $L^2([1,\,b)\times [0,\,1),\,{{\mathbb {C}}}^{Lq})$ if $\textrm{rank}({\varvec{\Psi }})(\cdot ,\,\cdot )<Lq$ on some $E\subset [1,\,b)\times [0,\,1)$ with $|E|>0$. Let $\{e_k:\,k\in {{\mathbb {N}}}_{Lq}\}$ be the canonical orthonormal basis for ${{\mathbb {C}}}^{Lq}$. If $P_{ker(\Psi )(x,\,\xi )}e_{n}=0$ for $n\in {{\mathbb {N}}}_{Lq}$ and $(x,\,\xi )\in E$, then $\textrm{rank}({\varvec{\Psi }})^*(x,\,\xi )=Lq$ for $(x,\,\xi )\in E$, this is a contradiction. Therefore, there exists $n_0\in {{\mathbb {N}}}_{Lq}$ and $E'\subset E$ with $|E'|>0$ such that

$$\begin{aligned} P_{ker({\varvec{\Psi }}(x,\,\xi ))}e_{n_0}\ne 0{ \text{ for } }(x,\,\xi )\in E'. \end{aligned}$$

Take $c\in l^2(\{1,\,2,\ldots ,\,L\}\times {{\mathbb {Z}}^2})$ by $C(x,\,\xi )=\left\{ \begin{array}{ll} P_{ker({\varvec{\Psi }}(x,\,\xi ))}e_{n_0} &{}\quad \hbox {if} \,\,(x,\xi )\in E'; \\ 0 &{}\quad \hbox {otherwise} \end{array} \right.$ for $(x,\,\xi )\in [1,\,b)\times [0,\,1)$. Then, $C(x,\,\xi )$ is measurable on $[1,\,b)\times [0,\,1)$ by Lemma 2.8 and

$$\begin{aligned} \Vert C(x,\xi )\Vert _{{{\mathbb {C}}}^{Lq}}\le 1{ \text{ for } \text{ a.e. } }(x,\xi )\in [1,\,b)\times [0,\,1). \end{aligned}$$

Thus, $C\in L^2([1,b)\times [0,1),\,{{\mathbb {C}}}^{Lq})$ and C is a nonzero solution to (2.24) in $L^2([1,b)\times [0,1),\,{{\mathbb {C}}}^{Lq})$. The proof is completed. $\square$

By the same procedures as in ([24], Theorem 4.1) and Lemma 2.5, we have the following completeness and frame characterization of an ${{\mathcal{M}\mathcal{D}}}$-system ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$.

Theorem 2.1

Let $a,\,b$ and $\Psi$ be as in the general setup. Then, (i) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is complete in $L^{2}({\mathbb {R}}_+)$ if and only if

$$\begin{aligned} \textrm{rank}({\varvec{\Psi }}(x,\,\xi ))=p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

(ii) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^{2}({\mathbb {R}}_+)$ with bound B if and only if

$$\begin{aligned} {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\le BI_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

(2.27)

(iii) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^{2}({\mathbb {R}}_+)$ with bounds A and B if and only if

$$\begin{aligned} a^{\frac{q-1}{p}}AI_p\le {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\le BI_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1).\nonumber \\ \end{aligned}$$

(2.28)

Remark 2.3

(i) Observe that ${\varvec{\Psi }}(x,\,\xi )$ is a $Lq\times p$ matrix. Theorem 2.1 (i) shows that the inequality

$$\begin{aligned} p\le Lq{ \text{( } \text{ equlvalently, } }\log _{b}a\le L) \end{aligned}$$

is necessary for the existence of complete ${\mathcal{M}\mathcal{D}}$-systems in $L^2({\mathbb {R}}_+)$. (ii) By Theorem 2.1 (iii), the lower frame bound A and upper frame bound B of a frame ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ for $L^2({\mathbb {R}}_+)$ always satisfy that

$$\begin{aligned} a^{\frac{q-1}{p}}A\le B. \end{aligned}$$

In particular, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a tight frame for $L^2({\mathbb {R}}_+)$ only if $q=1$.

Observe that a sequence is a Riesz basis if and only if it is a complete Riesz sequence, and if and only if it is a frame with Riesz property, and that an orthonormal basis (sequence) is exactly a Riesz basis (sequence) with bound 1. By Lemma 2.9 and Theorem 2.1, we have the following theorem.

Theorem 2.2

Let a, b and $\Psi$ be as in the general setup. Suppose ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^2({\mathbb {R}}_+)$. Then, the following are equivalent:

(i) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Riesz basis (an orthonormal basis) for $L^2({\mathbb {R}}_+)$.

(ii) $\log _ba=L$, and ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Riesz sequence (an orthonormal sequence) in $L^2({\mathbb {R}}_+)$.

(iii) $\log _ba=L$, and ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame (a Parseval frame) for $L^2({\mathbb {R}}_+)$.

By Remark 2.3, if a and b are as in the general setup, $p\le Lq$ is necessary for the existence of complete ${\mathcal{M}\mathcal{D}}$-system of the form ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$. In view of Lemma 2.7, the following theorem provides a parametric expression of all complete ${\mathcal{M}\mathcal{D}}$-systems and all ${\mathcal{M}\mathcal{D}}$-frames in $L^2({\mathbb {R}}_+)$ of the form ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$.

Theorem 2.3

Let $a,\,b$ be as in the general setup, and $\log _ba\le L$. Assume that $\lambda _s(x,\,\xi )\in L^2([1,\,a^{\frac{1}{p}})\times [0,\,1))$ with $s\in {\mathbb {N}}_{p}$. Define

$$\begin{aligned} \Lambda (x,\,\xi )=\textrm{diag}\left( \lambda _0(x,\,\xi ),\lambda _1(x,\,\xi ),\cdots ,\lambda _{p-1}(x,\,\xi )\right) \end{aligned}$$

and $\Psi =\{\psi _1,\,\psi _2,\,\ldots ,\,\psi _L\}$ by an $Lq\times p$ matrix-valued function

$$\begin{aligned} {\varvec{\Psi }}(x,\,\xi )=U(x,\,\xi )\left( \begin{array}{c} \Lambda (x,\,\xi ) \\ 0 \\ \end{array} \right) V(x,\,\xi ) \end{aligned}$$

for $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$, where $U(x,\,\xi )$ and $V(x,\,\xi )$ are $Lq\times Lq$ and $p\times p$ unitary matrix-valued measurable function defined on $[1,\,a^{\frac{1}{p}})\times [0,\,1)$, respectively. Then,

(i) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is complete in $L^{2}({\mathbb {R}}_+)$ if and only if

$$\begin{aligned} \lambda _0(x,\,\xi )\lambda _1(x,\,\xi )\cdots \lambda _{p-1}(x,\,\xi )\ne 0{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

(ii) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^{2}({\mathbb {R}}_+)$ if and only if

$$\begin{aligned} \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|<\infty . \end{aligned}$$

In this case, the Bessel bound is

$$\begin{aligned} \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|^2. \end{aligned}$$

(iii) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^{2}({\mathbb {R}}_+)$ if and only if

$$\begin{aligned} 0<\mathop {\min }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{essinf}}\limits _{(x,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\le \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|<\infty . \end{aligned}$$

In this case, the frame bounds are

$$\begin{aligned} a^{-\frac{q-1}{p}}\left( \mathop {\min }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{essinf}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\right) ^2{\text{ and }} \left( \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\right) ^2. \end{aligned}$$

In particular, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Riesz basis for $L^{2}({\mathbb {R}}_+)$ if $\log _{b}a=L$ in addition.

(iv) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a tight frame for $L^{2}({\mathbb {R}}_+)$ if and only if $q=1$ and there exists a constant $A>0$ such that

$$\begin{aligned} |\lambda _s(x,\,\xi )|=\sqrt{A}{ \text{ for } }s\in {{\mathbb {N}}}_p{ \text{ and } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

In this case, the frame bound is A.

Proof

(i) and (iv) follow from Theorem 2.1. Next we prove (ii) and (iii). By a direct computation,

$$\begin{aligned}{} & {} {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )=V^*(x,\,\xi )\textrm{diag}(|\lambda _0(x,\,\xi )|^2,\\{} & {} \quad \,|\lambda _1(x,\,\xi )|^2,\,\cdots ,\,|\lambda _{p-1}(x,\,\xi )|^2)V(x,\,\xi ) \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1).$ Observe that $V(x,\,\xi )$ is unitary. It follows that

$$\begin{aligned} \inf \limits _{c\in {{\mathbb {C}}}^p,\,\Vert c\Vert =1}\langle {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )c,\,c \rangle =\min \limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|^2, \end{aligned}$$

and

$$\begin{aligned} \sup \limits _{c\in {{\mathbb {C}}}^p,\,\Vert c\Vert =1}\langle {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )c,\,c \rangle =\max \limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|^2 \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1).$ Again by Theorem 2.1, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Bessel sequence in $L^2({\mathbb {R}}_+)$ if and only if

$$\begin{aligned} \mathop {\textrm{esssup}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}\mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|^2<\infty , \end{aligned}$$

and the Bessel bound is

$$\begin{aligned} \mathop {\textrm{esssup}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}\mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|^2 \end{aligned}$$

in this case; ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^2({\mathbb {R}}_+)$ if and only if

$$\begin{aligned} 0<\mathop {\min }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{essinf}}\limits _{(x,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\le \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|<\infty , \end{aligned}$$

and the frame bounds are

$$\begin{aligned} a^{-\frac{q-1}{p}}\left( \mathop {\min }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{essinf}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\right) ^2{ \text{ and } }\left( \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\right) ^2 \end{aligned}$$

in this case. Next we prove

$$\begin{aligned} \left( \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\right) ^2=\left( \mathop {\textrm{esssup}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}\mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|\right) ^2 \nonumber \\ \end{aligned}$$

(2.29)

and

$$\begin{aligned} \left( \mathop {\min }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{essinf}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\right) ^2=\left( \mathop {\textrm{essinf}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}\mathop {\min }\limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|\right) ^2 \nonumber \\ \end{aligned}$$

(2.30)

to finish the proof. Since

$$\begin{aligned} \mathop {\textrm{esssup}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\le \mathop {\textrm{esssup}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}\mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|, \end{aligned}$$

we have

$$\begin{aligned} \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|{\le }\mathop {\textrm{esssup}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}\mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|. \end{aligned}$$

On the other hand, since

$$\begin{aligned} \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\ge \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}}){\times }[0,\,1), \end{aligned}$$

we have

$$\begin{aligned} \mathop {\max }{s\in {{\mathbb {N}}}_p}\mathop {\textrm{esssup}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\ge \mathop {\textrm{esssup}}\limits _{(x,\xi )\in [1,a^{\frac{1}{p}})\times [0,\,1)}\mathop {\max }\limits _{s\in {{\mathbb {N}}}_p}|\lambda _s(x,\,\xi )|. \end{aligned}$$

Hence, (2.29) holds. (2.30) may be proved similarly. And ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Riesz basis for $L^{2}({\mathbb {R}}_+)$ if $\log _{b}a=L$ in addition by Theorem 2.2. The proof is completed. $\square$

Collecting Theorem 2.3 and Remark 2.3, we obtain the following density theorem.

Theorem 2.4

Let $a,\,b$ be as in the general setup. Then, the following are equivalent: (i) $\log _ba\le L$. (ii) There exists a sequence $\Psi =\{\psi _1,\,\psi _2,\,\ldots ,\,\psi _L\}$ in $L^2({\mathbb {R}}_+)$ such that ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is complete in $L^2({\mathbb {R}}_+)$. (iii) There exists a sequence $\Psi =\{\psi _1,\,\psi _2,\,\ldots ,\,\psi _L\}$ in $L^2({\mathbb {R}}_+)$ such that ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^2({\mathbb {R}}_+)$.

3 ${\mathcal{M}\mathcal{D}}$-dual characterization

This section focuses on proving $\log _ba\in {\mathbb {N}}$ being the necessary and sufficient condition for an ${\mathcal{M}\mathcal{D}}$-frame to admit ${\mathcal{M}\mathcal{D}}$-duals, and a parametric expression of ${\mathcal{M}\mathcal{D}}$-duals. We first give some lemmas for later use. The first two represent the composition of the mixed frame operator $S_{\Psi ,\,\Phi }$ and $\Gamma$ ($\Upsilon$, respectively) using $\Theta _\beta$-transform matrices.

Lemma 3.1

Let $a,\,b$ and $\Psi$ be as in the general setup, and $\Phi =\{\phi _1,\,\phi _2,\,\ldots ,\,\phi _L\}$ be a sequence in $L^2({\mathbb {R}}_+)$. Suppose ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ and ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ are both Bessel sequences in $L^{2}({\mathbb {R}}_+)$. Then,

$$\begin{aligned} \Gamma S_{\Psi ,\,\Phi }f(b^jx,\xi )=b^j\overline{{\varvec{\Psi }}^{*}(b^jx,\,\xi ){\varvec{\Phi }}(b^jx,\,\xi )}\Gamma f(b^jx,\,\xi ) \end{aligned}$$

(3.1)

for $f\in L^2({\mathbb {R}}_+)$, $j\in {\mathbb {Z}}$ and a.e. $(x,\,\xi )\in [1,\,b)\times [0,\,1)$.

Proof

Fix an arbitrary $f\in L^2({\mathbb {R}}_+)$. First we prove

$$\begin{aligned} \Gamma S_{\Psi ,\,\Phi }f(x,\xi )=\overline{{\varvec{\Psi }}^{*}(x,\,\xi )\mathbf{\Phi }(x,\,\xi )}\Gamma f(x,\,\xi ){ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,b)\times [0,\,1).\nonumber \\ \end{aligned}$$

(3.2)

Since ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ and ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ are both Bessel sequences in $L^{2}({\mathbb {R}}_+)$, all entries of ${\varvec{\Psi }}(x,\,\xi ),\,\mathbf{\Phi }(x,\,\xi )$ belong to $L^{\infty }({{\mathbb {R}}}_+\times {{\mathbb {R}}})$ by Remark 2.2. This implies that (2.18) holds for $f,\,g\in L^2({{\mathbb {R}}}_+)$ by Lemma 2.2 (v). Thus, (2.17) holds, equivalently,

$$\begin{aligned} \langle S_{\Psi ,\,\Phi }f,\,g\rangle =\langle \overline{{\varvec{\Psi }}^{*}{\varvec{\Phi }}}\Gamma f,\,\Gamma g\rangle _{L^2([1,\,b)\times [0,\,1),\,{{\mathbb {C}}}^p)}{ \text{ for } }f,\,g\in L^2({{\mathbb {R}}}_+). \end{aligned}$$

On the other hand,

$$\begin{aligned} \langle S_{\Psi ,\,\Phi }f,\,g\rangle =\langle \Gamma S_{\Psi ,\Phi }f,\,\Gamma g \rangle _{L^2([1,\,b)\times [0,\,1),\,{{\mathbb {C}}}^p)}{ \text{ for } }f,\,g\in L^2({{\mathbb {R}}}_+) \end{aligned}$$

by Lemma 2.2 (v). Thus, (3.2) holds.

Next we prove (3.1). Since ${{\mathbb {Z}}}={{\mathbb {N}}}_p+p{\mathbb {Z}}$, (3.1) is equivalent to, given $(s,\,j)\in {{\mathbb {N}}}_p\times {\mathbb {Z}}$ and $f\in L^2({{\mathbb {R}}}_+)$,

$$\begin{aligned} \Gamma S_{\Psi ,\,\Phi }f(b^{jp+s}x,\,\xi )=b^{jp+s}\overline{{\varvec{\Psi }}^{*}(b^{jp+s}x,\xi ){\varvec{\Phi }}(b^{jp+s}x,\xi )}\Gamma f(b^{jp+s}x,\,\xi )\nonumber \\ \end{aligned}$$

(3.3)

for a.e. $(x,\,\xi )\in [1,\,b)\times [0,\,1)$. Now we prove (3.3). By Lemma 2.3 (iii), we have

$$\begin{aligned} {\varvec{\Psi }}^{*}(x,\,\xi ){\varvec{\Phi }}(x,\,\xi )=b^{jp+s}{{\mathcal {M}}}_s(\xi ) {\varvec{\Psi }}^{*}(b^{jp+s}x,\,\xi ){\varvec{\Phi }}(b^{jp+s}x,\,\xi ){{\mathcal {M}}}_s^*(\xi ), \end{aligned}$$

(3.4)

and

$$\begin{aligned} \Gamma f(x,\,\xi )=b^{\frac{jp+s}{2}}e^{-2\pi ij\xi }\overline{{{\mathcal {M}}}_s(\xi )}\Gamma f(b^{jp+s}x,\,\xi ) \end{aligned}$$

(3.5)

for a.e. $(x,\,\xi )\in [1,\,b)\times [0,\,1)$. On the other hand,

$$\begin{aligned}{} & {} \Gamma S_{\Psi ,\,\Phi }f(b^{jp+s}x,\,\xi ) = b^{-\frac{jp+s}{2}}e^{2\pi i j\xi }\overline{{{\mathcal {M}}^*}_s(\xi )}\Gamma S_{\Psi }f(x,\,\xi )\nonumber \\{} & {} \quad =b^{-\frac{jp+s}{2}}e^{2\pi i j\xi }\overline{{{\mathcal {M}}^*}_s(\xi )}\overline{{\varvec{\Psi }}^{*}(x,\,\xi ){\varvec{\Phi }} (x,\,\xi )}\Gamma f(x,\,\xi ) \end{aligned}$$

(3.6)

for a.e. $(x,\,\xi )\in [1,\,b)\times [0,\,1)$ by Lemma 2.3 (iii) and (3.2). Substituting (3.4) and (3.5) into (3.6) gives (3.3). The proof is completed. $\square$

Lemma 3.2

Let $a,\,b$ and $\Psi$ be as in the general setup, $\log _ba\in {\mathbb {N}}$ and $\Phi =\{\phi _1,\,\phi _2,\,\ldots ,\,\phi _L\}$ be a sequence in $L^2({\mathbb {R}}_+)$. Suppose ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ and ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ are both Bessel sequences in $L^{2}({\mathbb {R}}_+)$. Then,

$$\begin{aligned} \Upsilon S_{\Psi ,\,\Phi }f(x,\xi )= & {} \Upsilon f(x,\,\xi ){\varvec{\Phi }}^{*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi ){ \text{ for } }f\nonumber \\\in & {} L^2({{\mathbb {R}}}_+){ \text{ and } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

(3.7)

Proof

Suppose $\log _ba\in {\mathbb {N}}$. Then, $q=1$ and ${\mathbb {N}}_q=\{0\}$. And by Lemma 3.1, we have

$$\begin{aligned} \Upsilon S_{\Psi ,\,\Phi }f(x,\xi )= & {} \left( \Gamma S_{\Psi ,\,\Phi }f(x,\xi )\right) ^{t}\\ {}= & {} \left( \Gamma f(x,\,\xi )\right) ^{t}{\varvec{\Phi }}^{*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\\= & {} \Upsilon f(x,\xi ){\varvec{\Phi }}^{*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi ) \end{aligned}$$

for $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1).$ The proof is completed. $\square$

The following lemma shows that $\log _ba \in {\mathbb {N}}$ is the necessary and sufficient condition for dilation-and-modulation operation and frame operator to commute.

Lemma 3.3

Let $a,\,b$ and $\Psi$ be as in the general setup, and $\psi _1,\,\psi _2,\ldots ,\psi _L$ be not all zero. Suppose ${{\mathcal{M}\mathcal{D}}}(\Psi , \,a,\,b)$ is a Bessel sequence in $L^2({\mathbb {R}}_+)$. Then,

$$\begin{aligned} \Lambda _mD_{a^j}S_{\Psi }=S_{\Psi }\Lambda _mD_{a^j}{ \text{ for } }m,\,j\in {\mathbb {Z}} \end{aligned}$$

(3.8)

if and only if $\log _ba \in {\mathbb {N}}$. In particular, if ${{\mathcal{M}\mathcal{D}}}(\Psi , \,a,\,b)$ is a frame for $L^2({\mathbb {R}}_+)$, then

$$\begin{aligned} \Lambda _mD_{a^j}S_{\Psi }^{-1}=S_{\Psi }^{-1}\Lambda _mD_{a^j}{ \text{ for } }m,\,j\in {\mathbb {Z}}. \end{aligned}$$

Proof

Sufficiency. Suppose $\log _ba\in {{\mathbb {N}}}$. Then, $a=b^p$ for some positive integer p. Since $\Lambda _m$ is b-dilation periodic, for arbitrary $m,\,j\in {\mathbb {Z}}$ and $f\in L^2({{\mathbb {R}}}_+)$,

$$\begin{aligned} \Lambda _mD_{a^j}f(\cdot )= & {} a^{\frac{j}{2}}\Lambda _m(b^{jp}\cdot )f(a^j\cdot )\\= & {} a^{\frac{j}{2}}\Lambda _m(a^j\cdot )f(a^j\cdot )\\= & {} D_{a^j}\Lambda _mf(\cdot ) \end{aligned}$$

a.e. on ${{\mathbb {R}}}_+$. Thus,

$$\begin{aligned} \Lambda _mD_{a^j}=D_{a^j}\Lambda _m{ \text{ for } }m,\,j\in {\mathbb {Z}}. \end{aligned}$$

(3.9)

Arbitrarily fix $m_0$, $j_0\in {\mathbb {Z}}$ and $f\in L^2({\mathbb {R}}_+)$. By (3.9), we have

$$\begin{aligned} S_{\Psi }\Lambda _{m_0}D_{a^{j_0}}f&= \sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}} \langle \Lambda _{m_0}D_{a^{j_0}}f,\,\Lambda _{m}D_{a^{j}}\phi _l\rangle \Lambda _{m}D_{a^{j}}\psi _l\\&=\sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\,D_{a^{-j_0}} \Lambda _{m-m_0}D_{a^{j}}\psi _l\rangle \Lambda _{m}D_{a^{j}}\psi _l\\&=\sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\, \Lambda _{m-m_0}D_{a^{j-j_0}}\psi _l\rangle \Lambda _{m}D_{a^{j}}\psi _l\\&=\Lambda _{m_0}D_{a^{j_0}}\sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle f,\, \Lambda _{m}D_{a^{j}}\psi _l\rangle \Lambda _{m}D_{a^{j}}\psi _l\\&=\Lambda _{m_0}D_{a^{j_0}}S_{\Psi }f. \end{aligned}$$

This leads to (3.8) by the arbitrariness of $m_0$, $j_0$ and f.

Necessity. By contradiction. Suppose (3.8) holds, but $\log _{b}a\notin {\mathbb {N}}$. Then, $q>1$ and $1\in {{\mathbb {N}}}_q$. Arbitrarily fix $m,\,j\in {\mathbb {Z}}$ and $f\in L^2({{\mathbb {R}}}_+)$. By Lemmas 3.1, 2.2 (iv) and 2.3 (i), we have

$$\begin{aligned}{} & {} \Gamma S_{\Psi }\Lambda _mD_{a^{jq+1}}f(x,\,\xi )=\overline{{\varvec{\Psi }^*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )}\Gamma \Lambda _mD_{a^{jq+1}}f(x,\,\xi )\nonumber \\{} & {} \quad =a^{\frac{1}{2}}e_{m,\,j}(x,\,\xi )\overline{{\varvec{\Psi }^*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )}\Gamma f(ax,\,\xi )\nonumber \\{} & {} \quad =a^{\frac{3}{2}}e_{m,\,j}(x,\,\xi )\overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}\Gamma f(ax,\,\xi ) \end{aligned}$$

(3.10)

for a.e. $(x,\,\xi )\in [1,\,b)\times [0,\,1)$. On the other hand, by Lemma 2.2 (iv),

$$\begin{aligned} \Gamma \Lambda _mD_{a^{jq+1}}S_{\Psi }f(x,\,\xi )= & {} a^{\frac{1}{2}}e_{m,\,j}(x,\xi )\Gamma S_{\Psi }f(ax,\,\xi ) \end{aligned}$$

(3.11)

for a.e. $(x,\,\xi )\in [1,\,b)\times [0,\,1)$. By (3.8), we have

$$\begin{aligned} \Gamma \Lambda _mD_{a^{jq+1}}S_{\Psi }=\Gamma S_{\Psi }\Lambda _mD_{a^{jq+1}}. \end{aligned}$$

This implies that

$$\begin{aligned} a\overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}\Gamma f(ax,\,\xi )= & {} \Gamma S_{\Psi }f(ax,\,\xi ){ \text{ for } \text{ a.e. } }(x,\,\xi )\nonumber \\\in & {} [1,\,b)\times [0,\,1) \end{aligned}$$

(3.12)

by (3.10) and (3.11). Since $\psi _1,\,\psi _2,\,\ldots ,\psi _L$ are not all zero, ${\varvec{\Psi }}(x,\,\xi )$ is a nonzero matrix-valued function on $[1,\,a^{\frac{1}{p}})\times [0,\,1)$ by Remark 1.1, equivalently, there exists $E_0\subset [1,\,a^{\frac{1}{p}})\times [0,\,1)$ with $|E_0|>0$ such that

$$\begin{aligned} {\varvec{\Psi }^*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\ne 0{ \text{ for } }(x,\,\xi )\in E_0. \end{aligned}$$

(3.13)

Let $E=\left\{ (ax,\,\xi ):\,(x,\,\xi )\in E_0\right\}$. Thus, $E\subset [a,\,a^{\frac{1}{p}+1})\times [0,\,1)$ with $|E|>0$ satisfies

$$\begin{aligned} {\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )\ne 0{ \text{ for } }(ax,\,\xi )\in E \end{aligned}$$

(3.14)

by (3.13) and Lemma 2.3 (i). Since

$$\begin{aligned} \left\{ b^{pk+s}[1,\,b):\,k\in {{\mathbb {Z}}},\,s\in {{\mathbb {N}}}_p\right\} \end{aligned}$$

is a partition of ${\mathbb {R}}_+$, there exist $k\in {\mathbb {Z}}$ and $s\in {{\mathbb {N}}}_p$ such that

$$\begin{aligned} \left| E\cap \left( [b^{kp+s},\,b^{kp+s+1})\times [0,\,1)\right) \right| >0. \end{aligned}$$

So, by Lemma 3.1,

$$\begin{aligned} \Gamma S_{\Psi }f(ax,\,\xi )= & {} b^{kp+s}\overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}\Gamma f(ax,\,\xi ) \end{aligned}$$

(3.15)

for a.e. $(ax,\,\xi )\in E\cap \left( [b^{kp+s},\,b^{kp+s+1})\times [0,\,1)\right)$. This implies that

$$\begin{aligned} a\overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}\Gamma f(ax,\,\xi )= & {} b^{kp+s}\overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}\Gamma f(ax,\,\xi ){ \text{ for } }f\nonumber \\{} & {} \in L^2({{\mathbb {R}}}_+) \end{aligned}$$

(3.16)

for a.e. $(ax,\,\xi )\in E\cap ([b^{kp+s},\,b^{kp+s+1})\times [0,\,1))$ by (3.12). Since (3.14) holds, there exists $s_0\in {{\mathbb {N}}}_p$ and $E_1\subset E\cap \left( [b^{kp+s},\,b^{kp+s+1})\times [0,\,1)\right)$ with $|E_1|>0$ such that the $s_0$th column $\left( \overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}\right) _{s_0}$ of $\overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}$ satisfies

$$\begin{aligned}{} & {} \left( \overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}\right) _{s_0}\nonumber \\{} & {} \ne 0{ \text{ for } \text{ a.e. } }(ax,\,\xi )\in E_1. \end{aligned}$$

(3.17)

Let $\{e_k:\,k\in {\mathbb {N}}_p\}$ be the canonical orthonormal basis for ${{\mathbb {C}}}^p$. Take f in (3.16) by

$$\begin{aligned} \Gamma f(x,\,\xi )=\chi _{_{F}}(x,\,\xi )e_{s_0}{ \text{ for } }(x,\,\xi )\in [b^{kp+s},\,b^{kp+s+1})\times [0,\,1). \end{aligned}$$

Thus, $f\in L^2({\mathbb {R}}_+)$ by Lemma 2.2 (v), and

$$\begin{aligned} \Gamma f(ax,\,\xi )=e_{s_0}{ \text{ for } }(ax,\,\xi )\in E_1. \end{aligned}$$

And then, by (3.16),

$$\begin{aligned} a\left( \overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}\right) _{s_0} =b^{kp+s}\left( \overline{{\varvec{\Psi }^*}(ax,\,\xi ){\varvec{\Psi }}(ax,\,\xi )}\right) _{s_0}{ \text{ for } \text{ a.e. } }(ax,\,\xi )\in E_1. \end{aligned}$$

This implies that

$$\begin{aligned} a=b^{kp+s} \end{aligned}$$

by (3.17), equivalently,

$$\begin{aligned} \log _ba=kp+s\in {\mathbb {Z}}. \end{aligned}$$

Thus, $\log _ba\in {{\mathbb {N}}}$ due to $\log _ba>0$. It is a contradiction. The other part follows from the fact that $S_{\Psi }$ is invertible if ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^2({{\mathbb {R}}}_+)$. The proof is completed. $\square$

The following theorem gives a characterization of the existence of ${\mathcal{M}\mathcal{D}}$-duals for a given ${\mathcal{M}\mathcal{D}}$-frame.

Theorem 3.1

Let $a,\,b$ and $\Psi$ be as in the general setup. Suppose ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^2({\mathbb {R}}_+)$. Then, there exists a sequence $\Phi =\{\phi _1,\,\phi _2,\,\ldots ,\,\phi _L\}$ in $L^2({\mathbb {R}}_+)$ such that ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ is a dual of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ if and only if $\log _ba\in {{\mathbb {N}}}$.

Proof

Sufficiency. Suppose $\log _ba\in {{\mathbb {N}}}$. Then,

$$\begin{aligned} S_{\Psi }^{-1}\Lambda _mD_{a^j}\psi _l=\Lambda _mD_{a^j}S_{\Psi }^{-1}\psi _l{ \text{ for } }m,\,j\in {{\mathbb {Z}}}{ \text{ and } }1\le l\le L \end{aligned}$$

by Lemma 3.3. Take $\Phi =\{\phi _1,\,\phi _2,\,\ldots ,\,\phi _L\}$ as $\phi _l=S_{\Psi }^{-1}\psi _l$ for $1\le l\le L$. Then, ${{\mathcal{M}\mathcal{D}}}({\Phi },\,a,\,b)$ is a dual of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$.

Necessity. By contradiction. Suppose $\log _ba\notin {{\mathbb {N}}}$ and ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ is a dual of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$. Then, $[a^{\frac{1}{p}},\,a^{\frac{2}{p}})\subset [1,\,b)$ due to $b=a^{\frac{p}{q}}$, and

$$\begin{aligned} \Gamma f(x,\,\xi )=\Gamma S_{\Psi ,\,\Phi }f(x,\,\xi )=\overline{{\varvec{\Psi }}^{*}(x,\,\xi )\mathbf{\Phi }(x,\,\xi )}\Gamma f(x,\,\xi ){ \text{ for } }f\in L^2({\mathbb {R}}_+)\nonumber \\ \end{aligned}$$

(3.18)

for a.e. $(x,\,\xi )\in [1,\,b)\times [0,\,1)$ by Lemma 3.1. Let $\{e_k:\,k\in {{\mathbb {N}}}_p\}$ is the canonical orthonormal basis for ${{\mathbb {C}}}^p$. Fix an arbitrary $k\in {{\mathbb {N}}}_p$. Take f in (3.18) by

$$\begin{aligned} \Gamma f (x,\,\xi )=e_k{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,b)\times [0,\,1). \end{aligned}$$

Then, it is well defined due to $\Gamma$ being a unitary operator from $L^2({\mathbb {R}}_+)$ onto $L^2([1,\,b)\times [0,\,1),\,{{\mathbb {C}}}^p)$ by Lemma 2.2 (v), and (3.18), the kth column of ${\varvec{\Psi }}^{*}(x,\,\xi ){\varvec{\Phi }}(x,\,\xi )$ is $e_k$. Thus,

$$\begin{aligned} {\varvec{\Psi }}^{*}(x,\,\xi ){\varvec{\Phi }}(x,\,\xi )=I_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1, \,b)\times [0,\,1). \end{aligned}$$

(3.19)

By Lemma 2.3 (ii), this implies that

$$\begin{aligned}{} & {} {\varvec{\Psi }}^{*}(a^{\frac{1}{p}}x,\,\xi ){\varvec{\Phi }}(a^{\frac{1}{p}}x,\,\xi )\\= & {} a^{-\frac{1}{p}}{{\mathcal {R}}}^*_{p,\,q}(\xi ){\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Phi }}(x,\,\xi ){{\mathcal {R}}}_{p,\,q}(\xi )\\= & {} a^{-\frac{1}{p}}I_p \end{aligned}$$

for a.e. $(x,\,\xi )\in [1, \,a^{\frac{1}{p}})\times [0,\,1)$, equivalently,

$$\begin{aligned} {\varvec{\Psi }}^{*}(x,\,\xi )\mathbf{\Phi }(x,\,\xi )=a^{-\frac{1}{p}}I_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [a^{\frac{1}{p}},\,a^{\frac{2}{p}})\times [0,\,1). \end{aligned}$$

This contradicts (3.19) by the fact that $b=a^{\frac{q}{p}}$ and $[a^{\frac{1}{p}},\,a^{\frac{2}{p}})\subset [1,\,b)$. The proof is completed. $\square$

Remark 3.1

By the proof of Theorem 3.1, if $\log _{b}a\in {\mathbb {N}}$ (this implies $b=a^{\frac{1}{p}}$), two Bessel sequences ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ and ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ in $L^2({\mathbb {R}}_+)$ form a pair of dual frames for $L^2({\mathbb {R}}_+)$ if and only if

$$\begin{aligned} {\varvec{\Psi }}^{*}(x,\xi ){\varvec{\Phi }}(x,\xi )=I_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

(3.20)

By Theorems 2.4 and 3.1, under the general setup, $\log _ba=p\le L$ is the sufficient and necessary condition for the existence of ${\mathcal{M}\mathcal{D}}$-duals of a given ${\mathcal{M}\mathcal{D}}$-frame. The following theorem presents a parametric expression of all ${\mathcal{M}\mathcal{D}}$-duals of an arbitrarily given ${\mathcal{M}\mathcal{D}}$-frame and shows that the window functions of the canonical dual have minimal norm among all ${\mathcal{M}\mathcal{D}}$-duals.

Theorem 3.2

Let $a,\,b$ and $\Psi$ be as in the general setup, $\log _{b}a=p\le L$, and ${{\mathcal{M}\mathcal{D}}}(\Psi , \,a,\,b)$ be a frame for $L^2({\mathbb {R}}_+)$. Then,

(i) All ${\mathcal{M}\mathcal{D}}$-dual frames ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ are precisely defined by

$$\begin{aligned} {\varvec{\Phi }}(x,\,\xi )= & {} {\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }} (x,\,\xi )\right) ^{-1}+{\varvec{\Omega }}(x,\,\xi )\nonumber \\{} & {} -{\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}{\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Omega }}(x,\,\xi ) \end{aligned}$$

(3.21)

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,1)$, where ${\varvec{\Omega }}(x,\,\xi )$ is an $Lq\times p$ matrix-valued function with all entries in $L^{\infty }([1,\,a^{\frac{1}{p}})\times [0,\,1))$. In particular, ${\varvec{\Psi }}(x,\,\xi )({\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi ))^{-1}$ is exactly the $\Theta _{\beta }$-transform matrix associated with $S_\Psi ^{-1}\Psi$, which is exactly the window function of the canonical dual ${{\mathcal{M}\mathcal{D}}}(S_\Psi ^{-1}\Psi ,\,a,\,b)$.

(ii) For each ${\mathcal{M}\mathcal{D}}$-dual frame ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$,

$$\begin{aligned} \sum \limits _{l=1}^{L}\Vert \phi _l\Vert ^2=\sum \limits _{l=1}^{L}\Vert S_{\Psi }^{-1}\psi _l\Vert ^2 +\sum \limits _{l=1}^{L}\Vert \phi _l-S_{\Psi }^{-1}\psi _l\Vert ^2. \end{aligned}$$

(3.22)

Proof

(i) Since ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^2({\mathbb {R}}_+)$,

$$\begin{aligned} AI_p\le {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\le BI_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1) \end{aligned}$$

(3.23)

for some $0<A\le B<\infty$ by Theorem 2.1 and the fact that $q=1$. First we prove that each $\Phi =\{\phi _1,\,\phi _2,\,\ldots ,\,\phi _L\}$ defined by (3.21) generates an ${\mathcal{M}\mathcal{D}}$-dual ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$. By (3.23) and Remark 2.2, all entries of ${\varvec{\Phi }}(x,\,\xi )$ defined by (3.21) belong to $L^{\infty }([1,\,a^{\frac{1}{p}})\times [0,\,1))$. Thus, ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ is a Bessel sequence in $L^{2}({\mathbb {R}}_+)$. A direct computation gives

$$\begin{aligned} {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Phi }}(x,\,\xi )=I_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

This implies that ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ is a dual frame of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$. Next we prove that the window function $\Phi =\{\phi _1,\,\phi _2,\,\ldots ,\,\phi _L\}$ of each ${{\mathcal{M}\mathcal{D}}}$-dual frame ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ may be defined by (3.21). Suppose ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ is a dual frame of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$. Then,

$$\begin{aligned} {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Phi }}(x,\,\xi )=I_p{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1) \end{aligned}$$

(3.24)

by Remark 3.1. Take ${\varvec{\Omega }}(x,\,\xi )={\varvec{\Phi }}(x,\,\xi )$. Then, all entries of ${\varvec{\Omega }}(x,\,\xi )$ belong to $L^{\infty }([1,\,a^{\frac{1}{p}})\times [0,\,1))$ by Remark 2.2, and ${\varvec{\Phi }}(x,\,\xi )$ has the form (3.21) by (3.24) and a direct computation.

In particular, by Lemma 3.2, we have

$$\begin{aligned}{} & {} \Upsilon f(x,\,\xi )=\Upsilon S_{\Psi }^{-1}f(x,\,\xi ){\varvec{\Psi }}^{*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi ){ \text{ for } }f\in L^2({{\mathbb {R}}}_+)\\{} & {} \quad { \text{ and } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

It follows that

$$\begin{aligned}{} & {} \Upsilon S_{\Psi }^{-1}\psi _l(x,\,\xi )=\Upsilon \psi _l(x,\,\xi )\left( {\varvec{\Psi }}^{*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}\\{} & {} \quad { \text{ for } }1\le l\le L{ \text{ and } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

Thus,

$$\begin{aligned}{} & {} \left( \begin{array}{c} \Upsilon S_{\Psi }^{-1}\psi _l(x,\,\xi ) \\ \Upsilon S_{\Psi }^{-1}\psi _2(x,\,\xi ) \\ \cdots \\ \Upsilon S_{\Psi }^{-1}\psi _L(x,\,\xi ) \\ \end{array} \right) =\left( \begin{array}{c} \Upsilon \psi _l(x,\,\xi ) \\ \Upsilon \psi _2(x,\,\xi ) \\ \cdots \\ \Upsilon \psi _L(x,\,\xi ) \\ \end{array} \right) \left( {\varvec{\Psi }}^{*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}\\{} & {} \qquad ={\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^{*}(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1} \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$.

(ii) Suppose ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ is a dual frame of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$. By Lemma 3.3, the canonical dual frame of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is ${{\mathcal{M}\mathcal{D}}}(S_\Psi ^{-1}\Psi ,\,a,\,b)$. This implies that

$$\begin{aligned} S_{\Psi }^{-1}\psi _l= & {} \sum \limits _{k=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle S_{\Psi }^{-1}\psi _l,\,\Lambda _mD_{a^j}\psi _k\rangle \Lambda _mD_{a^j}S_{\Psi }^{-1}\psi _k\\= & {} \sum \limits _{k=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle \Lambda _{-m}D_{a^{-j}}S_{\Psi }^{-1}\psi _l,\,\psi _k\rangle \Lambda _mD_{a^j}S_{\Psi }^{-1}\psi _k \end{aligned}$$

for $1\le l\le L$. Observe that

$$\begin{aligned} \Lambda _{-m}D_{a^{-j}}S_{\Psi }^{-1}=S_{\Psi }^{-1}\Lambda _{-m}D_{a^{-j}}{ \text{ for } }m,\,j\in {\mathbb {Z}} \end{aligned}$$

by Lemma 3.3. It follows that

$$\begin{aligned} S_{\Psi }^{-1}\psi _l= & {} \sum \limits _{k=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle \Lambda _{-m}D_{a^{-j}}\psi _l,\,S_{\Psi }^{-1}\psi _k\rangle \Lambda _mD_{a^j}S_{\Psi }^{-1}\psi _k, \end{aligned}$$

and thus

$$\begin{aligned} \langle S_{\Psi }^{-1}\psi _l,\,\phi _l\rangle= & {} \sum \limits _{k=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle \Lambda _{-m}D_{a^{-j}}\psi _l,\,S_{\Psi }^{-1}\psi _k\rangle \langle S_{\Psi }^{-1}\psi _k,\,\Lambda _{-m}D_{a^{-j}}\phi _l\rangle \\= & {} \sum \limits _{k=1}^{L}\left\langle S_{\Psi }^{-1}\psi _k,\,\sum \limits _{m,j\in {\mathbb {Z}}}\langle S_{\Psi }^{-1}\psi _k,\, \Lambda _{m}D_{a^{j}}\psi _l\rangle \Lambda _{m}D_{a^{j}}\phi _l\right\rangle \\ \end{aligned}$$

for $1\le l\le L$. This implies that

$$\begin{aligned} \sum \limits _{l=1}^{L}\langle S_{\Psi }^{-1}\psi _l,\,\phi _l\rangle= & {} \sum \limits _{k=1}^{L}\left\langle S_{\Psi }^{-1}\psi _k,\,\sum \limits _{l=1}^{L}\sum \limits _{m,j\in {\mathbb {Z}}}\langle S_{\Psi }^{-1} \psi _k,\, \Lambda _{m}D_{a^{j}}\psi _l\rangle \Lambda _{m}D_{a^{j}}\phi _l\right\rangle \\= & {} \sum \limits _{k=1}^{L}\Vert S_{\Psi }^{-1}\psi _k\Vert ^2 \end{aligned}$$

due to ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ being a dual frame of ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$. If follows that

$$\begin{aligned} \sum \limits _{l=1}^{L}\langle S_{\Psi }^{-1}\psi _l,\,\phi _l-S_{\Psi }^{-1}\psi _l\rangle =0. \end{aligned}$$

This leads to (3.22). The proof is completed. $\square$

Recall from the Gabor analysis theory that an arbitrary Gabor frame for $L^2({\mathbb {R}})$ has Gabor duals ([3], Theorem 12.3.2). Theorem 3.1 shows that this cannot be completely extended to ${\mathcal{M}\mathcal{D}}$-frames for $L^2({\mathbb {R}}_+)$, and it tells us that, under the general setup, an ${\mathcal{M}\mathcal{D}}$-frame has ${\mathcal{M}\mathcal{D}}$-duals if and only if $\log _ba\in {\mathbb {N}}$. Therefore, ${\mathcal{M}\mathcal{D}}$-frames in $L^2({\mathbb {R}}_+)$ are essentially different from Gabor frames in $L^2({\mathbb {R}})$. Recall that a Gabor frame for $L^2({\mathbb {R}})$ is a Riesz basis if and only if it has the unique Gabor dual. The following theorem shows that an ${\mathcal{M}\mathcal{D}}$-frame for $L^2({\mathbb {R}}_+)$ is a Riesz basis if and only if it admits the unique ${\mathcal{M}\mathcal{D}}$-dual. It presents that Gabor frames and ${\mathcal{M}\mathcal{D}}$-frames share some similarity.

Theorem 3.3

Let $a,\,b$ and $\Psi$ be as in the general setup, $\log _{b}a=p\le L$, and ${{\mathcal{M}\mathcal{D}}}(\Psi , \,a,\,b)$ be a frame for $L^2({\mathbb {R}}_+)$. Then, the following are equivalent:

(i) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Riesz basis for $L^2({\mathbb {R}}_+)$.

(ii) ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ has the unique ${\mathcal{M}\mathcal{D}}$-dual ${{\mathcal{M}\mathcal{D}}}(S_{\Psi }^{-1}\Psi ,\,a,\,b)$.

(iii) $\log _ba=L$.

Proof

By Theorem 2.2, (i) and (iii) are equivalent. Next we prove the equivalence between (ii) and (iii). First we prove that (ii) implies (iii). Suppose (ii) holds. By Theorem 3.2, for each $L\times p$ matrix-valued function ${\varvec{\Omega }}(x,\,\xi )$ with $L^{\infty }([1,\,a^{\frac{1}{p}})\times [0,\,1))$-entries,

$$\begin{aligned} {\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1} {\varvec{\Psi }}^*(x,\,\xi )\mathbf{\Omega }(x,\,\xi )= & {} {\varvec{\Omega }}(x,\,\xi ){ \text{ for } \text{ a.e. } }(x,\,\xi )\nonumber \\\in & {} [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

(3.25)

Let $\{e_k:\,k\in {{\mathbb {N}}}_L\}$ be the canonical orthonormal basis for ${{\mathbb {C}}}^L$. Fix an arbitrary $k\in {{\mathbb {N}}}_L$, take ${\varvec{\Omega }}(x,\,\xi )$ in (3.25) by

$$\begin{aligned} {\varvec{\Omega }}(x,\,\xi )=(e_k,\,0){ \text{ for } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

Then, the kth column of ${\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}{\varvec{\Psi }}^*(x,\,\xi )$ is exactly $e_k$. This implies that

$$\begin{aligned} {\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}{\varvec{\Psi }}^*(x,\,\xi )=I_L{ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1). \end{aligned}$$

Thus,

$$\begin{aligned} L=\textrm{rank}({\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}{\varvec{\Psi }}^*(x,\,\xi ))\le \textrm{rank}({\varvec{\Psi }}(x,\,\xi ))=p \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$ by Theorem 2.1. This leads to (iii) by the fact that $\log _ba=p\le L$.

Next, we prove that (iii) implies (ii). Suppose (iii) holds. Then, ${\varvec{\Psi }}(x,\,\xi )$ is an $L\times L$ matrix-valued function. Since $\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}$ exists for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$ by Theorem 2.1 (iii), so do ${\varvec{\Psi }}(x,\,\xi )$ and ${\varvec{\Psi }}^*(x,\,\xi )$. It follows that

$$\begin{aligned}{} & {} {\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}{\varvec{\Psi }}^*(x,\,\xi ) \\{} & {} \quad ={\varvec{\Psi }}(x,\,\xi ){\varvec{\Psi }}^{-1}(x,\,\xi )({\varvec{\Psi }}^{*})^{-1}(x,\,\xi ){\varvec{\Psi }}^*(x,\,\xi )=I_L \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$. Thus, for all ${\varvec{\Omega }}(x,\,\xi )$ with $L^{\infty }([1,\,a^{\frac{1}{p}})\times [0,\,1))$-entries,

$$\begin{aligned}{} & {} {\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }} (x,\,\xi )\right) ^{-1}\\{} & {} +{\varvec{\Omega }}(x,\,\xi )-{\varvec{\Psi }}(x,\,\xi ) \left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}{\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Omega }}(x,\,\xi )\\= & {} {\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1} \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$. Therefore, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ admits the unique ${{\mathcal{M}\mathcal{D}}}$-dual ${{\mathcal{M}\mathcal{D}}}(S_{\Psi }^{-1}\Psi ,\,a,\,b)$ by Theorem 3.2. The proof is completed. $\square$

Theorems 2.4 and 3.1 show that, under the general setup, $\log _ba=p\le L$ is necessary for the existence of ${\mathcal{M}\mathcal{D}}$-dual frames. Theorem 3.2 reduces constructing ${\mathcal{M}\mathcal{D}}$-dual of a given ${\mathcal{M}\mathcal{D}}$-frame ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ to computing ${\varvec{\Psi }}(x,\,\xi )({\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi ))^{-1}$ for $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$ which corresponds to the canonical dual. An alternate ${\mathcal{M}\mathcal{D}}$-dual may be obtained by choosing ${\varvec{\Omega }}(x,\,\xi )$ with $L^{\infty }([1,\,a^{\frac{1}{p}})\times [0,\,1))$-entries. The following theorem gives the expression of the canonical ${\mathcal{M}\mathcal{D}}$-dual of a given ${\mathcal{M}\mathcal{D}}$-frame. By Lemma 2.7, it covers all situations.

Theorem 3.4

Let a, b as be in the general setup with $\log _ba=p\le L$. Assume $\lambda _s(x,\,\xi )$ with $s\in {\mathbb {N}}_{p}$ are measurable functions satisfying

$$\begin{aligned} 0<\mathop {\min }\limits _{s\in {{\mathbb {N}}}_p}\mathop {\textrm{essinf}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|\le \mathop {\max }\limits _{s\in {{\mathbb {N}}}_p} \mathop {\textrm{esssup}}\limits _{(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)}|\lambda _s(x,\,\xi )|<\infty .\nonumber \\ \end{aligned}$$

(3.26)

Define

$$\begin{aligned}{} & {} \Lambda (x,\,\xi )=\textrm{diag}(\lambda _0(x,\,\xi ),\lambda _1(x,\,\xi ),\cdots ,\lambda _{p-1}(x,\,\xi ))\\{} & {} \quad { \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1), \end{aligned}$$

$\Psi =\{\psi _1,\,\psi _2,\,\ldots ,\,\psi _L\}$ by a $L\times p$ matrix-valued function

$$\begin{aligned} {\varvec{\Psi }}(x,\,\xi )=U(x,\,\xi )\left( \begin{array}{c} \Lambda (x,\,\xi ) \\ 0 \\ \end{array} \right) V(x,\,\xi ) \end{aligned}$$

(3.27)

for $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,1)$, where $U(x,\,\xi )$ and $V(x,\,\xi )$ are $L\times L$ and $p\times p$ unitary matrix-valued measurable function defined on $[1,\,a^{\frac{1}{p}})\times [0,1)$, respectively. Then, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^2({\mathbb {R}}_+)$, and the window functions of its canonical dual ${{\mathcal{M}\mathcal{D}}}(S_\Psi ^{-1}\Psi ,\,a,\,b)$ are defined by the $\Theta _\beta$-transform matrix

$$\begin{aligned} U(x,\,\xi )\left( \begin{array}{c} {\widetilde{\Lambda }}(x,\,\xi ) \\ 0 \\ \end{array} \right) V(x,\,\xi ){ \text{ for } \text{ a.e. } }(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1), \end{aligned}$$

(3.28)

where ${\widetilde{\Lambda }}(x,\xi )=\textrm{diag}\left( \overline{\lambda _0^{-1}(x,\,\xi )},\,\overline{\lambda _1^{-1}(x,\,\xi )},\,\cdots ,\, \overline{\lambda _{p-1}^{-1}(x,\,\xi )}\right) .$

Proof

By Theorem 2.3, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^2({\mathbb {R}}_+)$. By (3.27), we have

$$\begin{aligned} {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )=V^*(x,\,\xi )\Lambda ^*(x,\,\xi )\Lambda (x,\,\xi ) V(x,\,\xi ), \end{aligned}$$

and thus

$$\begin{aligned} {\varvec{\Psi }}(x,\,\xi )\left( {\varvec{\Psi }}^*(x,\,\xi ){\varvec{\Psi }}(x,\,\xi )\right) ^{-1}=U(x,\,\xi )\left( \begin{array}{c} {\widetilde{\Lambda }}(x,\,\xi ) \\ 0 \\ \end{array} \right) V(x,\,\xi ) \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{p}})\times [0,\,1)$. This leads to the theorem by Theorem 3.2. The proof is completed. $\square$

In what follows, we give some examples of Theorem 3.4.

Example 3.1

Given a, $b>1$ satisfying $\log _ba=3$, define $\lambda _0(x,\,\xi )=(1+x)e^{2\pi i\xi }$, $\lambda _1(x,\,\xi )=2x$, $\lambda _2(x,\,\xi )=(a-x)e^{-2\pi i\xi }$ for $(x,\,\xi )\in [1,\,a^{\frac{1}{3}})\times [0,\,1)$ in Theorem 3.4.

(i) Take

$$\begin{aligned} U(x,\,\xi )=\left( \begin{array}{ccc} \frac{\sqrt{3}}{3} &{} \frac{\sqrt{6}}{6} &{} -\frac{\sqrt{2}}{2} \\ -\frac{\sqrt{3}}{3} &{} \frac{\sqrt{6}}{3} &{} 0 \\ \frac{\sqrt{3}}{3} &{} \frac{\sqrt{6}}{6} &{} \frac{\sqrt{2}}{2} \\ \end{array} \right) { \text{ and } } V(x,\,\xi )=\left( \begin{array}{ccc} \frac{\sqrt{2}}{2} &{} -\frac{\sqrt{2}}{2} &{} 0 \\ \frac{\sqrt{2}}{2} &{} \frac{\sqrt{2}}{2} &{} 0\\ 0 &{} 0 &{} 1\\ \end{array} \right) \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{3}})\times [0,\,1)$. It is clearly that $U(x,\,\xi )$ and $V(x,\,\xi )$ are unitary matrix-valued measurable function defined on $[1,\,a^{\frac{1}{3}})\times [0,1)$. Define $\Psi =\{\psi _1,\,\psi _2,\,\psi _3\}$ by (3.27). Then,

$$\begin{aligned} {\varvec{\Psi }}(x,\,\xi )= \left( \begin{array}{ccc} \frac{\sqrt{6}}{6}(1+x)e^{2\pi i\xi }+\frac{\sqrt{3}}{3}x &{} -\frac{\sqrt{6}}{6}(1+x)e^{2\pi i\xi }+\frac{\sqrt{3}}{3}x &{} -\frac{\sqrt{2}}{2}(a-x)e^{-2\pi i\xi } \\ -\frac{\sqrt{6}}{6}(1+x)e^{2\pi i\xi }+\frac{2\sqrt{3}}{3}x &{} \frac{\sqrt{6}}{6}(1+x)e^{2\pi i\xi }+\frac{2\sqrt{3}}{3}x &{} 0 \\ \frac{\sqrt{6}}{6}(1+x)e^{2\pi i\xi }+\frac{\sqrt{3}}{3}x &{} -\frac{\sqrt{6}}{6}(1+x)e^{2\pi i\xi }+\frac{\sqrt{3}}{3}x &{} \frac{\sqrt{2}}{2}(a-x)e^{-2\pi i\xi } \\ \end{array} \right) \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{3}})\times [0,\,1)$, ${{\mathcal{M}\mathcal{D}}}(\Psi , \,a,\,b)$ is a Riesz basis for $L^2({\mathbb {R}}_+)$ by Theorem 2.2, and by Remark 1.1 and a standard computation, $\Psi =\{\psi _1,\,\psi _2,\,\psi _3\}$ has the form

$$\begin{aligned} \psi _1(x)= & {} \left\{ \begin{array}{ll} \frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}(1+ax) &{}\hbox {if } x\in a^{-1}[1,\,a^{\frac{1}{3}}),\\ -\frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}{b}^{-\frac{1}{2}}(1+ab^{-1}x) &{}\hbox {if } x\in a^{-1}b[1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{3}x &{}\hbox {if} x\in [1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{3}{b}^{-\frac{3}{2}}x &{}\hbox {if } x\in b[1,\,a^{\frac{1}{3}}),\\ -\frac{\sqrt{2}}{2}{a}^{-\frac{1}{2}}{b}^{-1}(a-a^{-1}b^{-2}x) &{}\hbox {if } x\in ab^2[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox {otherwise;} \end{array} \right. \\ \psi _2(x)= & {} \left\{ \begin{array}{ll} -\frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}(1+ax) &{}\hbox {if } x\in a^{-1}[1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}{b}^{-\frac{1}{2}}(1+ab^{-1}x) &{}\hbox {if } x\in a^{-1}b[1,\,a^{\frac{1}{3}}),\\ \frac{2\sqrt{3}}{3}x &{}\hbox {if } x\in [1,\,a^{\frac{1}{3}}),\\ \frac{2\sqrt{3}}{3}{b}^{-\frac{3}{2}}x &{}\hbox {if } x\in b[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox {otherwise; } \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \psi _3(x)= & {} \left\{ \begin{array}{ll} \frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}(1+ax) &{}\hbox {if }\, x\in a^{-1}[1,\,a^{\frac{1}{3}}),\\ -\frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}{b}^{-\frac{1}{2}}(1+ab^{-1}x) &{}\hbox {if }\, x\in a^{-1}b[1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{3}x &{}\hbox {if } x\in [1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{3}{b}^{-\frac{3}{2}}x &{}\hbox {if }\, x\in b[1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{2}}{2}{a}^{-\frac{1}{2}}{b}^{-1}(a-a^{-1}b^{-2}x) &{}\hbox {if }\, x\in ab^2[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox {otherwise.} \end{array} \right. \end{aligned}$$

By Theorem 3.4, the window functions $S_{\Psi }^{-1}\Psi =\left\{ S_{\Psi }^{-1}\psi _1,\,S_{\Psi }^{-1}\psi _2,\,S_{\Psi }^{-1}\psi _3\right\}$ of its canonical dual may be defined by the $\Theta _{\beta }$-transform

$$\begin{aligned} \left( \begin{array}{ccc} \frac{\sqrt{6}}{6}(1+x)^{-1}e^{2\pi i\xi }+\frac{\sqrt{3}}{12}x^{-1} &{} -\frac{\sqrt{6}}{6}(1+x)^{-1}e^{2\pi i\xi }+\frac{\sqrt{3}}{12}x^{-1} &{} -\frac{\sqrt{2}}{2}(a-x)^{-1}e^{-2\pi i\xi } \\ -\frac{\sqrt{6}}{6}(1+x)^{-1}e^{2\pi i\xi }+\frac{\sqrt{3}}{6}x^{-1} &{} \frac{\sqrt{6}}{6}(1+x)^{-1}e^{2\pi i\xi }+\frac{\sqrt{3}}{6}x^{-1} &{} 0 \\ \frac{\sqrt{6}}{6}(1+x)^{-1}e^{2\pi i\xi }+\frac{\sqrt{3}}{12}x^{-1} &{} -\frac{\sqrt{6}}{6}(1+x)^{-1}e^{2\pi i\xi }+\frac{\sqrt{3}}{12}x^{-1} &{} \frac{\sqrt{2}}{2}(a-x)^{-1}e^{-2\pi i\xi } \\ \end{array} \right) \end{aligned}$$

for $(x,\,\xi )\in [1,\,a^{\frac{1}{3}})\times [0,\,1)$. Applying Remark 1.1, we have

$$\begin{aligned} S_{\Psi }^{-1}\psi _1(x)= & {} \left\{ \begin{array}{ll} \frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}(1+ax)^{-1} &{}\hbox { if }\,\, x\in a^{-1}[1,\,a^{\frac{1}{3}}),\\ -\frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}{b}^{-\frac{1}{2}}(1+ab^{-1}x)^{-1} &{}\hbox { if }\,\, x\in a^{-1}b[1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{12}x^{-1} &{}\hbox { if }\,x\in [1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{12}{b}^{\frac{1}{2}}x^{-1} &{}\hbox { if }\,\, x\in b[1,\,a^{\frac{1}{3}}),\\ -\frac{\sqrt{2}}{2}{a}^{-\frac{1}{2}}{b}^{-1}(a-a^{-1}b^{-2}x)^{-1} &{}\hbox { if }\,\, x\in ab^2[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox { otherwise;} \end{array} \right. \\ S_{\Psi }^{-1}\psi _2(x)= & {} \left\{ \begin{array}{ll} -\frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}(1+ax)^{-1} &{}\hbox {if }\, x\in a^{-1}[1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}{b}^{-\frac{1}{2}}(1+ab^{-1}x)^{-1} &{}\hbox {if}\, x\in a^{-1}b[1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{6}x^{-1} &{}\hbox {if }\, x\in [1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{6}{b}^{\frac{1}{2}}x^{-1} &{}\hbox {if }\, x\in b[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox { otherwise; } \end{array} \right. \\ S_{\Psi }^{-1}\psi _3(x)= & {} \left\{ \begin{array}{ll} \frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}(1+ax)^{-1} &{}\hbox { if }\,\, x\in a^{-1}[1,\,a^{\frac{1}{3}}),\\ -\frac{\sqrt{6}}{6}{a}^{\frac{1}{2}}{b}^{-\frac{1}{2}}(1+ab^{-1}x)^{-1} &{}\hbox { if }\,\, x\in a^{-1}b[1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{12}x^{-1} &{}\hbox { if} x\in [1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{3}}{12}{b}^{\frac{1}{2}}x^{-1} &{}\hbox { if }\,\, x\in b[1,\,a^{\frac{1}{3}}),\\ \frac{\sqrt{2}}{2}{a}^{-\frac{1}{2}}{b}^{-1}(a-a^{-1}b^{-2}x)^{-1} &{}\hbox { if }\,\, x\in ab^2[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox { otherwise.} \end{array} \right. \end{aligned}$$

(ii) Take $U(x,\,\xi )=V(x,\,\xi )=I_3$ for $(x,\,\xi )\in [1,\,a^{\frac{1}{3}})\times [0,\,1)$, and define $\Psi =\{\psi _1,\,\psi _2,\,\psi _3\}$. Then, ${{\mathcal{M}\mathcal{D}}}(\Phi ,\,a,\,b)$ is a Riesz basis for $L^2({\mathbb {R}}_+)$ by Theorem 2.2. Similarly to (i), we have

$$\begin{aligned} \psi _1(x)= & {} \left\{ \begin{array}{ll} a^{\frac{1}{2}}(1+ax) &{}\hbox { if }\, x\in a^{-1}[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox { otherwise;} \end{array} \right. \\ \psi _2(x)= & {} \left\{ \begin{array}{ll} 2b^{-\frac{3}{2}}x &{}\hbox { if }\, x\in b[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox { otherwise; } \end{array} \right. \\ \psi _3(x)= & {} \left\{ \begin{array}{ll} a^{-\frac{1}{2}}b^{-1}(a-a^{-1}b^{-2}x) &{}\hbox { if }\, x\in ab^2[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox { otherwise, } \end{array} \right. \end{aligned}$$

and the window functions $S_{\Psi }^{-1}\Psi =\left\{ S_{\Psi }^{-1}\psi _1,\,S_{\Psi }^{-1}\psi _2,\,S_{\Psi }^{-1}\psi _3\right\}$ have the form

$$\begin{aligned} S_{\Psi }^{-1}\psi _1(x)= & {} \left\{ \begin{array}{ll} a^{\frac{1}{2}}(1+ax)^{-1} &{}\hbox { if }\, x\in a^{-1}[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox { otherwise;} \end{array} \right. \\ S_{\Psi }^{-1}\psi _2(x)= & {} \left\{ \begin{array}{ll} \frac{1}{2}b^{\frac{1}{2}}x^{-1} &{}\hbox { if }\, x\in b[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox { otherwise; } \end{array} \right. \\ S_{\Psi }^{-1}\psi _3(x)= & {} \left\{ \begin{array}{ll} a^{-\frac{1}{2}}b^{-1}(a-a^{-1}b^{-2}x)^{-1} &{}\hbox {if } x\in ab^2[1,\,a^{\frac{1}{3}}),\\ 0&{}\hbox {otherwise. } \end{array} \right. \end{aligned}$$

Example 3.2

Given $a,\,b>1$ satisfying $\log _ba=2$, define

$$\begin{aligned} U(x,\,\xi )=\left( \begin{array}{ccc} \frac{\sqrt{3}}{3} &{} \frac{\sqrt{6}}{6} &{} -\frac{\sqrt{2}}{2} \\ -\frac{\sqrt{3}}{3} &{} \frac{\sqrt{6}}{3} &{} 0 \\ \frac{\sqrt{3}}{3} &{} \frac{\sqrt{6}}{6} &{} \frac{\sqrt{2}}{2} \\ \end{array} \right) { \text{ and } } V(x,\,\xi )=\left( \begin{array}{cc} \frac{\sqrt{2}}{2} &{} -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} &{} \frac{\sqrt{2}}{2}\\ \end{array} \right) \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{2}})\times [0,\,1)$ in Theorem 3.4. Clearly, $U(x,\,\xi )$ and $V(x,\,\xi )$ are unitary.

(i) Take $\lambda _0(x,\,\xi )=e^{2\pi i\xi }$, $\lambda _1(x,\,\xi )=1$ for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{2}})\times [0,\,1)$. Define $\Psi =\{\psi _l:\,1\le l\le L\}$ by (3.27). Then,

$$\begin{aligned} {\varvec{\Psi }}(x,\,\xi )=\left( \begin{array}{cc} \frac{\sqrt{6}}{6}e^{2\pi i \xi }+\frac{\sqrt{3}}{6} &{}-\frac{\sqrt{6}}{6}e^{2\pi i \xi }+\frac{\sqrt{3}}{6} \\ -\frac{\sqrt{6}}{6}e^{2\pi i \xi }+\frac{\sqrt{3}}{3} &{}\frac{\sqrt{6}}{6}e^{2\pi i \xi }+\frac{\sqrt{3}}{3} \\ \frac{\sqrt{6}}{6}e^{2\pi i \xi }+\frac{\sqrt{3}}{6}&{} -\frac{\sqrt{6}}{6}e^{2\pi i \xi }+\frac{\sqrt{3}}{6} \end{array} \right) \end{aligned}$$

for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{2}})\times [0,\,1)$, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a Parseval frame for $L^2({\mathbb {R}}_+)$ by Theorem 2.3 (v). And by Remark 1.1 and a standard computation,

$$\begin{aligned} \psi _1(x)=\psi _3(x)= & {} \left\{ \begin{array}{ll} \frac{\sqrt{3}}{6} &{}\hbox { if }\, x\in [1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{6}}{6}a^{\frac{1}{2}} &{}\hbox { if }\, x\in a^{-1}[1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{3}}{6}b^{-\frac{1}{2}} &{}\hbox { if }\, x\in b[1,\,a^{\frac{1}{2}}),\\ -\frac{\sqrt{6}}{6}b^{\frac{1}{2}} &{}\hbox { if }\, x\in a^{-1}b[1,\,a^{\frac{1}{2}}),\\ 0&{}\hbox { otherwise, } \end{array} \right. \\ \psi _2(x)= & {} \left\{ \begin{array}{ll} \frac{\sqrt{3}}{3} &{}\hbox { if }\, x\in [1,\,a^{\frac{1}{3}}),\\ -\frac{\sqrt{6}}{6}{a}^{\frac{1}{2}} &{}\hbox { if }\, x\in a^{-1}[1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{3}}{3}{b}^{-\frac{1}{2}} &{}\hbox { if }\, x\in b[1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{6}}{6}{b}^{\frac{1}{2}} &{}\hbox { if }\, x\in a^{-1}b[1,\,a^{\frac{1}{2}}),\\ 0 &{}\hbox { otherwise. } \end{array} \right. \end{aligned}$$

(ii) Take $\lambda _0(x,\,\xi )=(1+x)e^{2\pi i\xi }$, $\lambda _1(x,\,\xi )=2x$ for a.e. $(x,\,\xi )\in [1,\,a^{\frac{1}{2}})\times [0,\,1)$, and define $\Psi =\{\psi _1,\,\psi _2,\,\psi _3\}$ by (3.27). Then, ${{\mathcal{M}\mathcal{D}}}(\Psi ,\,a,\,b)$ is a frame for $L^2({\mathbb {R}}_+)$ by Theorem 3.4. Similarly to (i), we have

$$\begin{aligned} \psi _1(x)=\psi _3(x)= & {} \left\{ \begin{array}{ll} \frac{\sqrt{3}}{3}x &{}\hbox {if } x\in [1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{6}}{6}a^{\frac{1}{2}}(1+ax) &{}\hbox {if } x\in a^{-1}[1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{3}}{3}b^{-\frac{3}{2}}x &{}\hbox {if }x\in b[1,\,a^{\frac{1}{2}}),\\ -\frac{\sqrt{6}}{6}a^{\frac{1}{2}}b^{-\frac{1}{2}}(1+ab^{-1}x) &{}\hbox {if } x\in a^{-1}b[1,\,a^{\frac{1}{2}}),\\ 0&{}\hbox {otherwise; } \end{array} \right. \\ \psi _2(x)= & {} \left\{ \begin{array}{ll} \frac{2\sqrt{3}}{3}x &{}\hbox { \,if } x\in [1,\,a^{\frac{1}{2}}),\\ -\frac{\sqrt{6}}{6}a^{\frac{1}{2}}(1+ax) &{}\hbox { \,if }x\in a^{-1}[1,\,a^{\frac{1}{2}}),\\ \frac{2\sqrt{3}}{3}b^{-\frac{3}{2}}x &{}\hbox { \,if } x\in b[1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{6}}{6}a^{\frac{1}{2}}b^{-\frac{1}{2}}(1+ab^{-1}x) &{}\hbox { \,if } x\in a^{-1}b[1,\,a^{\frac{1}{2}}),\\ 0&{}\hbox { \,otherwise,} \end{array} \right. \end{aligned}$$

and by Theorem 3.4 the window functions $S_{\Psi }^{-1}\Psi =\left\{ S_{\Psi }^{-1}\psi _1,\,S_{\Psi }^{-1}\psi _2,\,S_{\Psi }^{-1}\psi _3\right\}$ have the form

$$\begin{aligned} S_{\Psi }^{-1}\psi _1(x)=S_{\Psi }^{-1}\psi _3(x)= & {} \left\{ \begin{array}{ll} \frac{\sqrt{3}}{12}x^{-1} &{}\hbox {if } x\in [1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{6}}{6}a^{\frac{1}{2}}(1+ax)^{-1} &{}\hbox {if } x\in a^{-1}[1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{3}}{12}b^{\frac{1}{2}}x^{-1} &{}\hbox {if }x\in b[1,\,a^{\frac{1}{2}}),\\ -\frac{\sqrt{6}}{6}a^{\frac{1}{2}}b^{-\frac{1}{2}}(1+ab^{-1}x)^{-1} &{}\hbox {if } x\in a^{-1}b[1,\,a^{\frac{1}{2}}),\\ 0&{}\hbox {otherwise; } \end{array} \right. \\ S_{\Psi }^{-1}\psi _2(x)= & {} \left\{ \begin{array}{ll} \frac{\sqrt{3}}{6}x^{-1} &{}\hbox { \,if } x\in [1,\,a^{\frac{1}{2}}),\\ -\frac{\sqrt{6}}{6}a^{\frac{1}{2}}(1+ax)^{-1} &{}\hbox { \,if } x\in a^{-1}[1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{3}}{6}b^{\frac{1}{2}}x^{-1} &{}\hbox { \,if } x\in b[1,\,a^{\frac{1}{2}}),\\ \frac{\sqrt{6}}{6}a^{\frac{1}{2}}b^{-\frac{1}{2}}(1+ab^{-1}x)^{-1} &{}\hbox { \,if } x\in a^{-1}b[1,\,a^{\frac{1}{2}}),\\ 0&{}\hbox { \,otherwise. } \end{array} \right. \end{aligned}$$