The Zak transform and Wiener estimates on Gelfand-Shilov and modulation spaces with applications to operator theory

TL;DR

This paper characterizes Gelfand-Shilov and modulation spaces via Zak transform estimates, explores quasi-periodic functions, and provides conditions for operators to be conjugated by Zak transforms, with implications for operator theory.

Contribution

It introduces new characterizations of function spaces using Zak transform estimates and establishes criteria for operator conjugation in this context.

Findings

Characterization of Gelfand-Shilov and modulation spaces via Zak transform estimates

Necessary and sufficient conditions for operator conjugation by Zak transform

Applications to quasi-periodic functions and distributions

Abstract

We characterize Gelfand-Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these result for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by the Zak transform.

Equations

$$F\mapsto c\cdot \overline{Z\varphi }\cdot Z\psi \cdot F$$

$$(Z_{1}f)(x,\xi )\equiv \sum _{j\in {\mathbf{Z}}^d}f(x-j)e^{i⟨j,\xi ⟩},$$

$$F(x+k,\xi )=e^{i⟨k,\xi ⟩}F(x,\xi )\text{and}F(x,\xi +2\pi \kappa )=F(x,\xi ),k,\kappa \in {\mathbf{Z}}^d.$$

$$\Vert Z_{1}f{\Vert }_{L^2(Q_{d,1}\times Q_{d,2\pi })}=(2\pi {)}^{\frac{d}{2}}\Vert f{\Vert }_{L^2},f\in L^2({\mathbf{R}}^d).$$

$$\Vert Z_{1}f{\Vert }_{W^{\infty ,p}}\asymp \Vert f{\Vert }_{M^p}$$

$$f\in M^p({\mathbf{R}}^d)\iff V_{\Phi }(Z_{1}f)\in L^p(Q_{d,1}\times Q_{d,2\pi }\times {\mathbf{R}}^d\times {\mathbf{R}}^d).$$

$$f\in M^2({\mathbf{R}}^d)\iff Z_{1}f\in L^2(Q_{d,1}\times Q_{d,2\pi }),$$

$$T:{\mathcal{S}}_s^{\sigma }\to ({\mathcal{S}}_s^{\sigma }{)}^{\prime }$$

$$T_Z:{\mathcal{E}}_{Z,1}^{\sigma ,s}\to ({\mathcal{E}}_{Z,1}^{\sigma ,s}{)}^{\prime }$$

$$(U_{y,\eta }F)(x,\xi )=e^{-i⟨y,\xi +\eta ⟩}F(x+y,\xi +\eta ),y,\eta \in {\mathbf{R}}^d.$$

$$\Vert f{\Vert }_{{\mathcal{S}}_{s,h}^{\sigma }}\equiv \sup \frac{|x^{\alpha }{\partial }^{\beta }f(x)|}{h^{|\alpha +\beta |}\alpha {!}^s\beta {!}^{\sigma }}$$

$${\mathcal{S}}_s={\mathcal{S}}_s^s,{\mathcal{S}}_s^{\prime }=({\mathcal{S}}_s^s{)}^{\prime },{\Sigma }_s={\Sigma }_s^s\text{and}{\Sigma }_s^{\prime }=({\Sigma }_s^s{)}^{\prime }.$$

$$(\mathcal{F}f)(\xi )=\hat{f}(\xi )\equiv (2\pi {)}^{-\frac{d}{2}}{\int }_{{\mathbf{R}}^d}f(x)e^{-i⟨x,\xi ⟩}dx$$

$$\Vert f{\Vert }_{{\mathcal{S}}_{s_1,s_2,h}^{{\sigma }_1,{\sigma }_2}}\equiv \sup \frac{|x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}{\partial }_{x_1}^{{\beta }_1}{\partial }_{x_2}^{{\beta }_2}f(x_1,x_2)|}{h^{|{\alpha }_1+{\alpha }_2+{\beta }_1+{\beta }_2|}{\alpha }_1{!}^{s_1}{\alpha }_2{!}^{s_2}{\beta }_1{!}^{{\sigma }_1}{\beta }_2{!}^{{\sigma }_2}}$$

$$\begin{gathered} {\mathcal{S}}_{s_{0,1},s_{0,2}}^{{\sigma }_{0,1},{\sigma }_{0,2}}({\mathbf{R}}^{d_1+d_2})\hookrightarrow {\Sigma }_{s_1,s_2}^{{\sigma }_1,{\sigma }_2}({\mathbf{R}}^{d_1+d_2})\hookrightarrow {\mathcal{S}}_{s_1,s_2}^{{\sigma }_1,{\sigma }_2}({\mathbf{R}}^{d_1+d_2}) \\ \hookrightarrow \mathcal{S}({\mathbf{R}}^{d_1+d_2})\hookrightarrow {\mathcal{S}}^{\prime }({\mathbf{R}}^{d_1+d_2})\hookrightarrow ({\mathcal{S}}_{s_1,s_2}^{{\sigma }_1,{\sigma }_2}{)}^{\prime }({\mathbf{R}}^{d_1+d_2}) \\ \hookrightarrow ({\Sigma }_{s_1,s_2}^{{\sigma }_1,{\sigma }_2}{)}^{\prime }({\mathbf{R}}^{d_1+d_2})\hookrightarrow ({\mathcal{S}}_{s_{0,1},s_{0,2}}^{{\sigma }_{0,1},{\sigma }_{0,2}}{)}^{\prime }({\mathbf{R}}^{d_1+d_2}), \end{gathered}$$

$${\mathcal{F}}_1$$

$${\mathcal{F}}_2$$

$${\mathcal{F}}_1$$

$${\mathcal{F}}_2$$

$$|f(x)|\lesssim e^{-r|x{|}^{\frac{1}{s}}}\text{and}|\hat{f}(\xi )|\lesssim e^{-r|\xi {|}^{\frac{1}{\sigma }}},$$

$$|f(x_1,x_2)|\lesssim e^{-r(|x_1{|}^{\frac{1}{s_1}}+|x_2{|}^{\frac{1}{s_2}})}\text{and}|\hat{f}({\xi }_1,{\xi }_2)|\lesssim e^{-r(|{\xi }_1{|}^{\frac{1}{{\sigma }_1}}+|{\xi }_2{|}^{\frac{1}{{\sigma }_2}})},$$

$$V_{\varphi }f(x,\xi )=\mathcal{F}(f\overline{\varphi (\cdot -x)})(\xi ).$$

$$V_{\varphi }f(x,\xi )=(2\pi {)}^{-\frac{d}{2}}\int f(y)\overline{\varphi (y-x)}{e}^{-i⟨y,\xi ⟩}dy.$$

$$s_j,{\sigma }_j>0\text{and}\varphi \in {\mathcal{S}}_{s_1,s_2}^{{\sigma }_1,{\sigma }_2}({\mathbf{R}}^{d_1+d_2})\setminus 0(\varphi \in {\Sigma }_{s_1,s_2}^{{\sigma }_1,{\sigma }_2}({\mathbf{R}}^{d_1+d_2})\setminus 0).$$

$$|V_{\varphi }f(x_1,x_2,{\xi }_1,{\xi }_2)|\lesssim e^{-r(|x_1{|}^{\frac{1}{s_1}}+|x_2{|}^{\frac{1}{s_2}}+|{\xi }_1{|}^{\frac{1}{{\sigma }_1}}+|{\xi }_2{|}^{\frac{1}{{\sigma }_2}})},$$

$$|V_{\varphi }f(x_1,x_2,{\xi }_1,{\xi }_2)|\lesssim e^{r(|x_1{|}^{\frac{1}{s_1}}+|x_2{|}^{\frac{1}{s_2}}+|{\xi }_1{|}^{\frac{1}{{\sigma }_1}}+|{\xi }_2{|}^{\frac{1}{{\sigma }_2}})},$$

$$\Vert f{\Vert }_{K,h,\sigma }\equiv \underset{\alpha \in {\mathbf{N}}^d}{\sup }\left(\frac{\Vert {\partial }^{\alpha }f{\Vert }_{L^{\infty }(K)}}{h^{|\alpha |}\alpha {!}^{\sigma }}\right).$$

$${\mathcal{E}}^{\sigma }({\mathbf{R}}^d)$$

$$⟨e_j,e_k^{\prime }⟩=2\pi {\delta }_{jk}\text{for every}j,k=1,\dots ,d.$$

$${\Lambda }_E$$

$${\Lambda }_E^{\prime }$$

$$V_1=\{(x,0)\in {\mathbf{R}}^{2d};x\in {\mathbf{R}}^d\},V_2=\{(0,\xi )\in {\mathbf{R}}^{2d};\xi \in {\mathbf{R}}^d\}$$

Full text

0. Introduction

In the paper we characterise Gelfand-Shilov spaces of functions and distributions, modulation spaces and Gevrey classes in background of mapping properties of the Zak transforms. We apply these results to deduce duality properties of spaces of quasi-periodic functions and distributions and for investigating transitions of linear operators under the Zak transform.

The Zak transforms are unpredictable and exciting in several ways. They appear in natural ways when dealing with Gabor frame operators in the cases of "critical sampling", where the Gabor theory cease to work properly. This ought to be the reason why the transform possess several exciting and almost magical properties, useful in Gabor theory.

For example, in critical sampling cases, the Zak transform $Z$, adapted to the sampling parameters, takes the Gabor frame operator $S_{\phi,\psi}$ into the multiplication operator

[TABLE]

for some constant $c$ which depends on the sampling parameters. (See [20, 36] and Section 1 for notations.) We remark that this property is heavily used when showing that Gabor atoms and their canonical dual atoms often belong to the same function classes. (See [3, 4, 17].)

An other example concerns the fact that if $Zf$ is continuous, then it has zeros. This property is important when deducing various kinds of Balian-Low theorems, which are essential when finding limitations for bases and Gabor frames in Gabor analysis (see Theorem 8.4.1 and its consequences in[17]).

Before entering the Gabor theory, Zak transforms were first introduced and used in a problem in differential equation by Gelfand in [15]. Subsequently, the transforms were applied in various contexts, especially in solid state physics by Zak in [37] and in differential equations by Brezin in [2].

In these considerations it is essential to understand various kinds of mapping properties of the Zak transforms. The transforms take suitable functions, defined on the configuration space $\mathbf{R}^{d}$ into quasi-periodic functions, defined on the phase space $\mathbf{R}^{2d}$. Hence, in similar ways as for periodic functions, Zak transformed functions are completely described by their behaviour on suitable rectangles.

For example, the (standard) Zak transform is given by

[TABLE]

when $f$ is a suitable function or distribution (see (1.15) for the general definition of the Zak transform). By the definition it follows that if $F=Z_{1}f$ and $Q_{d,r}$ is the cube $[0,r]^{d}$, then $F$ is quasi-periodic (with respect to $Q_{d,1}\times Q_{d,2\pi}$). That is,

[TABLE]

It follows from these equalities that $F$ is completely reconstructable from its data on $Q_{d,1}\times Q_{d,2\pi}$.

It is well-known that $Z_{1}$ is bijective from $L^{2}(\mathbf{R}^{d})$ to the set of quasi-periodic elements in $L^{2}(Q_{d,1}\times Q_{d,2\pi})$ and that

[TABLE]

(Cf. e. g. [17, Theorem 8.2.3].) Consequently, $L^{2}(\mathbf{R}^{d})$ can be characterized in a convenient way by its image under the Zak transform.

An other space that can be characterized by related mapping properties concerns the Schwartz space $\mathscr{S}(\mathbf{R}^{d})$. In fact, it is proved in [21] by Janssen that $Z_{1}$ is continuous and bijective from $\mathscr{S}(\mathbf{R}^{d})$ to the set of quasi-periodic elements in $C^{\infty}(\mathbf{R}^{2d})$.

In [28, 29], Heil and Tinaztepe deduce some important mapping properties for the Zak transform on modulation spaces, and apply these results to deduce Balian-Low properties in the framework of such spaces. These mapping properties on modulation spaces seems not to be (complete) characterizations, because of absence of bijectivity. In fact, apart from the spaces $L^{2}(\mathbf{R}^{d})$ and $\mathscr{S}(\mathbf{R}^{d})$, the whole theory seems to lack characterizations of essential function and distribution spaces via the Zak transform as remarked in Subsection 8.2 (f) in [17].

In Section 2 we make this part more complete and furnish the theory with various kinds of characterizations. Especially we characterize modulation and Lebesgue spaces by suitable Lebesgue estimates of short-time Fourier transforms of the Zak transforms of the involved functions. We also characterize the dual $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ of $\mathscr{S}(\mathbf{R}^{d})$, the (standard) Gelfand-Shilov spaces and their distribution spaces by their images under the Zak transform.

For example we prove that $Z_{1}$ is continuous and bijective from $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ to the set of all quasi-periodic distributions on $Q_{d,1}\times Q_{d,2\pi}$. (See Theorem 2.1.) In Theorems 2.2 and 2.5 we deduce similar characterizations for Gelfand-Shilov spaces and their distribution spaces. As a consequence of Theorem 2.2 we have that the Zak transform $Z_{1}$ maps the Gelfand-Shilov space $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ bijectively to $\mathcal{E}_{Z,1}^{\sigma,s}(\mathbf{R}^{2d})$, the set of all quasi-periodic functions (quasi-periodic distributions) on $Q_{d,1}\times Q_{d,2\pi}$ in the Gevrey class $\mathcal{E}^{\sigma,s}(\mathbf{R}^{2d})$. In the same way it follows from Theorem 2.5 that the Zak transform $Z_{1}$ maps the Gelfand-Shilov distribution space $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ bijectively to $(\mathcal{E}_{Z,1}^{\sigma,s})^{\prime}(\mathbf{R}^{2d})$, the set of all quasi-periodic distributions on $Q_{d,1}\times Q_{d,2\pi}$ in $(\mathcal{S}_{s,\sigma}^{\sigma,s})^{\prime}(\mathbf{R}^{2d})$. As a consequence, if $s+\sigma<1$, then there are no non-trivial quasi-periodic functions in $\mathcal{E}^{\sigma,s}(\mathbf{R}^{2d})$ (cf. Corollary 2.3).

An other consequence of our results is that $Z_{1}$ maps the modulation space $M^{p}(\mathbf{R}^{d})$ continuously and bijectively to the set of all elements in $W^{\infty,p}(\mathbf{R}^{2d})$ which are quasi-periodic on $Q_{d,1}\times Q_{d,2\pi}$. Furthermore,

[TABLE]

(see Theorem 2.13 and Corollary 2.14).

We also use some recent results in [35] on Wiener estimates to deduce different versions of the latter characterization. For example we show that (0.2) in combination with results in [35, Section 2] give

[TABLE]

If $p=2$, then an application of Parseval’s formula implies that (0.3) is the same as

[TABLE]

which is a slightly weaker form of (0.1).

In Section 3 we apply the mapping results of the Zak transform to deduce duality properties for spaces of quasi-periodic functions and distributions. For example, if $p\in[1,\infty)$ and $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, then we prove that the dual of quasi-periodic elements in $\mathcal{E}^{\sigma,s}$ and in $W^{\infty,p}$ can be identified with the set of quasi-periodic elements in the Gelfand-Shilov distribution space $(\mathcal{S}_{s,\sigma}^{\sigma,s})^{\prime}$, respective in $W^{\infty,p^{\prime}}$. An essential part of these investigations concern characterizations of quasi-periodic elements in terms of estimates of their short-time Fourier transforms, given in the end of Section 2 and the beginning of Section 3.

Finally, in Section 4 we show how linear operators, $T$ are transformed under conjugation of the Zak transform, $T_{Z}=Z_{1}\circ T\circ Z_{1}^{-1}$. It follows from our investigations that the map $T\mapsto T_{Z}$ is a bijection between the set of all continuous linear mappings

[TABLE]

the set of all continuous linear mappings

[TABLE]

(cf. Theorems 4.1 and 4.2). At the same time we prove that a map $T_{Z}$ maps quasi-periodic functions or distributions into quasi-periodic functions or distributions, if and only if $T_{Z}$ commutes with each operator $U_{y,\eta}$, $y,\eta\in\mathbf{R}^{d}$, where

[TABLE]

Acknowledgement

I am very grateful to Professor Hans Feichtinger for reading parts of the paper and giving valuable comments, leading to improvements of the content and the style.

1. Preliminaries

In this section we recall some basic facts. We start by discussing Gelfand-Shilov spaces and their properties. Thereafter we recall some properties of modulation spaces and discuss different aspects of periodic distributions

1.1. Gelfand-Shilov spaces and Gevrey classes

Let $0<s,\sigma\in\mathbf{R}$ be fixed. Then the Gelfand-Shilov space $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ ($\Sigma_{s}^{\sigma}(\mathbf{R}^{d})$) of Roumieu type (Beurling type) with parameters $s$ and $\sigma$ consists of all $f\in C^{\infty}(\mathbf{R}^{d})$ such that

[TABLE]

is finite for some $h>0$ (for every $h>0$). Here the supremum should be taken over all $\alpha,\beta\in\mathbf{N}^{d}$ and $x\in\mathbf{R}^{d}$. We equip $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ ($\Sigma_{s}^{\sigma}(\mathbf{R}^{d})$) by the canonical inductive limit topology (projective limit topology) with respect to $h>0$, induced by the semi-norms in (1.1).

The Gelfand-Shilov distribution spaces $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ and $(\Sigma_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ are the dual spaces of $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ and $\Sigma_{s}^{\sigma}(\mathbf{R}^{d})$, respectively. As for the Gelfand-Shilov spaces there is a canonical projective limit topology (inductive limit topology) for $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ ($(\Sigma_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$).(Cf. [16, 22, 24].) For conveniency we set

[TABLE]

From now on we let $\mathscr{F}$ be the Fourier transform which takes the form

[TABLE]

when $f\in L^{1}(\mathbf{R}^{d})$. Here $\langle\,\cdot\,,\,\cdot\,\rangle$ denotes the usual scalar product on $\mathbf{R}^{d}$. The map $\mathscr{F}$ extends uniquely to homeomorphisms on $\mathscr{S}^{\prime}(\mathbf{R}^{d})$, from $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ to $(\mathcal{S}_{\sigma}^{s})^{\prime}(\mathbf{R}^{d})$ and from $(\Sigma_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ to $(\Sigma_{\sigma}^{s})^{\prime}(\mathbf{R}^{d})$. Furthermore, $\mathscr{F}$ restricts to homeomorphisms on $\mathscr{S}(\mathbf{R}^{d})$, from $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ to $\mathcal{S}_{\sigma}^{s}(\mathbf{R}^{d})$ and from $\Sigma_{s}^{\sigma}(\mathbf{R}^{d})$ to $\Sigma_{\sigma}^{s}(\mathbf{R}^{d})$, and to a unitary operator on $L^{2}(\mathbf{R}^{d})$.

Next we consider a more general class of Gelfand-Shilov spaces and their distribution spaces. Let $0\leq s_{1},s_{2},\sigma_{1},\sigma_{2}\in\mathbf{R}$ be fixed. Then the Gelfand-Shilov space $\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$ ($\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$) of Roumieu type (Beurling type) with parameters $s_{1}$, $s_{2}$, $\sigma_{1}$ and $\sigma_{2}$ consists of all $f\in C^{\infty}(\mathbf{R}^{d_{1}+d_{2}})$ such that

[TABLE]

is finite for some $h>0$ (for every $h>0$). Here the supremum should be taken over all $\alpha_{j},\beta_{j}\in\mathbf{N}^{d_{j}}$ and $x_{j}\in\mathbf{R}^{d_{j}}$, $j=1,2$. We equip $\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$ ($\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$) by the canonical inductive limit topology (projective limit topology) with respect to $h>0$, induced by the semi-norms in (1.2).

The space $\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$ is a Fréchet space when the topology is induced by the seminorms $\|\,\cdot\,\|_{\mathcal{S}_{s_{1},s_{2},h}^{\sigma_{1},\sigma_{2}}}$, $h>0$.

The Gelfand-Shilov distribution spaces $(\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$ and $(\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$ are the dual spaces of $\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$ and $\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$, respectively. Evidently, $\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$, $\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$ and their duals possess similar topological properties as $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$, $\Sigma_{s}^{\sigma}(\mathbf{R}^{d})$ and their duals. The space $\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$ is a Fréchet space with seminorms $\|\,\cdot\,\|_{\mathcal{S}_{s_{1},s_{2},h}^{\sigma_{1},\sigma_{2}}}$, $h>0$. Moreover, $\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})\neq\{0\}$, if and only if $s_{j}+\sigma_{j}\geq 1$ and $(s_{j},\sigma_{j})\neq(\frac{1}{2},\frac{1}{2})$, $j=1,2$, and $\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})\neq\{0\}$, if and only if $s_{j}+\sigma_{j}\geq 1$, $j=1,2$. By $(j+k)!\leq 2^{j+k}j!k!$ when $j,k\geq 0$ are integers we get $\mathcal{S}_{s,s}^{\sigma,\sigma}=\mathcal{S}_{s}^{\sigma}$ and $\Sigma_{s,s}^{\sigma,\sigma}=\Sigma_{s}^{\sigma}$.

For any $s_{j},\sigma_{j},s_{0,j},\sigma_{0,j}>0$ such that $s_{j}>s_{0,j}$, $\sigma_{j}>\sigma_{0,j}$ and $s_{0,j}+\sigma_{0,j}\geq 1$, $j=1,2$, we have

[TABLE]

with dense embeddings. Here and in what follows we use the notation $A\hookrightarrow B$ when the topological spaces $A$ and $B$ satisfy $A\subseteq B$ with continuous embeddings.

Let $\mathscr{F}_{j}f$ denote the partial Fourier transform of $f(x_{1},x_{2})\in\mathscr{S}(\mathbf{R}^{d_{1}+d_{2}})$ with respect to $x_{j}$, $j=1,2$. Then $\mathscr{F}_{1}$ and $\mathscr{F}_{2}$ extend uniquely to homeomorphisms

[TABLE]

The same holds true after each Gelfand-Shilov function or distribution space of Roumieu type have been replaced by corresponding Beurling type space.

Gelfand-Shilov spaces can in convenient ways be characterized in terms of estimates of the involved functions and their Fourier transforms. More precisely, in [5, 7] it is proved that if $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and $s,\sigma>0$, then $f\in\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ ($f\in\Sigma_{s}^{\sigma}(\mathbf{R}^{d})$), if and only if

[TABLE]

for some $r>0$ (for every $r>0$). Here $f(\theta)\lesssim g(\theta)$ means that $f(\theta)\leq cg(\theta)$ for some constant $c>0$ which is independent of $\theta$ in the domain of $f$ and $g$. We also set $f(\theta)\asymp g(\theta)$ when $f(\theta)\lesssim g(\theta)$ and $g(\theta)\lesssim f(\theta)$. More generally, it follows from [5] that if $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$ and $s_{1},s_{2},\sigma_{1},\sigma_{2}>0$, then $f\in\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$ ($f\in\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$), if and only if

[TABLE]

for some $r>0$ (for every $r>0$).

Gelfand-Shilov spaces and their distribution spaces can also be characterized by estimates of short-time Fourier transforms, (see e. g. [19, 32]). More precisely, let $\phi\in\mathcal{S}_{s}(\mathbf{R}^{d})$ be fixed. Then the short-time Fourier transform $V_{\phi}f$ of $f\in\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ with respect to the window function $\phi$ is the Gelfand-Shilov distribution on $\mathbf{R}^{2d}$, defined by

[TABLE]

If $f,\phi\in\mathcal{S}_{s}(\mathbf{R}^{d})$, then it follows that

[TABLE]

By [30, Theorem 2.3] it follows that the map $(f,\phi)\mapsto V_{\phi}f$ from $\mathscr{S}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d})$ to $\mathscr{S}(\mathbf{R}^{2d})$ is uniquely extendable to a continuous map from $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})\times(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ to $(\mathcal{S}_{s,\sigma}^{\sigma,s})^{\prime}(\mathbf{R}^{2d})$, and restricts to a continuous map from $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})\times\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ to $\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})$.

The same conclusion holds with $\Sigma_{s}^{\sigma}$ and $\Sigma_{s,\sigma}^{\sigma,s}$ in place of $\mathcal{S}_{s}^{\sigma}$ and $\mathcal{S}_{s,\sigma}^{\sigma,s}$, respectively, at each place.

The following results characterize Gelfand-Shilov spaces and their distribution spaces in terms of estimates of short-time Fourier transform.

Proposition 1.1**.**

Let $f$ be a Gelfand-Shilov distribution on $\mathbf{R}^{d_{1}+d_{2}}$,

[TABLE]

Then the following is true:

1. (1)

$f\in\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$* ($f\in\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$), if and only if*

[TABLE]

for some $r>0$ (for every $r>0$). 2. (2)

$f\in(\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$* ($f\in(\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$), if and only if*

[TABLE]

for every $r>0$ (for some $r>0$).

A proof of Proposition 1.1 (1) can be found in e. g. [19] (cf. [19, Theorem 2.7]) and a proof of Proposition 1.1 (2) in the case $d_{2}=0$ can be found in [32]. The general case of Proposition 1.1 (2) follows by similar arguments as in [32] and is left for the reader. See also [6] for related results.

Next we consider Gevrey classes on $\mathbf{R}^{d}$. Let $\sigma\geq 0$. For any compact set $K\subseteq\mathbf{R}^{d}$, $h>0$ and $f\in C^{\infty}(K)$ let

[TABLE]

The Gevrey class $\mathcal{E}^{\sigma}(K)$ ($\mathcal{E}^{\sigma;0}(K)$) of order $\sigma$ and of Roumieu type (of Beurling type) is the set of all $f\in C^{\infty}(K)$ such that (1.7) is finite for some (for every) $h>0$. We equipp $\mathcal{E}^{\sigma}(K)$ ($\mathcal{E}^{\sigma;0}(K)$) by the inductive (projective) limit topology with respect to $h>0$, supplied by the seminorms in (1.7). Finally if $\{K_{j}\}_{j\geq 1}$ is an exhausted set of compact subsets of $\mathbf{R}^{d}$, then let

[TABLE]

It is clear that $\mathcal{E}^{0;0}(\mathbf{R}^{d})$ contains all constant functions on $\mathbf{R}^{d}$, and that $\mathcal{E}^{0}(\mathbf{R}^{d})\setminus\mathcal{E}^{0;0}(\mathbf{R}^{d})$ contains all non-constant trigonometric polynomials.

1.2. Ordered, dual and phase split bases

Our discussions involving Zak transforms, periodicity, modulation spaces and Wiener spaces are done in terms of suitable bases.

Definition 1.2**.**

Let $E=\{e_{1},\dots,e_{d}\}$ be an ordered basis of $\mathbf{R}^{d}$. Then $E^{\prime}$ denotes the basis of $e_{1}^{\prime},\dots,e_{d}^{\prime}$ in $\mathbf{R}^{d}$ which satisfies

[TABLE]

The corresponding lattices are given by

[TABLE]

The sets $E^{\prime}$ and $\Lambda^{\prime}_{E}$ are called the dual basis and dual lattice of $E$ and $\Lambda_{E}$, respectively. If $E_{1},E_{2}$ are ordered bases of $\mathbf{R}^{d}$ such that a permutation of $E_{2}$ is the dual basis for $E_{1}$, then the pair $(E_{1},E_{2})$ are called permuted dual bases (to each others on $\mathbf{R}^{d}$).

Remark 1.3*.*

Evidently, if $E$ is the same as in Definition 1.2, then there is a matrix $T_{E}$ with $E$ as the image of the standard basis in $\mathbf{R}^{d}$. Then $E^{\prime}$ is the image of the standard basis under the map $T_{E^{\prime}}=2\pi(T^{-1}_{E})^{t}$.

Definition 1.4**.**

Let $E_{1},E_{2}$ be ordered bases of $\mathbf{R}^{d}$,

[TABLE]

and let $\pi_{j}$ from $\mathbf{R}^{2d}$ to $\mathbf{R}^{d}$, $j=1,2$, be the projections

[TABLE]

Then $E_{1}\times E_{2}$ is the ordered basis $\{e_{1},\dots,e_{2d}\}$ of $\mathbf{R}^{2d}$ such that

[TABLE]

In the phase space it is convenient to consider phase split bases, which are defined as follows.

Definition 1.5**.**

Let $V_{1}$, $V_{2}$, $\pi_{1}$ and $\pi_{2}$ be as in Definition 1.4, $E$ be an ordered basis of the phase space $\mathbf{R}^{2d}$ and let $E_{0}\subseteq E$. Then $E$ is called phase split (with respect to $E_{0}$), if the following is true:

1. (1)

the span of $E_{0}$ and $E\setminus E_{0}$ equal $V_{1}$ and $V_{2}$, respectively; 2. (2)

if $E_{1}=\pi_{1}(E_{0})$ and $E_{2}=\pi_{2}(E\setminus E_{0})$, then $(E_{1},E_{2})$ are permuted dual bases.

If $E$ is a phase split basis with respect to $E_{0}$ and that $E_{0}$ consists of the first $d$ vectors in $E$, then $E$ is called strongly phase split (with respect to $E_{0}$).

In Definition 1.5 it is understood that the orderings of $E_{0}$ and $E\setminus E_{0}$ are inherited from the ordering in $E$.

Remark 1.6*.*

Let $E$ and $E_{j}$, $j=0,1,2$ be the same as in Definition 1.5. It is evident that $E_{0}$ and $E\setminus E_{0}$ consist of $d$ elements, and that $E_{1}$ and $E_{2}$ are uniquely defined if the orders of $E_{0}$ and $E\setminus E_{0}$ are preserved. The pair $(E_{1},E_{2})$ is called the pair of permuted dual bases, induced by $E$ and $E_{0}$.

On the other hand, suppose that $(E_{1},E_{2})$ is a pair of permuted dual bases to each others on $\mathbf{R}^{d}$. Then it is clear that for $E_{1}\times E_{2}=\{e_{1},\dots,e_{2d}\}$ in Definition 1.4 and $E_{0}=\{e_{1},\dots,e_{d}\}$, we have that $E_{0}$ and $E$ fullfils all properties in Definition 1.5. In this case, $E_{1}\times E_{2}$ is called the phase split basis (of $\mathbf{R}^{2d}$) induced by $(E_{1},E_{2})$.

It follows that if $E^{\prime}$, $E_{1}^{\prime}$ and $E_{2}^{\prime}$ are the dual bases of $E$, $E_{1}$ and $E_{2}$, repsectively, then $E^{\prime}=E_{1}^{\prime}\times E_{2}^{\prime}$.

1.3. Invariant quasi-Banach spaces and spaces of

mixed quasi-normed spaces of Lebesgue types

We recall that a quasi-norm $\|\,\cdot\,\|_{\mathscr{B}}$ of order $r\in(0,1]$ on the vector-space $\mathscr{B}$ over $\mathbf{C}$ is a nonnegative functional on $\mathscr{B}$ which satisfies

[TABLE]

The space $\mathscr{B}$ is then called a quasi-normed space. A complete quasi-normed space is called a quasi-Banach space. If $\mathscr{B}$ is a quasi-Banach space with quasi-norm satisfying (1.8) then by [1, 27] there is an equivalent quasi-norm to $\|\,\cdot\,\|_{\mathscr{B}}$ which additionally satisfies

[TABLE]

From now on we always assume that the quasi-norm of the quasi-Banach space $\mathscr{B}$ is chosen in such way that both (1.8) and (1.9) hold.

Before giving the definition of $v$-invariant spaces, we recall some facts on weight functions.

A weight or weight function on $\mathbf{R}^{d}$ is a positive function $\omega\in L^{\infty}_{loc}(\mathbf{R}^{d})$ such that $1/\omega\in L^{\infty}_{loc}(\mathbf{R}^{d})$. The weight $\omega$ is called moderate, if there is a positive weight $v$ on $\mathbf{R}^{d}$ such that

[TABLE]

If $\omega$ and $v$ are weights on $\mathbf{R}^{d}$ such that (1.10) holds, then $\omega$ is also called $v$-moderate. We note that (1.10) implies that $\omega$ fulfills the estimates

[TABLE]

We let $\mathscr{P}_{E}(\mathbf{R}^{d})$ be the set of all moderate weights on $\mathbf{R}^{d}$.

By [18] it follows that if $\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$, then

[TABLE]

for some $r>0$.

We say that $v$ is submultiplicative if $v$ is even and (1.10) holds with $\omega=v$. In the sequel, $v$ and $v_{j}$ for $j\geq 0$, always stand for submultiplicative weights if nothing else is stated.

In the following we define a broad family of mixed quasi-normed Lebesgue spaces. Here $Q_{E}$ denotes the closed parallelepiped (or the $E$-cube) is spanned by the ordered basis $E$ of $\mathbf{R}^{d}$.

Definition 1.7**.**

Let $E=\{e_{1},\dots,e_{d}\}$ be an ordered basis of $\mathbf{R}^{d}$, $Q_{E}$ be the parallelepiped spanned by $E$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$ $\boldsymbol{q}=(q_{1},\dots,q_{d})\in(0,\infty]^{d}$ and $r=\min(1,\boldsymbol{q})$. If $f\in L^{r}_{loc}(\mathbf{R}^{d})$, then

[TABLE]

where $g_{k}\,:\,\mathbf{R}^{d-k}\to\mathbf{R}$, $k=0,\dots,d-1$, are inductively defined as

[TABLE]

If $\Omega\subseteq\mathbf{R}^{d}$ is measurable, then $L^{\boldsymbol{q}}_{E,(\omega)}(\Omega)$ consists of all $f\in L^{r}_{loc}(\Omega)$ with finite quasi-norm

[TABLE]

The space $L^{\boldsymbol{q}}_{E,(\omega)}(\Omega)$ is called $E$-split Lebesgue space (with respect to $\omega$, $\boldsymbol{q}$ and $\Omega$).

We let $\ell_{0}^{\prime}(\Lambda^{\prime}_{E})$ be the set of all formal sequences $a=\{a(j)\}_{j\in\Lambda_{E}}\subseteq\mathbf{C}$ on $\Lambda_{E}$ and $\ell^{\boldsymbol{p}}_{E,(\omega)}(\Lambda_{E})$ be the discrete version of $L^{\boldsymbol{p}}_{E,(\omega)}(\mathbf{R}^{d})$ when ${\boldsymbol{p}}\in(0,\infty]^{d}$. That is, $\ell^{\boldsymbol{p}}_{E,(\omega)}(\Lambda_{E})$ consists of all formal sequences $a\in\ell_{0}^{\prime}(\Lambda^{\prime}_{E})$ such that

[TABLE]

is finite. Here $\chi_{\Omega}$ is the characteristic function of the set $\Omega$.

Remark 1.8*.*

Evidently, $L^{\boldsymbol{q}}_{E,(\omega)}(\Omega)$ and $\ell^{\boldsymbol{q}}_{E,(\omega)}(\Lambda)$ in Definition 1.7 are quasi-Banach spaces of order $\min({\boldsymbol{p}},1)$. We set

[TABLE]

when $\omega=1$. For conveniency we identify $\boldsymbol{q}=(q,\dots,q)\in(0,\infty]^{d}$ with $q\in(0,\infty]$ when considering spaces involving Lebesgue exponents. In particular,

[TABLE]

respectively, with equivalent quasi-norms.

1.4. Modulation and Wiener spaces

We consider the following broad family of modulation spaces which contains the classical modulation spaces, introduced by Feichtinger in [9].

Definition 1.9**.**

Let ${\boldsymbol{p}},\boldsymbol{q}\in(0,\infty]^{d}$, $E_{1}$ and $E_{2}$ be ordered bases of $\mathbf{R}^{d}$, $E=E_{1}\times E_{2}$, $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$ and let $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$. For any $f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ set

[TABLE]

and

[TABLE]

The modulation space $M^{{\boldsymbol{p}},\boldsymbol{q}}_{E,(\omega)}(\mathbf{R}^{d})$ ($W^{{\boldsymbol{p}},\boldsymbol{q}}_{E,(\omega)}(\mathbf{R}^{d})$) consist of all $f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ such that $\|f\|_{M^{{\boldsymbol{p}},\boldsymbol{q}}_{E,(\omega)}}$ ($\|f\|_{W^{{\boldsymbol{p}},\boldsymbol{q}}_{E,(\omega)}}$) is finite.

The theory of modulation spaces has developed in different ways since they were introduced in [9] by Feichtinger. (Cf. e. g. [10, 14, 17, 31].) For example, let ${\boldsymbol{p}}$, $\boldsymbol{q}$, $E$, $\omega$ and $v$ be the same as in Definition 1.9, and let $L^{{\boldsymbol{p}},\boldsymbol{q}}_{E}(\mathbf{R}^{2d})$ and $r=\min(1,{\boldsymbol{p}},\boldsymbol{q})$. Then $M^{{\boldsymbol{p}},\boldsymbol{q}}_{E,(\omega)}(\mathbf{R}^{d})$ is a quasi-Banach space. Moreover, $f\in M^{{\boldsymbol{p}},\boldsymbol{q}}_{E,(\omega)}(\mathbf{R}^{d})$ if and only if $V_{\phi}f\cdot\omega\in L^{{\boldsymbol{p}},\boldsymbol{q}}_{E}(\mathbf{R}^{2d})$, and different choices of $\phi$ give rise to equivalent quasi-norms in Definition 1.9. We also note that

[TABLE]

Similar facts hold for the space $W^{{\boldsymbol{p}},\boldsymbol{q}}_{E,(\omega)}(\mathbf{R}^{d})$. (Cf. [14, 31].)

Definition 1.10**.**

Let ${\boldsymbol{p}},\boldsymbol{r}\in(0,\infty]^{d}$, $\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{d})$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$, $E\subseteq\mathbf{R}^{d}$ be an ordered basis, and let $Q_{E}$ be the closed parallelepiped spanned by $E$. Also let $f$ and $F$ be measurable on $\mathbf{R}^{d}$ respective $\mathbf{R}^{2d}$, and let $F_{\omega}=F\cdot\omega$. Then $\|f\|_{\mathsf{W}^{\boldsymbol{r}}_{E}(\omega_{0},\ell^{{\boldsymbol{p}}}_{E})}$ is given by

[TABLE]

The set $\mathsf{W}^{\boldsymbol{r}}_{E}(\omega,\ell^{{\boldsymbol{p}}}_{E})$ consists of all measurable $f$ on $\mathbf{R}^{d}$ such that $\|f\|_{\mathsf{W}^{\boldsymbol{r}}_{E}(\omega_{0},\ell^{{\boldsymbol{p}}}_{E})}<\infty$.

We observe that $\mathsf{W}^{\boldsymbol{r}}_{E}(\omega_{0},\ell_{E}^{{\boldsymbol{p}}})$ is equal to $W(L^{r},L^{{\boldsymbol{p}}}_{(\omega_{0})})$ in [11, 17, 14, 25, 26] when $E$ is the standard basis. In particular, $\mathsf{W}^{\boldsymbol{r}}_{E}(\omega_{0},\ell_{E}^{{\boldsymbol{p}}})$ is related to so-called coorbit spaces. (See [8, 11, 12, 13, 25, 26].)

Remark 1.11*.*

Let {\boldsymbol{p}},$$\boldsymbol{q}, $\omega_{0}$, $\omega$, $E$, $f$ and $F$ be the same as in Definition 1.10. Evidently, by using the fact that $\omega_{0}$ is $v_{0}$-moderate for some $v_{0}$, it follows that

[TABLE]

for $k=1,2$.

Remark 1.12*.*

For the spaces in Definition 1.10 we set $\mathsf{W}^{q_{0},\boldsymbol{r}_{0}}=\mathsf{W}^{\boldsymbol{r}}$, when

[TABLE]

and similarly for other types of exponents and for the spaces in Definition 1.9. (See also Remark 1.8.) We also set

[TABLE]

when $E_{1},E_{2}$ are ordered bases of $\mathbf{R}^{d}$ and $E=E_{1}\times E_{2}$, for spaces in Definition 1.9, since these spaces are independent of $E_{1}$.

The following result is essential when characterizing elements in modulation spaces in terms of estimates of their Zak transforms. We omit the proof since the result is a consequence of [35, Proposition 2.6].

Proposition 1.13**.**

Let $E_{0}$ be a basis for $\mathbf{R}^{d}$, $E_{0}^{\prime}$ be its dual basis, $E=E_{0}\times E_{0}^{\prime}$, $\boldsymbol{q},\boldsymbol{r}\in(0,\infty]^{d}$, $\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{d})$ and $\omega(x,\xi)=\omega_{0}(\xi)$. Then

[TABLE]

The next result is a restatement of Proposition 1.15*′* in [35]. The proof is therefore omitted. Here

[TABLE]

Proposition 1.14**.**

Let $E$ be a phase split basis for $\mathbf{R}^{2d}$, ${\boldsymbol{p}},\boldsymbol{r}\in(0,\infty]^{2d}$, $r\in(0,\min(1,{\boldsymbol{p}})]$, $\omega,v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that $\omega$ is $v$-moderate, $\rho$ and $\Theta_{\rho}v$ $\rho$ be as in (1.12) with strict inequality when $r<1$, and let $\phi_{1},\phi_{2}\in M^{1}_{(\Theta_{\rho}v)}(\mathbf{R}^{d})\setminus 0$. Then

[TABLE]

In particular, if $f\in\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$, then

[TABLE]

1.5. Classes of periodic elements

Let $s,\sigma\in\mathbf{R}_{+}$ be such that $s+t\geq 1$, $f\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$, $E$ be a basis of $\mathbf{R}^{d}$ and let $E_{0}\subseteq E$. Then $f$ is called $E_{0}$-periodic if $f(x+y)=f(x)$ for every $x\in\mathbf{R}^{d}$ and $y\in E_{0}$.

We note that for any $E$-periodic function $f\in C^{\infty}(\mathbf{R}^{d})$, we have

[TABLE]

For any $s\geq 0$ and basis $E\subseteq\mathbf{R}^{d}$ we let $\mathcal{E}^{\sigma;0}_{E}(\mathbf{R}^{d})$ and $\mathcal{E}^{\sigma}_{E}(\mathbf{R}^{d})$ be the sets of all $E$-periodic elements in $\mathcal{E}^{\sigma;0}(\mathbf{R}^{d})$ and in $\mathcal{E}^{\sigma}(\mathbf{R}^{d})$, respectively.

Let $s,s_{0},\sigma,\sigma_{0}>0$ be such that $s+\sigma\geq 1$, $s_{0}+\sigma_{0}\geq 1$ and $(s_{0},\sigma_{0})\neq(\frac{1}{2},\frac{1}{2})$. Then we recall that the duals $(\mathcal{E}^{\sigma}_{E})^{\prime}(\mathbf{R}^{d})$ and $(\mathcal{E}^{\sigma_{0};0}_{E})^{\prime}(\mathbf{R}^{d})$ of $\mathcal{E}^{\sigma}_{E}(\mathbf{R}^{d})$ and $\mathcal{E}^{\sigma_{0};0}_{E}(\mathbf{R}^{d})$, respectively, can be identified with the $E$-periodic elements in $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ and $(\Sigma_{s_{0}}^{\sigma_{0}})^{\prime}(\mathbf{R}^{d})$ respectively via unique extension of the form

[TABLE]

on $\mathcal{E}^{\sigma_{0};0}_{E}(\mathbf{R}^{d})\times\mathcal{E}^{\sigma_{0};0}_{E}(\mathbf{R}^{d})$. We also let $(\mathcal{E}^{0}_{E})^{\prime}(\mathbf{R}^{d})$ be the set of all formal expansions in (1.13) and $\mathcal{E}^{0}_{E}(\mathbf{R}^{d})$ be the set of all formal expansions in (1.13) such that at most finite numbers of $c(f,{\alpha})$ are non-zero (cf. [36]). We refer to [23, 36] for more characterizations of $\mathcal{E}^{\sigma}_{E}$, $\mathcal{E}^{\sigma;0}_{E}$ and their duals.

The following definition takes care of spaces of formal expansions (1.13) with coefficients obeying specific quasi-norm estimates.

Definition 1.15**.**

Let $E$ be a basis of $\mathbf{R}^{d}$, $\mathscr{B}$ be a quasi-Banach space continuously embedded in $\ell_{0}^{\prime}(\Lambda^{\prime}_{E})$ and let $\omega_{0}$ be a weight on $\mathbf{R}^{d}$. Then $\mathcal{E}_{E}(\omega_{0},\mathscr{B})$ consists of all $f\in(\mathcal{E}^{0}_{E})^{\prime}(\mathbf{R}^{d})$ such that

[TABLE]

is finite.

The next result is a reformulation of Proposition 1.18*′* in [35]. The proof is therefore omitted.

Proposition 1.16**.**

Let $E$ be an ordered basis of $\mathbf{R}^{d}$, $E=E_{0}\times E_{0}^{\prime}$, $\boldsymbol{q},\boldsymbol{r}\in(0,\infty]^{d}$, $\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{d})$ and let $\omega(x,\xi)=\omega_{0}(\xi)$, $x,\xi\in\mathbf{R}^{d}$. Then

[TABLE]

Remark 1.17*.*

Let $E_{0}$, $\boldsymbol{q}$, $\boldsymbol{r}$ and $\omega$ be the same as in Proposition 1.16. Also let $v_{0}\in\mathscr{P}_{E}(\mathbf{R}^{d})$ be such that $\omega_{0}$ is $v_{0}$-moderate, $v(x,\xi)=v_{0}(\xi)$, $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$, $r_{0}\leq\min(\boldsymbol{r})$, $f\in(\mathcal{E}^{0}_{E})^{\prime}(\mathbf{R}^{d})$ with Fourier series expansion (1.13),

[TABLE]

Then it follows from Proposition 2.7 and Remark 2.8 in [35] that

[TABLE]

1.6. The Zak transform

For any ordered basis $E$ of $\mathbf{R}^{d}$ and $f\in\mathscr{S}(\mathbf{R}^{d})$, the Zak transform is defined by

[TABLE]

Several properties for the Zak transform can be found in [17]. For example, by the definition it follows that $Z_{E}$ is continuous from $\mathscr{S}(\mathbf{R}^{d})$ to the set of all smooth functions on $\mathbf{R}^{2d}$ which are bounded together with all their derivatives. It is also clear that $Z_{E}f$ is quasi-periodic of order $E$. Here, if $F$ is a function or an ultra-distribution, then $F$ is called quasi-periodic of order $E$, when

[TABLE]

By interpreting (1.15) as a Fourier series in the $\xi$ variable, we regain $f(x)$ as the zero order Fourier coefficient, which is evaluated by

[TABLE]

For conveniency we set $Z_{1}=Z_{E}$ when $E$ is the standard basis of $\mathbf{R}^{d}$, and recall the following important mapping properties on $L^{2}(\mathbf{R}^{d})$ and $\mathscr{S}(\mathbf{R}^{d})$.

Proposition 1.18**.**

Let $E$ be an ordered basis of $\mathbf{R}^{d}$. Then the operator $Z_{E}$ is homeomorphic from $L^{2}(\mathbf{R}^{d})$ to the set of all quasi-periodic elements of order $E$ in $L^{2}_{loc}(\mathbf{R}^{2d})$, and

[TABLE]

Proof.

Let $T_{E}$ be as in Remark 1.3. By straight-forward computations it follows that

[TABLE]

The assertion now follows from (0.1), (1.18) and suitable changes of variables in the involved integrals. The details are left for the reader. ∎

2. Zak transform on Gelfand-Shilov spaces,

Lebesgue spaces and modulation spaces

In this section we deduce characterizations of Lebesgue spaces, modulation spaces, and Gelfand-Shilov spaces and their distribution spaces in terms of suitable estimates of the Zak transforms of the involved elements. The characterizations on modulation spaces are related to results given in [28, 29].

2.1. Spaces of quasi-periodic functions and

distributions

Since quasi-periodic functions depend on the phase space variable $(x,\xi)\in\mathbf{R}^{2d}$, it is suitable that the Gevrey regularity with respect to $x\mathbf{R}^{d}$ for such functions might be different to the Gevrey regularity with respect to $\xi\in\mathbf{R}^{d}$. We therefore consider two parameters analogies of $\mathcal{E}^{s}$ and $\mathcal{E}^{s;0}$, where the parameter $s$ is replaced by the pair $\sigma,s$. More preceisely, for any compact $K\subseteq\mathbf{R}^{2d}$ and $s,\sigma\geq 0$, $\mathcal{E}^{\sigma,s}(K)$ ($\mathcal{E}^{\sigma,s;0}(K)$) is the set of all $F\in C^{\infty}(K)$ such that

[TABLE]

is finite for some $h>0$ (for every $h>0$). The two parameter Gevrey classes, $\mathcal{E}^{\sigma,s}(\mathbf{R}^{d})$ and $\mathcal{E}^{\sigma,s;0}(\mathbf{R}^{d})$, are the projective limits of $\mathcal{E}^{\sigma,s}(K_{j})$ respective $\mathcal{E}^{\sigma,s;0}(K_{j})$, when $\{K_{j}\}_{j\geq 1}$ is an exhausted set of compact subsets of $\mathbf{R}^{2d}$. Furthermore we let

[TABLE]

respectively, with respect to the ordered basis $E$ on $\mathbf{R}^{d}$. For conveniency we also set

[TABLE]

when $E$ is the standard basis of $\mathbf{R}^{d}$.

Next we introduce spaces of quasi-periodic functions and distributions which correspond to Lebesgue spaces and modulation spaces. We let $L^{p}_{Z,E}(\mathbf{R}^{2d})$ for $p\in(0,\infty]$ be the set of all quasi-periodic measurable functions $F$ on $\mathbf{R}^{2d}$ with respect to the ordered basis $E$ such that

[TABLE]

is finite. Evidently, we may identify $L^{p}_{Z,E}(\mathbf{R}^{2d})$ by $L^{p}(Q_{E\times E^{\prime}})$, and the scalar product on $L^{2}_{Z,E}(\mathbf{R}^{2d})$ is given by

[TABLE]

when $F,G\in L^{2}_{Z,E}(\mathbf{R}^{2d})$.

Let ${\boldsymbol{p}},\boldsymbol{r}\in(0,\infty]^{2d}$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{4d})$ and $\Phi\in\Sigma_{1}(\mathbf{R}^{2d})\setminus 0$ be fixed, and let $E_{0}$ be an ordered basis in $\mathbf{R}^{d}$. Then set

[TABLE]

when $F$ is a quasi-periodic Gelfand-Shilov distribution with respect to the ordered basis $E$. We let $W_{Z,E,E_{0},(\omega)}^{\boldsymbol{r},{\boldsymbol{p}}}(\mathbf{R}^{2d})$ be the set of all quasi-periodic Gelfand-Shilov distributions $F$ with respect to the ordered basis $E$ such that

[TABLE]

is finite. We also let the topology of $W_{Z,E,E_{0},(\omega)}^{\boldsymbol{r},{\boldsymbol{p}}}(\mathbf{R}^{2d})$ be induced by the quasi-norm $\|\,\cdot\,\|_{W_{Z,E,E_{0},(\omega)}^{\boldsymbol{r},{\boldsymbol{p}}}}$. Usually we assume that $\omega$ is given by

[TABLE]

for some $\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$.

2.2. The Zak transform on test function spaces

and their distribution spaces

For the classical spaces $\mathscr{S}(\mathbf{R}^{d})$ and its distribution space $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ we have the following.

Theorem 2.1**.**

Let $E$ be an ordered basis of $\mathbf{R}^{d}$. Then the following is true:

1. (1)

The operator $Z_{E}$ is a homeomorphism from $\mathscr{S}(\mathbf{R}^{d})$ to $C^{\infty}_{Z,E}(\mathbf{R}^{2d})$; 2. (2)

The operator $Z_{E}$ from $\mathscr{S}(\mathbf{R}^{d})$ to $C^{\infty}(\mathbf{R}^{2d})$ is uniquely extendable to a homeomorphism from $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ to $\mathscr{S}^{\prime}_{Z,E}(\mathbf{R}^{2d})$.

The assertion (1) in Theorem 2.1 follows from (1.18) and [17, Theorem 8.2.5], and (2) in the same theorem follows by similar arguments as in the proof of Theorem 2.5 below. The verifications of Theorem 2.1 are therefore left for the reader.

The analogous result of the previous theorem for Gelfand-Shilov functions and their distributions, are given in Theorems 2.2 and 2.5 below.

Theorem 2.2**.**

Let $s,\sigma>0$ and $E$ be an ordered basis. Then the operator $Z_{E}$ from $\mathscr{S}(\mathbf{R}^{d})$ to $C^{\infty}(\mathbf{R}^{2d})$ restricts to a homeomorphism from $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ to $\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$.

The same holds true with $\Sigma_{s}^{\sigma}(\mathbf{R}^{d})$ and $\mathcal{E}^{\sigma,s;0}_{Z,E}(\mathbf{R}^{2d})$ in place of $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ and $\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$, respectively at each occurrence.

By the previous result and the facts that $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ is trivially equal to $\{0\}$ when $s+\sigma<1$, and $\Sigma_{s}^{\sigma}(\mathbf{R}^{d})$ is trivial when $s+\sigma<1$ or $s=\sigma=\frac{1}{2}$, we get the following.

Corollary 2.3**.**

Let $s,\sigma>0$ and $E$ be an ordered basis. Then the following is true:

1. (1)

if $s+\sigma<1$ and $F\in\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$, then $F=0$; 2. (2)

if $s+\sigma<1$ or $s=\sigma=\frac{1}{2}$ and $F\in\mathcal{E}^{\sigma,s;0}_{Z,E}(\mathbf{R}^{2d})$, then $F=0$.

Remark 2.4*.*

Let $\sigma\geq 0$, $E$ be a basis of $\mathbf{R}^{d}$ and let $f$ be an $E$-periodic distribution on $\mathbf{R}^{d}$. Then $f\in\mathcal{E}^{\sigma}(\mathbf{R}^{d})$, if and only if its Fourier coefficients $c(f,\alpha)$ in (1.13) satisfies

[TABLE]

for some $r>0$. In particular, $\mathcal{E}^{\sigma}_{E}(\mathbf{R}^{d})\neq\{0\}$ for every $\sigma\geq 0$ (cf. [23, 36]). Consequently, the conclusions in Corollary 2.3 are not true for periodic functions in place of quasi-periodic functions.

Theorem 2.5**.**

Let $s,\sigma>0$ and $E$ be an ordered basis of $\mathbf{R}^{d}$. Then the operator $Z_{E}$ from $\mathscr{S}(\mathbf{R}^{d})$ to $C^{\infty}(\mathbf{R}^{2d})$ extends uniquely to a homeomorphism from $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ to $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$.

The same holds true with $(\Sigma_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ and $(\mathcal{E}^{\sigma,s;0}_{Z,E})^{\prime}(\mathbf{R}^{2d})$ in place of $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ and $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$, respectively at each occurrence.

We need the following lemma for the proof of Theorem 2.2.

Lemma 2.6**.**

Let $r,s>0$. Then there is a constant $h>0$ such that

[TABLE]

Proof.

By reasons of symmetry and Stirling’s formula, the result follows if we prove

[TABLE]

for some $h>0$. By taking the logaritm it follows that we need to prove that for some constant $C>0$,

[TABLE]

is bounded from above by a constant which is independent of $t,\tau>0$.

By differentiation and analysing the sign of $g^{\prime}(\tau)$, it follows that $g(\tau)$ has a global maximum for

[TABLE]

with value

[TABLE]

By choosing

[TABLE]

it follows that $g(\tau)$ is negative, giving the result. ∎

Proof of Theorem 2.2.

Let $T_{E}$ be the same as in (1.18). Then the map $F(x,\xi)\mapsto F(T_{E}^{-1}x,T_{E}^{*}\xi)$ maps quasi-periodc elements of order $E$ to quasi-periodic elements with respect to the standard basis. Since $f\mapsto f\circ T$ maps $E$-periodic elements to $1$-periodic functions, it follows from these observations and (1.18) that it suffices to prove the result when $E$ is the standard basis.

The assertion (1) is the same as Theorem 8.2.5 in [17].

In order to prove (2) we shall follow the proof of Theorem 8.2.5 in [17]. In fact, assume first that $f\in\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$, $x\in k_{0}+Q_{d,1}$ for some fixed $k_{0}\in\mathbf{Z}^{d}$, and let $F=Z_{1}f$. Then

[TABLE]

for some positive constant $C$ which only depends on $k_{0}$ and some positive constants $h>0$ and $r>0$ which are independent of $x$, $k_{0}$, $k$ and $\alpha$.

The series in (1.15) is absolutely convergent together with all its derivatives. This gives

[TABLE]

for some constant $h_{1}>0$. By Lemma 2.6 it follows that

[TABLE]

for some constant $h>0$. A combination of these estimates give

[TABLE]

and it follows that $F\in\mathcal{E}^{\sigma,s}(\mathbf{R}^{2d})$. This shows that $Z_{1}$ is continuous from $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ to $\mathcal{E}^{\sigma,s}_{Z}(\mathbf{R}^{2d})$.

Next we show that any $F$ in $\mathcal{E}^{\sigma,s}_{Z}(\mathbf{R}^{2d})$ is the Zak transform of an element in $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$. By Theorem 8.2.5 in [17] it follows that $F=Z_{1}f$ when

[TABLE]

We need to prove that $f\in\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$.

Since $k\mapsto f(x-k)$ is the Fourier coefficient of order $k$ for the function $\xi\mapsto F(x,\xi)$, we have

[TABLE]

By applying the operator $k^{\alpha}\partial_{x}^{\beta}$ and integrating by parts we get

[TABLE]

This gives

[TABLE]

which is the same as $f\in\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$. This gives (2). The assertion (3) follows by similar arguments and is left for the reader. ∎

For the proof of Theorem 2.5 we need the following lemma on tensor product of Gelfand-Shilov distributions.

Lemma 2.7**.**

Let $s_{j},\sigma_{j}>0$ and $f_{j}\in(\mathcal{S}_{s_{j}}^{\sigma_{j}})^{\prime}(\mathbf{R}^{d_{j}})$, $j=1,2$. Then there is a unique $f\in(\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$ such that

[TABLE]

Moreover, if $\varphi\in\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}(\mathbf{R}^{d_{1}+d_{2}})$,

[TABLE]

then

[TABLE]

The same holds true with $\Sigma_{s_{j}}^{\sigma_{j}}$, $\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}$, $(\Sigma_{s_{j}}^{\sigma_{j}})^{\prime}$ and $(\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}$ in place of $\mathcal{S}_{s_{j}}^{\sigma_{j}}$, $\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}}$, $(\mathcal{S}_{s_{j}}^{\sigma_{j}})^{\prime}$ and $(\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}$, respectively, $j=1,2$.

Lemma 2.7 is essentially a restatement of Theorem 2.4 in [34]. The proof is therefore omitted.

Remark 2.8*.*

We notice that the uniqueness assertions in Lemma 2.7 is an immediate consequence of [34, Lemma 2.3] which asserts that if $f\in(\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$ ($f\in(\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$) satisfies

[TABLE]

for every $\varphi_{j}\in\mathcal{S}_{s_{j}}^{\sigma_{j}}(\mathbf{R}^{d_{j}})$ ($\varphi_{j}\in\Sigma_{s_{j}}^{\sigma_{j}}(\mathbf{R}^{d_{j}})$), then $f=0$ (as an element in $(\mathcal{S}_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$ ($(\Sigma_{s_{1},s_{2}}^{\sigma_{1},\sigma_{2}})^{\prime}(\mathbf{R}^{d_{1}+d_{2}})$)).

Proof of Theorem 2.5.

By similar arguments as in the proof of Theorem 2.2 we may assume that $E$ is the standard basis for $\mathbf{R}^{d}$.

We begin to prove (2). Let $\Phi\in\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})$. Then

[TABLE]

If $f\in\mathscr{S}(\mathbf{R}^{d})$, then

[TABLE]

where $\Phi_{2}=\mathscr{F}_{2}^{-1}\Phi$ and $\check{f}(x)=f(-x)$.

Assume instead that $f\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ is arbitrary. We claim that the series on the right-hand side of (2.5) converges absolutely for every $\Phi$ as above.

In fact, since $f\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$, we have

[TABLE]

for every $r>0$, giving that for some $r_{0}>0$ and $c\geq 1$ we have

[TABLE]

Hence, if $r$ is chosen small enough and $x=y=j$, then

[TABLE]

when $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$. The absolutely convergence of the series of the right-hand side of (2.5) now follows from (2.6).

If $f\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$, then $Z_{1}f$ is defined as the element in $(\mathcal{S}_{s,\sigma}^{\sigma,s})^{\prime}(\mathbf{R}^{2d})$, given by the right-hand side of (2.5). The previous estimates show that this definition makes sense, and that the map $f\mapsto Z_{1}f$ is continuous from from $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ to $(\mathcal{E}^{\sigma,s}_{Z})^{\prime}(\mathbf{R}^{2d})$. By approximating elements in $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ by sequences of elements in $\mathscr{S}(\mathbf{R}^{d})$, it also follows that the continuous extension of $Z_{1}$ to such distribution is unique.

We need to prove that any element in $(\mathcal{E}^{\sigma,s}_{Z})^{\prime}(\mathbf{R}^{2d})$ is the Zak transform of an element in $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$. Therefore, let $\varphi\in\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$, $F\in(\mathcal{E}^{\sigma,s}_{Z})^{\prime}(\mathbf{R}^{2d})$, and let $g_{\varphi}\in(\mathcal{S}_{\sigma}^{s})^{\prime}(\mathbf{R}^{d})$ be defined by

[TABLE]

Then $g_{\varphi}$ is $2\pi$-periodic, and it follows from Remark 2.3 and Proposition 2.5 in [36] that if $\phi\in\mathcal{S}_{\sigma}^{s}(\mathbf{R}^{d})\setminus 0$, then

[TABLE]

By straight-forward computations we get

[TABLE]

and it is clear that the map which takes $\varphi$ into the right-hand side in (2.7) defines a continuous linear form on $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$. Hence

[TABLE]

for some $f\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$. Furthermore, by the quasi-periodicity of $F$ we obtain

[TABLE]

that is,

[TABLE]

A combination of these facts now gives

[TABLE]

giving that

[TABLE]

Hence, if $F_{0}=F-Z_{1}f$, then $\langle F_{0},\varphi\otimes\psi\rangle=0$ when $\varphi\in\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ and $\psi\in\mathcal{S}_{\sigma}^{s}(\mathbf{R}^{d})$. By Remark 2.8 it now follows that $F=Z_{1}f$, which gives the result. ∎

For completeness we also show that all quasi-periodic distributions are tempered or Gelfand-Shilov distributions. (Cf. [20, Section 7.2].) Here $(\mathcal{D}^{\sigma,s})^{\prime}(\mathbf{R}^{2d})$ and $(\mathcal{D}^{\sigma,s;0})^{\prime}(\mathbf{R}^{2d})$ are the duals of $\mathcal{E}^{\sigma,s}(\mathbf{R}^{2d})\cap C^{\infty}_{0}(\mathbf{R}^{2d})$ and $\mathcal{E}^{\sigma,s;0}(\mathbf{R}^{2d})\cap C^{\infty}_{0}(\mathbf{R}^{2d})$, respectively.

Proposition 2.9**.**

Let $s,\sigma>1$ and $E$ be an ordered basis of $\mathbf{R}^{d}$. Then the following is true:

1. (1)

The set of all quasi-periodic elements of order $E$ in $\mathscr{D}^{\prime}(\mathbf{R}^{2d})$ is equal to $\mathscr{S}^{\prime}_{Z,E}(\mathbf{R}^{2d})$; 2. (2)

The set of all quasi-periodic elements of order $E$ in $(\mathcal{D}^{\sigma,s})^{\prime}(\mathbf{R}^{2d})$ is equal to $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$; 3. (3)

The set of all quasi-periodic elements of order $E$ in $(\mathcal{D}^{\sigma,s;0})^{\prime}(\mathbf{R}^{2d})$ is equal to $(\mathcal{E}^{\sigma,s;0}_{Z,E})^{\prime}(\mathbf{R}^{2d})$.

Proof.

We only prove (2). The other assertions follow by similar arguments and are left for the reader.

Let $F\in C^{\infty}_{Z,E}(\mathbf{R}^{2d})$, $\Phi\in\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})$ and let $\chi\in C_{0}^{\infty}(\mathbf{R}^{2d})\cap\mathcal{E}^{\sigma,s}(\mathbf{R}^{2d})$ be such that

[TABLE]

If $\Phi\in\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})$, then it follows by the quasi-periodisity of $F$ and some straight-forward computations that

[TABLE]

and that $T_{\chi}$ in (2.9) is continuous from $\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})$ to $C_{0}^{\infty}(\mathbf{R}^{2d})\cap\mathcal{E}^{\sigma,s}(\mathbf{R}^{2d})$.

By letting $\langle F,\Phi\rangle$ be defined by the right-hand side of (2.8) when $F\in(\mathcal{D}^{\sigma,s})^{\prime}(\mathbf{R}^{2d})$ and $\Phi\in\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})$, it follows that $\Phi\mapsto\langle F,T_{\chi}\Phi\rangle$ in (2.8) defines a linear and continuous form on $\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})$ which agree with the usual distribution action, $\Phi\mapsto\langle F,\Phi\rangle$ when $\Phi\in C_{0}^{\infty}(\mathbf{R}^{2d})\cap\mathcal{E}^{\sigma,s}(\mathbf{R}^{2d})$. ∎

The mapping properties of Gelfand-Shilov distributions also lead to some quieries concerning the inversion formula (1.17) for the Zak transform. Evidently, if $F$ is a general quasi-periodic distribution or even Gelfand-Shilov distribution, then the integral on the right-hand side of (1.17) might not be defined. On the other hand, since $Z_{E}^{-1}F(x)$ is the zero order Fourier coefficient of the expansion (1.15), it follows from [36, Remark 2.3] that the following is true. The details are left for the reader.

Proposition 2.10**.**

Let $s,\sigma>0$ and $\phi\in\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})\setminus 0$ ($\phi\in\Sigma_{s}^{\sigma}(\mathbf{R}^{d})\setminus 0$) be fixed, $f\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ ($f\in(\Sigma_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$), and let $F=Z_{E}f$. Then

[TABLE]

2.3. The Zak transform on Lebesgue and

modulation spaces

For completeness we begin the subsection by making a review of the Zak transform when acting on Lebesgue spaces. Here we let

[TABLE]

Proposition 2.11**.**

Let $E$ be an ordered basis of $\mathbf{R}^{d}$. Then the following is true:

1. (1)

if $p\in(0,2]$, then $Z_{E}$ from $\mathscr{S}(\mathbf{R}^{d})$ to $C^{\infty}(\mathbf{R}^{2d})$ is uniquely extendable to a continuous map from $L^{p}(\mathbf{R}^{d})$ to $L^{p}_{Z,E}(\mathbf{R}^{2d})$, and

[TABLE] 2. (2)

if $p\in[2,\infty]$, then $Z_{E}^{-1}$ from $C^{\infty}_{Z,E}(\mathbf{R}^{2d})$ to $\mathscr{S}(\mathbf{R}^{d})$ is uniquely extendable to a continuous map from $L^{p}_{Z,E}(\mathbf{R}^{2d})$ to $L^{p}(\mathbf{R}^{d})$, and

[TABLE] 3. (3)

if $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$, $p\in[1,\infty]$ and $v\in\mathscr{P}_{E}(\mathbf{R}^{d})$ satisfy $1/v\in L^{1}(\mathbf{R}^{d})$, then

[TABLE]

Proposition 2.11 (1) and (2) are evidently true for $p=2$, in view of Proposition 1.18, and is presented in [28, Lemma 3.1.2], without any proof in the case $p=1$. In order to be self-contained, we give a proof in Appendix A.

When investigation mapping properties of the Zak transform on modulation spaces, we need to deduce various kinds of estimates on short-time Fourier transforms and partial short-time Fourier transforms of Zak transforms. Especially we search suitable estimates on $V_{\Phi}(Z_{E}f)$, and on

[TABLE]

which are compositions of the Zak transform and the partial short-time Fourier transforms with respect to the first and second variable, respectively.

From the previous subsection it is clear that there is a one-to-one correspondence between quasi-periodic functions and distributions, and Zak transforms of functions and distributions. For a quasi-periodic function or distribution $F$ on $\mathbf{R}^{2d}$ which satisfies (1.16), and a suitable function or distribution $\Phi$ on $\mathbf{R}^{2d}$, we have

[TABLE]

which follows by straight-forward computations. We remark that functions and distributions which satisfy conditions given in (2.13) are so-called echo-periodic functions and distributions, considered in [33].

First we have the following result concerning identifying Lebesgue spaces via estimates of corresponding Zak transforms.

Theorem 2.12**.**

Let $E$ be an ordered basis of $\mathbf{R}^{d}$, $p,r\in(0,\infty]$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$, $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$, and let $f$ be a Gelfand-Shilov distribution on $\mathbf{R}^{d}$. Then

[TABLE]

Proof.

We only prove the result for $p<\infty$. The case $p=\infty$ follows by similar arguments and is left for the reader.

The distribution $\xi\mapsto Z_{E}f(x,\xi)$ is $E^{\prime}$-periodic, and it follows from (1.14) that

[TABLE]

The result now follows by apply the $L^{p}(Q_{E})$ quasi-norm with respect to the $x$-variable. ∎

In the same way we may identify modulation spaces by using the Zak transform as in the next result. We also recall Definition 1.9 and Remark 1.12 for definitions and notions concerning the Wiener amalgam space $W^{{\boldsymbol{p}}}_{E,(\omega)}(\mathbf{R}^{d})$.

Theorem 2.13**.**

Let $E,E_{0}$ be an ordered bases of $\mathbf{R}^{d}$, ${\boldsymbol{p}},\boldsymbol{r}\in(0,\infty]^{2d}$, $\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ and $\omega\in\mathscr{P}_{E}(\mathbf{R}^{4d})$ be such that (2.4) holds. Then $Z_{E}$ from $\Sigma_{1}(\mathbf{R}^{d})$ to $C^{\infty}(\mathbf{R}^{2d})$ is uniquely extendable to a homeomorphism from $M^{{\boldsymbol{p}}}_{E\times E_{0}^{\prime},(\omega_{0})}(\mathbf{R}^{d})$ to $W^{\boldsymbol{r},{\boldsymbol{p}}}_{Z,E,E_{0},(\omega)}(\mathbf{R}^{2d})$, and

[TABLE]

Proof.

First we prove the result for $\boldsymbol{r}=\infty$. Let $\Phi=\phi_{1}\otimes\phi_{2}$ with $\phi_{1},\phi_{2}\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$. By straight-forward computations we get

[TABLE]

Let

[TABLE]

when

[TABLE]

and consider the functions

[TABLE]

Since $\xi\mapsto(\mathsf{ZV}_{1,\phi_{1}}^{(1)}f)(x,\xi,\eta)$ is $E^{\prime}$-periodic with Fourier coefficients

[TABLE]

(cf. (2.18)), and the (partial) short-time Fourier transform of that distribution equals $V_{\Phi}(Z_{1}f)$, it follows from (1.14) that

[TABLE]

First let $r_{0}\leq\min({\boldsymbol{p}})$. If we apply the $L^{{\boldsymbol{p}}_{2}}_{E_{0}^{\prime}}$ norm on (2.19) with respect to the $\eta$ variable and using Hölder’s inequality we get

[TABLE]

If

[TABLE]

then the fact that $r_{0}\leq\min({\boldsymbol{p}})$ and Jensen’s inequality give $g_{1}\lesssim g_{0}$.

By applying the $L^{r_{0}}_{E}(Q_{E})$ norm on the latter inequality, using the fact that

[TABLE]

and Jensen’s inequality again we obtain

[TABLE]

where $E_{1}=E\times E_{0}^{\prime}$. That is,

[TABLE]

which is the same as

[TABLE]

in view of Proposition 1.14.

In order to estimate $h_{0,r_{0}}$ we apply (2.13) to get

[TABLE]

By first applying the $L^{{\boldsymbol{p}}_{1}}_{E}(\mathbf{R}^{d})$ norm with respect to the $y$ variable and then the $L^{{\boldsymbol{p}}_{2}}_{E_{0}^{\prime}}(\mathbf{R}^{d})$ norm with respect to the $\eta$ variable we get

[TABLE]

Hence, by applying the $L^{r_{0}}_{E}(Q_{E})$ norm on $h_{0,r_{0}}$ and using Hölder’s and Jensen’s inequalities we get

[TABLE]

A combination of (2.20)–(2.22) and Proposition 1.14 now gives

[TABLE]

In order to get the reversed estimate we again apply the $L^{{\boldsymbol{p}}_{2}}_{E_{0}^{\prime}}$ norm on (2.19) with respect to the $\eta$ variable and use Hölder’s inequality to get

[TABLE]

If

[TABLE]

then Jensen’s inequality give $g_{0}\lesssim g_{2}$.

By applying the $L^{\infty}_{E}(Q_{E})$ norm on the latter inequality and using Jensen’s inequality again we obtain

[TABLE]

That is,

[TABLE]

which is the same as

[TABLE]

in view of Proposition 1.14.

By applying the $L^{\infty}_{E}(Q_{E})$ norm on $h_{0,\infty}$ and using (2.22) we get

[TABLE]

where the last relation follows from Proposition 1.14. A combination of (2.24), (2.25) and (2.26) now gives

[TABLE]

and the result in the case $\boldsymbol{r}=\infty$ follows by combining (2.27) with (2.23).

For general $\boldsymbol{r}\in(0,\infty]^{2d}$, we notice that the echo-periodicity (2.13) implies that $H_{2,F,\omega,E,E_{0},{\boldsymbol{p}}}$ in (2.3) is $E\times E^{\prime}$ periodic. The general case now follows from Proposition 1.14, Theorem 2.13 and the previous observation. The details are left for the reader. ∎

A consequence of the previous result is that $W_{Z,E,E_{0},(\omega)}^{\boldsymbol{r},{\boldsymbol{p}}}(\mathbf{R}^{2d})$ is independent of $\boldsymbol{r}$ (also in topological sense), and for this reason we set

[TABLE]

As a special case of the previous result we have the following.

Corollary 2.14**.**

Let $E$ be an ordered basis in $\mathbf{R}^{d}$, $\Phi\in\Sigma_{1}(\mathbf{R}^{2d})\setminus 0$ and let $p\in(0,\infty]$. Then

[TABLE]

3. Duality properties and some further characterizations

of quasi-peridic elements

In this section we discuss various aspects concerning duality and characterizations for quasi-periodic elements, as well as transitions of linear operators under the Zak transform. In Subsection 3.1 we show that the elements in $\mathcal{E}^{\sigma,s}_{Z,E}$ and $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}$ can be completely characterized by estimates on their short-time Fourier transform. Thereafter we use these characterizations in Subsection 3.2 to show that the $L^{2}_{Z,E}$ form on $\mathcal{E}^{\sigma,s}_{Z,E}$ is uniquely extendable to a continuous sesqui-linear form on $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}\times\mathcal{E}^{\sigma,s}_{Z,E}$, and that the dual of $\mathcal{E}^{\sigma,s}_{Z,E}$ can be identified by $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}$ through this form. We conclude the section by showing in which ways linear operators are transformed by the Zak transform (cf. Subsection 3.3).

3.1. Characterizations of quasi-periodic elements via estimates on

their short-time Fourier transform

The following results are analogous to Propositions 2.7 and 2.8 in [36] concerning characterizations of periodic elements in Gelfand-Shilov distribution spaces, and to Proposition 1.1. Here and in what follows we set

[TABLE]

when $E$ is an ordered basis of $\mathbf{R}^{d}$.

Proposition 3.1**.**

Let $s,\sigma>0$, $E$ be an ordered basis of $\mathbf{R}^{d}$, $\Omega_{E}$ be as in (3.1), $F$ be a quasi-periodic Gelfand-Shilov distribution with respect to $E$ and let $\phi\in\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})\setminus 0$ ($\phi\in\Sigma_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})\setminus 0$). Then the following conditions are equivalent:

1. (1)

$F\in\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$* ($F\in\mathcal{E}^{\sigma,s;0}_{Z,E}(\mathbf{R}^{2d})$);* 2. (2)

for some $r>0$ (for every $r>0$), it holds

[TABLE] 3. (3)

for some $r>0$ (for every $r>0$), it holds

[TABLE]

Proposition 3.2**.**

Let $s,\sigma>0$, $E$ be an ordered basis of $\mathbf{R}^{d}$, $\Omega_{E}$ be as in (3.1), $F$ be a quasi-periodic Gelfand-Shilov distribution with respect to $E$ and let $\phi\in\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})\setminus 0$ ($\phi\in\Sigma_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})\setminus 0$). Then the following conditions are equivalent:

1. (1)

$F\in(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$* ($F\in(\mathcal{E}^{\sigma,s;0}_{Z,E})^{\prime}(\mathbf{R}^{2d})$);* 2. (2)

for every $r>0$ (for some $r>0$), it holds

[TABLE] 3. (3)

for every $r>0$ (for some $r>0$), it holds

[TABLE]

We only prove Proposition 3.2, and then only in the Roumieu case. The Beurling case of Proposition 3.2 as well as Proposition 3.1 follow by similar arguments and are left for the reader.

Proof.

Since it is obvious that (3) implies (2), the result follows if we prove that (2) implies (1) and (1) implies (3).

Suppose (3.4) holds for every $r>0$, and let $k=k_{x}\in\Lambda_{E}$ and $\kappa=\kappa_{\xi}\in\Lambda_{E}^{\prime}$ be chosen such that $x_{0}=x-k\in Q_{E}$ and $\xi-\kappa\in Q_{E^{\prime}}$, when $x,\xi\in\mathbf{R}^{d}$ are given. Then (2.13) and (3.4) give

[TABLE]

for every $r>0$. By Proposition Proposition 1.1, it now follows that $F\in(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$, and we have proved that (2) implies (1).

It remains to prove that (1) implies (3). Suppose that (1) holds, let $x,\xi,\eta,y\in\mathbf{R}^{d}$ be arbitrary, and choose $x_{0}\in Q_{E}$, $\xi_{0}\in Q_{E^{\prime}}$, $k\in\Lambda_{E}$ and $\kappa\in\Lambda_{E}^{\prime}$ such that

[TABLE]

Then (1) and Proposition 1.1 give

[TABLE]

and we have proved that (1) implies (3). ∎

3.2. Duality properties of Gevrey type

quasi-period elements

We shall next use the previous characterizations in Propositions 3.1 and 3.2 to show that the form (2.1) can be written as

[TABLE]

when $F,G\in L^{2}_{Z,E}(\mathbf{R}^{2d})$. Here $\phi\in\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})\setminus 0$ is fixed and $\Omega_{E}$ is given by (3.1). We use this identity to extend the definition of this form to permit

[TABLE]

We also show that the dual of $\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$ is equal to $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$ through this form, and similarly when $\mathcal{E}^{\sigma,s}_{Z,E}$ are replaced by $\mathcal{E}^{\sigma,s;0}_{Z,E}$ at each occurrence.

Remark 3.3*.*

By Propositions 3.1 and 3.2 it follows that for $V_{\phi}F$ and $V_{\phi}G$ in (3.6) we have

[TABLE]

Hence the right-hand side of

[TABLE]

makes sense and we may evaluate the integrals with respect to $x,\xi,\eta,y$ in any order. It also follows that the map $(F,G)\mapsto(F,G)_{Z,E}$ defines a continuous map from $\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})\times(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$ to $\mathbf{C}$.

Theorem 3.4**.**

Let $(\,\cdot\,,\,\cdot\,)_{Z,E}$ be given by (3.6). Then the following is true:

1. (1)

$(\,\cdot\,,\,\cdot\,)_{Z,E}$* on $\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})\times\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$ is uniquely extendable to a continuous sesqui-linear map from $L^{2}_{Z,E}(\mathbf{R}^{2d})\times L^{2}_{Z,E}(\mathbf{R}^{2d})$ and from $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})\times\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$ to $\mathbf{C}$, and*

[TABLE] 2. (2)

if $f\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$, $f\in\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$, $F=Z_{E}f$ and $G=Z_{E}g$, then $(F,G)_{Z,E}=(f,g)_{L^{2}(\mathbf{R}^{d})}$; 3. (3)

the dual of $\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$ is equal to $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$ through the form $(\,\cdot\,,\,\cdot\,)_{Z,E}$;

The same holds true with $\mathcal{E}^{\sigma,s;0}_{Z,E}$ and $\Sigma_{s}^{\sigma}$ in place of $\mathcal{E}^{\sigma,s}_{Z,E}$ respective $\mathcal{S}_{s}^{\sigma}$ at each occurrence.

Proof.

By Proposition 1.18, Theorems 2.2, 2.5, and the facts that $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ is dense in $L^{2}(\mathbf{R}^{d})$ and the dual of $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ equals $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ through the form $(\,\cdot\,,\,\cdot\,)_{L^{2}(\mathbf{R}^{d})}$, it suffices to prove (2).

Let

[TABLE]

Then it follows by straight-forward computations that

[TABLE]

where $X=(x,\eta,y)\in Q_{E}\times\mathbf{R}^{2d}$, and the Fourier coefficients $c_{X}(f,j)$ and $c_{X}(g,j)$ are given by

[TABLE]

Since the short-time Fourier transforms $V_{\phi}(Z_{E}f)$ and $V_{\phi}(Z_{E}g)$ are smooth, it follows that $\xi\mapsto F_{X}(\xi)$ and $\xi\mapsto G_{X}(\xi)$ are smooth periodic functions for every $X$. Hence the Fourier coefficients in (3.8) and (3.9) satisfies

[TABLE]

for every $N\geq 0$, when $X\in Q_{E}\times\mathbf{R}^{2d}$ is fixed. By integrating

[TABLE]

with respect to the $\xi$ variable and using (3.10), we obtain

[TABLE]

Hence, integrating with respect to $x,\eta,y$, using Moyal’s identity and Remark 3.3, we obtain

[TABLE]

which gives (2). Here we observe that the estimates in Proposition 1.1 implies that the involved expressions in (3.11) possess suitable $L^{1}$ properties, which allow us to swap the orders of summations and integrations. This gives the result. ∎

3.3. Duality properties of Banach spaces of

quasi-periodic elements

In the following we use the links Theorem 2.13 and (3.7) to carry over duality properties of Lebesgue and modulation spaces to quasi-periodic elements in Lebesgue and Wiener type spaces. Here $p^{\prime}\in[1,\infty]$ denotes the conjugate exponent to $p\in[1,\infty]$, i. e. $p$ and $p^{\prime}$ should satisfy $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Furthermore, we let ${\boldsymbol{p}}^{\prime}=(p_{1}^{\prime},\dots,p_{d}^{\prime})$ when ${\boldsymbol{p}}=(p_{1},\dots,p_{d})$.

Theorem 3.5**.**

Let ${\boldsymbol{p}}\in[1,\infty]^{2d}$, $E$ and $E_{0}$ be ordered bases of $\mathbf{R}^{d}$, $\omega\in\mathscr{P}_{E}(\mathbf{R}^{4d})$ and $\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (2.4) holds. Then the following is true:

1. (1)

the map $(F,G)\mapsto(F,G)_{Z,E}$ from $\mathcal{E}_{Z,E}^{1,1;0}(\mathbf{R}^{2d})\times\mathcal{E}_{Z,E}^{1,1;0}(\mathbf{R}^{2d})$ to $\mathbf{C}$ is uniquely extendable to a continuous mapping from $W^{{\boldsymbol{p}}}_{Z,E,E_{0},(\omega)}(\mathbf{R}^{2d})\times W^{{\boldsymbol{p}}^{\prime}}_{Z,E,E_{0},(1/\omega)}(\mathbf{R}^{2d})$ to $\mathbf{C}$. If in addition $\max({\boldsymbol{p}})<\infty$, then the dual of $W^{{\boldsymbol{p}}}_{Z,E,E_{0},(\omega)}(\mathbf{R}^{2d})$ can be identified with $W^{{\boldsymbol{p}}^{\prime}}_{Z,E,E_{0},(1/\omega)}(\mathbf{R}^{2d})$ through the form $(\,\cdot\,,\,\cdot\,)_{Z,E}$; 2. (2)

if $f\in M^{{\boldsymbol{p}}}_{E\times E_{0}^{\prime},(\omega_{0})}(\mathbf{R}^{d})$, $g\in M^{{\boldsymbol{p}}^{\prime}}_{E\times E_{0}^{\prime},(1/\omega_{0})}(\mathbf{R}^{d})$, then $F=Z_{E}f\in W^{{\boldsymbol{p}}}_{Z,E,E_{0},(\omega)}(\mathbf{R}^{2d})$, $G=Z_{E}g\in W^{{\boldsymbol{p}}^{\prime}}_{Z,E,E_{0},(\omega)}(\mathbf{R}^{2d})$ and $(F,G)_{Z,E}=(f,g)_{L^{2}(\mathbf{R}^{d})}$.

Proof.

In order to prove (1) we first observe that if $F\in W^{{\boldsymbol{p}}}_{Z,E,E_{0},(\omega)}(\mathbf{R}^{2d})$ and $G\in W^{{\boldsymbol{p}}^{\prime}}_{Z,E,E_{0},(1/\omega)}(\mathbf{R}^{2d})$, then the integrand on the right-hand side of (3.6)′ belongs to $L^{1}(\Omega_{E})$, in view of Theorem 2.13. This proves the extension assertions for $(\,\cdot\,,\,\cdot\,)_{Z,E}$ on $W^{{\boldsymbol{p}}}_{Z,E,E_{0},(\omega)}(\mathbf{R}^{2d})\times W^{{\boldsymbol{p}}^{\prime}}_{Z,E,E_{0},(1/\omega)}(\mathbf{R}^{2d})$. The duality assertion will follow after we have proved (3), using the fact that the dual of $M^{{\boldsymbol{p}}}_{E\times E_{0}^{\prime},(\omega_{0})}(\mathbf{R}^{d})$ is equal to $M^{{\boldsymbol{p}}^{\prime}}_{E\times E_{0}^{\prime},(1/\omega_{0})}(\mathbf{R}^{d})$ (see [17, Theorem 11.3.6]).

It remains to prove (2). Let $f\in M^{{\boldsymbol{p}}}_{E\times E_{0}^{\prime},(\omega_{0})}(\mathbf{R}^{d})$, $g\in M^{{\boldsymbol{p}}^{\prime}}_{E\times E_{0}^{\prime},(1/\omega_{0})}(\mathbf{R}^{d})$, $F=Z_{E}f$ and $G=Z_{E}g$. Then Theorem 2.13 shows that $F\in W^{{\boldsymbol{p}}}_{Z,E,E_{0},(\omega)}(\mathbf{R}^{2d})$ and $G\in W^{{\boldsymbol{p}}^{\prime}}_{Z,E,E_{0},(1/\omega)}(\mathbf{R}^{2d})$. By straight-forward computations it follows that (3.11) holds for our choices of $f$, $g$, $F$ and $G$. This gives (2). ∎

Remark 3.6*.*

Let $p\in[1,\infty]$ and $E$ be an ordered basis of $\mathbf{R}^{d}$. Then recall that we may identify $L^{p}_{Z,E}(\mathbf{R}^{2d})$ with $L^{p}(Q_{E\times E^{\prime}})$. Hence, by (2.1) and standard duality properties for Lebesgue spaces show that the map $(F,G)\mapsto(F,G)_{Z,E}$ from $C^{\infty}_{Z,E}(\mathbf{R}^{2d})\times C^{\infty}_{Z,E}(\mathbf{R}^{2d})$ to $\mathbf{C}$ is uniquely extendable to a continuous mapping from $L^{p}_{Z,E}(\mathbf{R}^{2d})\times L^{p^{\prime}}_{Z,E}(\mathbf{R}^{2d})$ to $\mathbf{C}$. If in addition $p<\infty$, then the dual of $L^{p}_{Z,E}(\mathbf{R}^{2d})$ can be identified with $L^{p^{\prime}}_{Z,E}(\mathbf{R}^{2d})$ through the form $(\,\cdot\,,\,\cdot\,)_{Z,E}$;

Remark 3.7*.*

Let $E_{0}$ be an ordered basis of $\mathbf{R}^{d}$, $E=E_{0}\times E_{0}^{\prime}$, $\omega_{0}\in\mathscr{P}_{E}(\mathbf{R}^{d})$, $\omega(x,\xi)=\omega_{0}(\xi)$, $x,\xi\in\mathbf{R}^{d}$ and $\boldsymbol{q}\in[1,\infty)^{d}$. For periodic functions and distributions, Theorem 3.5 (2) together with Proposition 1.16 correspond to [36, Theorem 3.2], which among others asserts that the dual of $\mathcal{E}_{E}(\omega_{0},\ell_{E_{0}^{\prime}}^{\boldsymbol{q}}(\Lambda_{E_{0}^{\prime}}))$ is equal to $\mathcal{E}_{E}(1/\omega_{0},\ell_{E_{0}^{\prime}}^{\boldsymbol{q}^{\prime}}(\Lambda_{E_{0}^{\prime}}))$ through a unique extension of the $L^{2}(Q_{E_{0}})$ form on $\mathcal{E}_{E}^{0}(\mathbf{R}^{d})\times\mathcal{E}_{E}^{0}(\mathbf{R}^{d})$.

Here we observe the misprint in (2) in [36, Theorem 3.2], where it stays $\mathcal{E}^{E}(1/\omega,\ell^{\boldsymbol{q}}_{\kappa(E^{\prime})})$ instead of $\mathcal{E}^{E}(1/\omega,\ell^{\boldsymbol{q}^{\prime}}_{\kappa(E^{\prime})})$

4. Transitions of operators under the Zak

transform

In this section we show how linear operators are transformed by the Zak transform into corresponding operators acting on quasi-periodic functions or distributions. We also present a condition on linear operators which is both necessary and sufficient in order for these operators should map quasi-periodic elements into quasi-periodic elements.

Our results are described in the following two theorems, which explain how linear operators acting on functions and distributions on $\mathbf{R}^{d}$ are transfered by the Zak transform. Especially the operator representation

[TABLE]

is important for characterizing such operators.

Theorem 4.1**.**

Let $s,\sigma>0$, $U_{y,\eta}$ be as in (4.1) and $T$ be a linear operator from $\mathcal{S}_{s}^{\sigma}(\mathbf{R}^{d})$ to $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ with kernel $K\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{2d})$. Then there is a unique linear and continuous operator $T_{Z}$ from $\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$ to $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$ such that $Z_{E}\circ T=T_{Z}\circ Z_{E}$, for every ordered basis $E$ of $\mathbf{R}^{d}$. The kernel of $T_{Z}$ is given by

[TABLE]

The same holds true with $\mathcal{E}_{Z,E}^{\sigma,s;0}$ and $(\mathcal{E}_{Z,E}^{\sigma,s;0})^{\prime}$, or with $C^{\infty}_{Z,E}$ and $\mathscr{S}_{Z,E}^{\prime}$ in place of $\mathcal{E}_{Z,E}^{\sigma,s}$ and $(\mathcal{E}_{Z,E}^{\sigma,s})^{\prime}$, respectively at each occurrence.

The converse of Theorem 4.1 is the following

Theorem 4.2**.**

Let $s,\sigma>0$, $T_{Z}$ be a linear and continuous map from $\mathcal{S}_{s,\sigma}^{\sigma,s}(\mathbf{R}^{2d})$ to $(\mathcal{S}_{s,\sigma}^{\sigma,s})^{\prime}(\mathbf{R}^{2d})$ with kernel $K_{Z}$ and such that (4.3) holds. Then the following is true:

1. (1)

there is a unique $K\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{2d})$ such that (4.2) holds; 2. (2)

if $E$ is an ordered basis of $\mathbf{R}^{d}$, then $T_{Z}$ is uniquely extendable to a linear and continuous operator from $\mathcal{E}^{\sigma,s}_{Z,E}(\mathbf{R}^{2d})$ to $(\mathcal{E}^{\sigma,s}_{Z,E})^{\prime}(\mathbf{R}^{2d})$; 3. (3)

If $T$ is the linear operator with kernel $K$ in (1) and $E$ is an ordered basis of $\mathbf{R}^{d}$, then $Z_{E}\circ T=T_{Z}\circ Z_{E}$.

The same holds true with $\Sigma_{s,\sigma}^{\sigma,s}$, $\mathcal{E}_{Z,E}^{\sigma,s;0}$ and $(\mathcal{E}_{Z,E}^{\sigma,s;0})^{\prime}$, or with $\mathscr{S}$, $C^{\infty}_{Z,E}$ and $\mathscr{S}_{Z,E}^{\prime}$ in place of $\mathcal{S}_{s,\sigma}^{\sigma,s}$, $\mathcal{E}_{Z,E}^{\sigma,s}$ and $(\mathcal{E}_{Z,E}^{\sigma,s})^{\prime}$, respectively at each occurrence.

Proof of Theorem 4.1.

We only prove the result when the involved spaces are given by $\mathcal{E}_{Z,E}^{\sigma,s}$ or $(\mathcal{E}_{Z,E}^{\sigma,s})^{\prime}$. The other cases follow by similar arguments and are left for the reader.

First suppose that $K_{Z}$ is given by (4.2). Since pull-back results of the type [20, Theorem 6.1] for usual distribution, hold true for Gelfand-Shilov distributions concerning linear pull-backs, it follows that $K_{1}(x,y,z)\equiv K(x-z,y-z)$ belongs to $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{3d})$ when $K\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{2d})$. By Fourier transformation, it follows that $K_{2}(x,y,\xi)\equiv\mathscr{F}(K_{1}(x,y,\,\cdot\,))(\xi)$ $(\mathcal{S}_{s,s,\sigma}^{\sigma,\sigma,s})^{\prime}(\mathbf{R}^{3d})$. By similar pull-back results it now follows that

[TABLE]

Since $K_{Z}(x,\xi,y,\eta)=(2\pi)^{-\frac{d}{2}}K_{3}(x,y,\xi,\eta)$, we get

[TABLE]

and the last property in (4.2) follows.

Let $E$ be an ordered basis of $\mathbf{R}^{d}$. We only prove $Z_{E}\circ T=T_{Z}\circ Z_{E}$ when $K\in\mathscr{S}(\mathbf{R}^{2d})$. The general result follows by similar arguments and is left for the reader. We have

[TABLE]

This gives

[TABLE]

This shows that $Z_{E}\circ T=T_{Z}\circ Z_{E}$.

The continuity assertions of $T_{Z}$ now follows from the latter identity and Theorem 2.5. ∎

We need the following lemma for the proof of Theorem 4.2.

Lemma 4.3**.**

Let $s,\sigma>0$ and $K\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{2d})$. Then the following conditions are equivalent:

1. (1)

$K(\,\cdot\,+(z,z))=K$* for every $z\in\mathbf{R}^{d}$;* 2. (2)

there is a unique $K_{0}\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ such that $K(x,y)=K_{0}(x-y)$.

The same hold true with $(\Sigma_{s}^{\sigma})^{\prime}$, $\mathscr{S}^{\prime}$ or $\mathscr{D}^{\prime}$ in place of $(\mathcal{S}_{s}^{\sigma})^{\prime}$ at each occurrence.

Lemma 4.3 is at least implicitly available in the literature, e. g. in [20]. In order to be self-contained we give a proof in Appendix A.

Proof of Theorem 4.2.

Again we only prove the result when the involved spaces are given by $\mathcal{E}_{Z,E}^{\sigma,s}$ or $(\mathcal{E}_{Z,E}^{\sigma,s})^{\prime}$. The other cases follow by similar arguments and are left for the reader.

The condition (4.3) implies that

[TABLE]

for every $x,y,z,\xi,\eta,\zeta\in\mathbf{R}^{d}$. By Lemma 4.3 it follows that

[TABLE]

for some $K_{0}\in(\mathcal{S}_{s,\sigma}^{\sigma,s})^{\prime}(\mathbf{R}^{2d})$. It now follows that

[TABLE]

fullfils all required properties. ∎

Appendix A

In this appendix we present proofs of Proposition 2.11 and Lemma.

Proof of Proposition 2.11.

First suppose that $p\in(0,1]$. Then

[TABLE]

and (1) follows for $p\in(0,1]$.

Since the result is true for $p=2$ in view of Proposition 1.18 below and Proposition 1.18, the result now follows in the case $p\in[1,2]$ by interpolating the case $p=1$ above with the case $p=2$. This gives (1), and (2) now follows from (1) and duality.

Finally, by the assumptions we have that $\sum_{j\in\Lambda_{E}}v(x+j)^{-1}$ is bounded. Hence, by the inversion formula (1.17) we obtain

[TABLE]

and (3) follows in the case $p=1$ by combining the previous estimate with (1). Since (3) is true for $p=2$ in view of Proposition 1.18, it follows that it is true also for $p\in[1,2]$ by interpolating between the cases $p=1$ and $p=2$. For $p\in[2,\infty]$, (3) now follows from the case $p\in[2,\infty]$ and duality. ∎

Proof of Lemma 4.3.

We only prove the result when the involved spaces are of the forms $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$ and $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{2d})$. The other cases follow by similar arguments and are left for the reader.

It is evident that (2) implies (1). Suppose that (1) is true. Then $(x,y)\mapsto K(2^{-1}(x+y),2^{-1}(x-y))$ is an element in $(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{2d})$ which is constant with respect to the $x$ variable. Hence,

[TABLE]

for some $K_{0}\in(\mathcal{S}_{s}^{\sigma})^{\prime}(\mathbf{R}^{d})$. By taking $2^{-1}(x+y)$ and $2^{-1}(x-y)$ as new variables, we obtain (2). ∎