\section{Introduction}\label{sec0}

\par

In the paper we characterise
Gelfand-Shilov spaces of functions and distributions,
modulation spaces and Gevrey classes in background of
various kinds of Wiener estimates, and mapping
properties of the Zak transforms. Especially we perform such
investigations for periodic and quasi-periodic functions 
and distributions.

\par

The Zak transforms are whimsical 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.

\par

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
$$
F\mapsto c\cdot \overline {Z\phi}\cdot Z\psi \cdot F
$$
for some constant $c$ which depends on the sampling
parameters. (See \cite{Ho1,ToNa} and Section \ref{sec1} 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 \cite{Bol,BoJa,Gc2}.)

\par

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\cite{Gc2}).

\par

Before entering the Gabor theory, Zak transforms were
first introduced and used in a problem in differential
equation by Gelfand in \cite{Gel}. Subsequently, the transforms
were rediscovered in various contexts, especially 
in solid state physics by Zak in \cite{Za1} and in
differential equations by Brezin in \cite{Bre}.

\par

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

\par

For example, the (standard) Zak transform is given by
$$
(Z_1 f)(x,\xi ) \equiv \sum _{j\in \zz d}
f(x-j)e^{i\scal j\xi},
$$
when $f$ is a suitable function or distribution (see \eqref{Eq:DefZak} for the
general definition of the Zak transform). By the definition
it follows that if $F=Z_1f$ 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,
$$
F(x+k,\xi ) = e^{i\scal k\xi}F(x,\xi )
\quad \text{and}\quad
F(x,\xi +2\pi \kappa) = F(x,\xi ),\quad k,\kappa \in \zz d.
$$
It follows from these equalities that $F$
is completely reconstructable
from its data on $Q_{d,1}\times Q_{d,2\pi}$.

\par

It is well-known that $Z_1$ is bijective from $L^2(\rr d)$ to
the set of quasi-periodic elements in $L^2(Q_{d,1}
\times Q_{d,2\pi})$.
Furthermore,
by a straight-forward application of Parseval's
formula we have
\begin{equation}\label{Eq:ZakonL2}
\nm {Z_1 f}{L^2(Q_{d,1}\times Q_{d,2\pi})} =
(2\pi) ^{\frac d2}\nm f{L^2},
\qquad f\in L^2(\rr d).
\end{equation}
(Cf. e.{\,}g. \cite[Theorem 8.2.3]{Gc2}.) Consequently,
$L^2(\rr d)$ can be characterized in a convenient
way by mapping properties of the Zak transform.

\par

An other space that can be characterized by related
mapping properties concerns the Schwartz space
$\mathscr S (\rr d)$. In fact, it is proved in
\cite{Jan} by Janssen that $Z_1$ is continuous and
bijective from $\mathscr S (\rr d)$ to the set of
quasi-periodic elements in $C^\infty (\rr {2d})$.

\par

In \cite{Ti,TiHe}, Heil and Tinaztepe deduce
some important mapping properties for the Zak transform
on modulation spaces, and apply these
results to deduce Balian-Low properties for
such spaces.
On the other hand, these mapping
properties on modulation spaces seems not to be (complete)
characterizations, because of absence of bijectivity.
In fact, apart from the spaces
$L^2(\rr d)$ and $\mathscr S (\rr d)$, it seems that
the whole theory lacks characterizations of
essential function and distribution spaces
via the Zak transform (cf. Subsection 8.2 (f) in
\cite{Gc2}).

\par

In Section \ref{sec3} 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 $\mathscr S '(\rr d)$ of
$\mathscr S (\rr d)$, the (standard) Gelfand-Shilov
spaces and their distribution spaces by their
images under the Zak transform.

\par

For example
we prove that $Z_1$ is continuous and bijective
from $\mathscr S '(\rr d)$ to the set of all
quasi-periodic distributions on
$Q_{d,1}\times Q_{d,2\pi}$. (See Theorem
\ref{Thm:ZakClassical}. For similar characterizations
of Gelfand-Shilov spaces and their distribution
spaces, see Theorems \ref{Thm:ZakTestFunctions}
and \ref{Thm:ZakDist}.) An other consequence of our 
results is that $Z_1$ maps the modulation space $M^p(\rr 
d)$ continuously and bijectively to the set of all
elements in $M^{\infty ,p}(\rr {2d})$ which are 
quasi-periodic on $Q_{d,1}\times Q_{d,2\pi}$.
Furthermore,
\begin{equation}\label{Eq:ZakModIntrRel}
\nm {Z_1f}{M^{\infty ,p}}\asymp \nm f{M^p}
\end{equation}
(see Theorem \ref{Thm:ZakModulation}
and Corollary \ref{Cor:ZakModulation}).

\medspace

Our investigations also include some extensions
in \cite{ToNa} concerning characterizations of periodic elements
in modulation spaces.
In fact, it follows from \cite{ToNa} that if $q\in 
(0,\infty ]$
and $f$ is a $2\pi$-periodic Gelfand-Shilov distribution
on $\rr d$ with
Fourier coefficients $c(f,\alpha )$, $\alpha \in \zz d$, 
then
\begin{alignat}{4}
&\{ c(f,\alpha )\} _{\alpha \in \zz d} \in \ell ^q &
\quad & &\quad 
&\Leftrightarrow & \quad f &\in M^{\infty ,q} .
\label{Eq:IntrEquiv3}
\end{alignat}
Here
$M^{\infty ,q}$ is the (unweighted) modulation
spaces with Lebesgue parameters $\infty$ and $q$.

\par

We note that a proof of \eqref{Eq:IntrEquiv3} in the case
$q\in [1,\infty]$ can be found in e.{\,}g. \cite{RuSuToTo},
and with some extensions in \cite{Re}.  

\par

In \cite{ToNa} related equivalences to
\eqref{Eq:IntrEquiv3} which involve
short-time Fourier transforms were deduced. In fact, let
$p,q\in (0,\infty ]$, $\phi \in \mathcal S _s^\sigma (\rr d)\setminus 0$ and $f$
be $2\pi$-periodic.
Then it is proved in \cite{ToNa} that
\begin{alignat}{3}
\{ c(f,\alpha )\} _{\alpha \in \zz d} &\in \ell ^q &
\qquad 
&\Leftrightarrow &\qquad \xi &\mapsto \nm {V_\phi f(\, \cdot \,  ,\xi )}{L^\infty (\rr d)} \in L^q .
\tag*{(\ref{Eq:IntrEquiv3})$'$}
\intertext{By observing that periodicity of $f$ induce the same periodicity
for $x\mapsto |V_\phi f(x,\xi )|$, it follows that
\eqref{Eq:IntrEquiv3}$'$ is the same as}
\{ c(f,\alpha )\} _{\alpha \in \zz d} &\in \ell ^q &
\qquad 
&\Leftrightarrow &\qquad \xi &\mapsto
\nm {V_\phi f(\, \cdot \,  ,\xi )}{L^\infty ([0,2\pi ]^d)} \in L^q .
\tag*{(\ref{Eq:IntrEquiv3})$''$}
\intertext{
\indent
In Section \ref{sec2} we show that the latter equivalence hold true
with $L^r([0,2\pi]^d)$ norm in place of $L^\infty ([0,2\pi]^d)$ norm for every
$r\in (0,\infty ]$. That is, we extend \eqref{Eq:IntrEquiv3}$''$ into}
\{ c(f,\alpha )\} _{\alpha \in \zz d} &\in \ell ^q &
\qquad 
&\Leftrightarrow &\qquad \xi &\mapsto
\nm {V_\phi f(\, \cdot \,  ,\xi )}{L^r ([0,2\pi ]^d)} \in L^q .
\tag*{(\ref{Eq:IntrEquiv3})$'''$}
\intertext{In particular, if $q<\infty$ and choosing $r=q$, then we obtain}
\sum _{\alpha \in \zz d}|c(f,\alpha ) |^q &<\infty &
\qquad 
&\Leftrightarrow &\quad 
&\iint _{[0,2\pi ]^d\times \rr d} |V_\phi f(x,\xi )|^q\, dxd\xi <\infty .
\label{Eq:IntrEquiv3A}
\end{alignat}

\par

The improved equivalence \eqref{Eq:IntrEquiv3}$'''$ may
under the additional assumptions, $q,r\ge 1$ be obtained from
\eqref{Eq:IntrEquiv3}$''$
by a suitable combination of H{\"o}lder's and Young's inequalities
and the inequality
\begin{align}
F(X) &\lesssim \int _\Omega 
\Phi (X-Y) F(Y )\, dY ,
\quad X\in \Omega =[0,2\pi]^d\times \rr d,
\label{Eq:ContConvPer}
\intertext{where}
\Phi (x,\xi ) &= \sum _{k\in \zz d}|V_\phi \phi (x-2\pi k,\xi )|
\quad \text{and}\quad F(X) = |V_\phi f (X)|,
\notag
\end{align}
which follows from Lemma 1.3.3 in \cite{Gc2} for $2\pi$-periodic
distributions $f$. It follows that this case can be handled by
straight-forward modifications of the methods that are used
when establishing basic results for classical modulation
spaces in \cite{F1} and in Chapter 11 in \cite{Gc2}.

\par

In our situation, the parameters $q$ and $r$ are, more generally, allowed
to belong to the full
interval $(0,\infty ]$ instead of $[1,\infty ]$. The classical approaches in
\cite{F1,FG1,FG2,Gc2}
are then insufficient because they require convex structures in the topology of
the involved vector spaces. This convexity is absent when $q<1$ or $r<1$.

\par

We manage our more general situation by
using techniques based on ideas in \cite{GaSa,Rau1,Rau2,Toft15}
and which can handle Lebesgue and Wiener spaces which are
quasi-Banach spaces but may fail to be Banach spaces. Especially
we shall follow a main idea in \cite{GaSa,Toft15}
and replace the usual convolution, used in
\cite{F1,FG1,FG2,Gc2}, by a semi-continuous version which is less
sensitive when convexity is lacking in the topological structures.
For the semi-continuous convolution we deduce in Section \ref{sec2}
the needed Lebesgue and Wiener estimates.
In the end we achieve in Section \ref{sec2}
various types of characterizations of modulation spaces in terms
of Wiener norm estimates on the short-time Fourier transforms of
the functions and (ultra-)distributions under considerations. For example,
as special case of Propositions \ref{Prop:WienerEquiv}$'$ after Proposition
\ref{Prop:SemiContConvEstNonPer}, we have for $p,q,r\in (0,\infty ]$ that
\begin{equation}\label{Eq:ModSpaceIntrChar}
\nm f{M^{p,q}} \asymp \nm a{\ell ^{p,q}}
\quad \text{when}\quad
a(j) = \nm {V_\phi f}{L^r(j+[0,1]^{2d})}.
\end{equation}
Similar facts hold true for those Wiener amalgam spaces which are Fourier images
of modulation spaces of the form $M^{p,q}$. In particular our results can be
used to deduce certain invarians properties concerning the choice of
local component in the Wiener amalgam quasi-norm. (See also Proposition
\ref{Prop:SpecCaseWienerRel1}.) Here we remark that for Wiener amalgam
spaces which at the same time are Banach spaces, the approaches are often
less complicated and there are several
examples on other Banach spaces (e.{\,}g. suitable modulation spaces)
to furnish the local component in the Wiener amalgam norms.
(See e.{\,}g. \cite{FeLu,FeZi} and the references therein.)

\par

We also present some applications on periodic elements which gives
\eqref{Eq:IntrEquiv3}$'''$ and \eqref{Eq:IntrEquiv3A} as special cases. (See
Propositions \ref{Prop:PeriodicMod} and \ref{Prop:PerMod}$'$.)

\par

The Wiener spaces under considerations
can also be described in terms of coorbit spaces, whose general
theory was founded
by Feichtinger and Gr{\"o}chenig in \cite{FG1,FG2} and further developed
in different ways, e.{\,}g. by Rauhut in \cite{Rau1,Rau2}. Since our
investigations in Section \ref{sec2} concern quasi-Banach spaces
which may fail to be Banach spaces, our investigations are especially
linked to Rauhut's analysis in \cite{Rau1,Rau2}. 
In this context, a part of our analysis on modulation spaces can be formulated
as coorbit norm estimates of short-time Fourier transforms with local component
in $L^r$-spaces with $r\in (0,\infty ]$ and global component in other Lebesgue 
spaces. Proposition \ref{Prop:WienerEquiv}$'$ in Section \ref{sec2} then
shows that different choices of $r$ give rise to equivalent norm estimates
on short-time Fourier transforms.
Again we remark that if $r$ belongs
to the subset $[1,\infty ]$ of $(0,\infty ]$ and that all involved spaces
are Banach spaces, then our results can be obtained in other less complicated
ways, given e.{\,}g. in Chapters 11 and 12 in \cite{Gc2}.

\par

As explained above, our results on Wiener spaces are essential for our results on
periodicity and the Zak transform. For example, the analysis behind
\eqref{Eq:ModSpaceIntrChar} also leads to that \eqref{Eq:ZakModIntrRel}
can be reformulated as
\begin{equation}\label{Eq:EquivZakMod}
f\in M^p(\rr d)\quad \Leftrightarrow \quad
V_\Phi (Z_1 f) \in L^p(Q_{d,1}\times Q_{d,2\pi}\times \rr d\times \rr d).
\end{equation}
If $p=2$, then an application of
Parseval's formula implies that \eqref{Eq:EquivZakMod} is the same as
$$
f\in M^2(\rr d)\quad \Leftrightarrow \quad
Z_1 f \in L^2(Q_{d,1}\times Q_{d,2\pi}),
$$
which is a slightly weaker form of \eqref{Eq:ZakonL2}.

\par

\section*{Acknowledgement}

\par

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.


\par

\section{Preliminaries}\label{sec1}

\par

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

\par

\subsection{Gelfand-Shilov spaces and Gevrey classes}\label{subsec1.1}
Let $0<s,\sigma \in \mathbf R$ be fixed. Then the Gelfand-Shilov
space $\mathcal S_{s}^\sigma (\rr d)$
($\Sigma _{s}^\sigma (\rr d)$) of Roumieu type (Beurling type) with parameters $s$
and $\sigma$ consists of all $f\in C^\infty (\rr d)$ such that
\begin{equation}\label{gfseminorm}
\nm f{\mathcal S_{s,h}^\sigma }\equiv \sup \frac {|x^\alpha \partial ^\beta
f(x)|}{h^{|\alpha  + \beta |}\alpha !^s \, \beta !^\sigma}
\end{equation}
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 $x\in \rr d$. We equip
$\mathcal S_{s}^\sigma (\rr d)$ ($\Sigma _{s}^\sigma (\rr d)$) by the canonical inductive limit
topology (projective limit topology) with respect to $h>0$, induced by
the semi-norms in \eqref{gfseminorm}.

\par


\medspace

The \emph{Gelfand-Shilov distribution spaces} $(\mathcal S_s^{\sigma})'(\rr d)$
and $(\Sigma _s^\sigma)'(\rr d)$ are the dual spaces of $\mathcal S_s^{\sigma}(\rr d)$
and $\Sigma _s^\sigma (\rr d)$, respectively.  As for the Gelfand-Shilov spaces there 
is a canonical projective limit topology (inductive limit topology) for $(\mathcal S _s^{\sigma})'(\rr d)$ 
($(\Sigma _s^\sigma)'(\rr d)$).(Cf. \cite{GS, Pil1, Pil3}.)
For conveniency we set
$$
\mathcal S _s=\mathcal S _s^s,\quad \mathcal S _s'=(\mathcal S _s^s)',\quad
\Sigma _s=\Sigma _s^s
\quad \text{and}\quad
\Sigma _s'=(\Sigma _s^s)'.
$$

\par

From now on we let $\mathscr F$ be the Fourier transform which
takes the form
$$
(\mathscr Ff)(\xi )= \widehat f(\xi ) \equiv (2\pi )^{-\frac d2}\int _{\rr
{d}} f(x)e^{-i\scal  x\xi }\, dx
$$
when $f\in L^1(\rr d)$. Here $\scal \, \cdot \,  \, \cdot \, $ denotes the usual
scalar product on $\rr d$. The map $\mathscr F$ extends 
uniquely to homeomorphisms on $\mathscr S'(\rr d)$,
from $(\mathcal S_s^\sigma )'(\rr d)$ to $(\mathcal S_\sigma ^s)'(\rr d)$ and
from $(\Sigma _s^\sigma)'(\rr d)$ to $(\Sigma _\sigma ^s)'(\rr d)$. Furthermore,
$\mathscr F$ restricts to
homeomorphisms on $\mathscr S(\rr d)$, from
$\mathcal S_s^\sigma (\rr d)$ to $\mathcal S_\sigma ^s(\rr d)$ and
from $\Sigma _s^\sigma (\rr d)$ to $\Sigma _\sigma ^s(\rr d)$,
and to a unitary operator on $L^2(\rr d)$. 

\medspace

Next we consider a more general class of Gelfand-Shilov spaces and their distribution
spaces. Let $0\le s_1,s_2,\sigma _1,\sigma _2\in \mathbf R$ be fixed. Then the Gelfand-Shilov
space $\mathcal S_{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$
($\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$) of Roumieu type (Beurling type)
with parameters $s_1$, $s_2$, $\sigma _1$ and $\sigma _2$ consists of all $f\in C^\infty
(\rr {d_1+d_2})$ such that
\begin{equation}\label{gfseminorm2}
\nm f{\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}}
\end{equation}
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 $x_j\in \rr {d_j}$, $j=1,2$. We
equip $\mathcal S_{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$
($\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$) by the canonical inductive limit
topology (projective limit topology) with respect to $h>0$, induced by
the semi-norms in \eqref{gfseminorm2}.

\par

The space $\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$ is a Fr{\'e}chet space
when the topology is induced by the seminorms
$\nm \, \cdot \, {\mathcal S_{s_1,s_2,h}^{\sigma _1,\sigma _2}}$, $h>0$.

\medspace

The \emph{Gelfand-Shilov distribution spaces} $(\mathcal S _{s_1,s_2}
^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})$
and $(\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})$ are the dual spaces of
$\mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$
and $\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$, respectively. Evidently,
$\mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$,
$\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$
and their duals possess similar topological properties as
$\mathcal S _{s}^{\sigma }(\rr {d})$, $\Sigma _{s}^{\sigma }(\rr {d})$ and their duals.
By $(j+k)!\le 2^{j+k}j!k!$ when $j,k\ge 0$ are integers we get
$\mathcal S _{s,s}^{\sigma ,\sigma } = \mathcal S _s^\sigma$ and $\Sigma _{s,s}^{\sigma ,\sigma}
= \Sigma _s^\sigma$.

\par

For any $s_j,\sigma _j,s_{0,j},\sigma _{0,j}>0$ such that $s_j>s_{0,j}$, $\sigma _j>\sigma _{0,j}$,
we have
\begin{multline}\label{GSembeddings}
\mathcal S _{s_{0,1},s_{0,2}}^{\sigma _{0,1},\sigma _{0,2}}(\rr {d_1+d_2})
\hookrightarrow
\Sigma _{s_{1},s_{2}}^{\sigma _{1},\sigma _{2}}(\rr {d_1+d_2})
\hookrightarrow 
\mathcal S _{s_{1},s_{2}}^{\sigma _{1},\sigma _{2}}(\rr {d_1+d_2})
\\[1ex]
\hookrightarrow 
\mathscr S (\rr {d_1+d_2})
\hookrightarrow 
\mathscr S '(\rr {d_1+d_2})
\hookrightarrow
(\mathcal S _{s_{1},s_{2}}^{\sigma _{1},\sigma _{2}})' (\rr {d_1+d_2})
\\[1ex]
\hookrightarrow
(\Sigma _{s_{1},s_{2}}^{\sigma _{1},\sigma _{2}})'(\rr {d_1+d_2})
\hookrightarrow
(\mathcal S _{s_{0,1},s_{0,2}}^{\sigma _{0,1},\sigma _{0,2}})'(\rr {d_1+d_2}),
\end{multline}
with dense embeddings, provided the parameters have been chosen such that
all spaces are non-trivial.
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.
The space $\Sigma _{s_{1},s_{2}}^{\sigma _{1},\sigma _{2}}(\rr {d_1+d_2})$
is a Fr{\'e}chet space
with seminorms $\nm \, \cdot \, {\mathcal S _{s_{1},s_{2},h}^{\sigma _{1},\sigma _{2}}}$, $h>0$. 
Moreover,
$\Sigma _{s_{1},s_{2}}^{\sigma _{1},\sigma _{2}}(\rr {d_1+d_2})\neq \{ 0\}$, if and only if 
$s_j+\sigma _j\ge 1$ and
$(s_j,\sigma _j)\neq (\frac 12,\frac 12)$, $j=1,2$, and
$\mathcal S _{s_{1},s_{2}}^{\sigma _{1},\sigma _{2}}
(\rr {d_1+d_2})\neq \{ 0\}$, if and only if $s_j+\sigma _j\ge 1$, $j=1,2$.

\par

Let $\mathscr F _jf$ denote the partial Fourier transform of
$f(x_1,x_2)\in \mathscr S (\rr {d_1+d_2})$
with respect to $x_j$, $j=1,2$.
Then $\mathscr F _1$ and $\mathscr F _2$ extend uniquely to homeomorphisms
\begin{alignat*}{4}
\mathscr F _1 & \ :\  &   &(\mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})
&   &\ \to \ &
&(\mathcal S _{\sigma _1,s_2}^{s_1,\sigma _2})'(\rr {d_1+d_2})
\\[1ex]
\mathscr F _2 & \ :\  &   &(\mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})
&   &\ \to \ &
&(\mathcal S _{s_1,\sigma _2}^{\sigma _1,s_2})'(\rr {d_1+d_2}),
\intertext{respectively, and restricts to homeomorphisms}
\mathscr F _1 & \ :\  &   &\mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})
&   &\ \to \ &
&\mathcal S _{\sigma _1,s_2}^{s_1,\sigma _2}(\rr {d_1+d_2})
\\[1ex]
\mathscr F _2 & \ :\  &   &\mathcal S _{s_1,s_2}^
{\sigma _1,\sigma _2}(\rr {d_1+d_2})
&   &\ \to \ &
&\mathcal S _{s_1,\sigma _2}^{\sigma _1,s_2}(\rr {d_1+d_2}),
\end{alignat*}
The same holds true after each Gelfand-Shilov function or 
distribution space
of Roumieu type have been replaced by corresponding Beurling 
type space.

\par

Gelfand-Shilov spaces can in convenient
ways be characterized in terms of estimates of the
involved functions and their Fourier
transforms. More precisely, in \cite{ChuChuKim, Eij}
it is proved that
if $f\in \mathscr S '(\rr d)$ and $s,\sigma >0$, then
$f\in \mathcal S _s^\sigma (\rr d)$
($f\in \Sigma _s^\sigma (\rr d)$), if and only if
\begin{equation}\label{Eq:GSFtransfChar}
|f(x)|\lesssim e^{-r|x|^{\frac 1s}}
\quad \text{and}\quad
|\widehat f(\xi )|\lesssim e^{-r|\xi |^{\frac 1\sigma }},
\end{equation}
for some $r>0$ (for every $r>0$).
Here $f(\theta )\lesssim g(\theta )$ means that
$f(\theta )\le cg(\theta)$ 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 \cite{ChuChuKim}
that if $f\in \mathscr S '(\rr {d_1+d_2})$ and
$s_1,s_2,\sigma _1,\sigma _2>0$, then
$f\in \mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$
($f\in \Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}
(\rr {d_1+d_2})$), if and only if
\begin{equation}\tag*{(\ref{Eq:GSFtransfChar})$'$}
|f(x_1,x_2)|
\lesssim
e^{-r(|x_1|^{\frac 1{s_1}} + |x_2|^{\frac 1{s_2}})}
\quad \text{and}\quad
|\widehat f(\xi _1,\xi _2)|\lesssim e^{-r(|\xi _1|^{\frac 1{\sigma _1}} + |\xi _2|^{\frac 1{\sigma _2}})},
\end{equation}
for some $r>0$ (for every $r>0$).

\par

Gelfand-Shilov spaces and their distribution spaces can also
be characterized by estimates of short-time Fourier
transforms, (see e.{\,}g. \cite{GZ,Toft18}).
More precisely, let $\phi \in \mathcal S _s (\rr d)$ be fixed. Then the \emph{short-time
Fourier transform} $V_\phi f$ of $f\in \mathcal S _s '
(\rr d)$ with respect to the \emph{window function} $\phi$ is
the Gelfand-Shilov distribution on $\rr {2d}$, defined by
$$
V_\phi f(x,\xi )  =
\mathscr F (f \, \overline {\phi (\, \cdot \,  -x)})(\xi ).
$$
If $f ,\phi \in \mathcal S _s (\rr d)$, then it follows that
$$
V_\phi f(x,\xi ) = (2\pi )^{-\frac d2}\int f(y)\overline {\phi
(y-x)}e^{-i\scal y\xi}\, dy .
$$

\par

By \cite[Theorem 2.3]{Toft10} it follows that the map
$(f,\phi)\mapsto V_{\phi} f$ from $\mathscr S (\rr d) \times \mathscr S (\rr d)$ 
to $\mathscr S(\rr {2d})$ is uniquely extendable to a continuous map from 
$(\mathcal S_{s}^{\sigma})'(\rr d)\times (\mathcal S_{s}^{\sigma})'(\rr d)$
to $(\mathcal S_{s,\sigma}^{\sigma ,s})'(\rr {2d})$, and restricts to a continuous map
from $\mathcal S_{s}^{\sigma}(\rr d)\times \mathcal S_{s}^{\sigma}(\rr d)$
to $\mathcal S_{s,\sigma}^{\sigma ,s}(\rr {2d})$.

\par

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.

\par

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

\par

\begin{prop}\label{stftGelfand2}
Let $s_j,\sigma _j>0$ be such that $s_j+ \sigma _j\ge 1$, $j=1,2$. Also let
$\phi \in \mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})\setminus 0$
($\phi \in \Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})\setminus 0$) and
$f$ be a Gelfand-Shilov distribution on $\rr {d_1+d_2}$.
Then $f\in  \mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$
($f\in  \Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$), if and only if
\begin{equation}\label{stftexpest2}
|V_\phi 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}})},
\end{equation}
for some $r > 0$ (for every $r>0$).
\end{prop}

\par

\begin{prop}\label{stftGelfand2dist}
Let $s_j,\sigma _j>0$ be such that $s_j+\sigma _j\ge 1$, $j=1,2$. Also let
$\phi \in \mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})\setminus 0$
($\phi \in \Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})\setminus 0$) and
$f$ be a Gelfand-Shilov distribution on $\rr {d_1+d_2}$.
Then $f\in (\mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})$
($f\in (\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})$), if and only if
\begin{equation}\label{stftexpest2dist}
|V_\phi 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}})},
\end{equation}
for every $r > 0$ (for some $r>0$).
\end{prop}

\par

We note that if $s_j=\sigma _j=\frac 12$ for some $j$ in Propositions \ref{stftGelfand2}
and \ref{stftGelfand2dist}, then it is not possible to find any $\phi \in
\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2}(\rr {d_1+d_2})\setminus 0$. Hence, these
results give no information in the Beurling case for such choices of $s_j$ and $\sigma _j$.

\par

A proof of Proposition \ref{stftGelfand2} can be found in
e.{\,}g. \cite{GZ} (cf. \cite[Theorem 2.7]{GZ}) and a proof of
Proposition \ref{stftGelfand2dist} in the case $d_2=0$
can be found in \cite{Toft18}. The general case of Proposition \ref{stftGelfand2dist}
follows by similar arguments as in \cite{Toft18} and is left for the reader.
See also \cite{CPRT10} for related results.
In \cite[Theorem 2.4]{ToNa} analogous characterizations for
periodic functions and distributions are obtained.

\par

Next we consider Gevrey classes on $\rr d$. Let $\sigma \ge 0$.
For any compact set $K\subseteq \rr d$, $h>0$ and $f\in C^{\infty}(K)$ let
\begin{equation}\label{e2}
\nm {f}{K,h,\sigma} \equiv \underset{\alpha\in \nn d}\sup
\left (
\frac{\nm {\partial^{\alpha}f}{L^{\infty}(K)}}{h^{\vert \alpha\vert}\alpha ! ^\sigma}
\right ) .
\end{equation} 
The Gevrey class $\mathcal E _\sigma (K)$ ($\mathcal E _{0,\sigma}(K)$) of order $\sigma$ and
of Roumieu type (of Beurling type) is the set of all
$f\in C^{\infty}(K)$ such that \eqref{e2} is finite for some (for every)
$h>0$. We equipp $\mathcal E _\sigma (K)$ ($\mathcal E _{0,\sigma}(K)$) by
the inductive (projective) limit topology with respect to $h>0$,
supplied by the seminorms in \eqref{e2}. Finally if $\lbrace
K_j\rbrace_{j\geq 1}$ is an exhausted
sets of compact subsets of $\rr d$, then let
\begin{alignat*}{3}
\mathcal E _\sigma (\rr d) &= \projlim{j} \mathcal E _\sigma (K_j)
& \quad &\text{and} &\quad
\mathcal E _{0,\sigma}(\rr d) &= \projlim{j} \mathcal E _{0,\sigma}(K_j).
\intertext{In particular,}
\mathcal E _\sigma (\rr d) &=\underset{j\geq 1}\bigcap \mathcal E _\sigma (K_j)
& \quad &\text{and} &\quad
\mathcal E _{0,\sigma}(\rr d) &= \underset{j\geq 1}\bigcap \mathcal E _{0,\sigma}(K_j).
\end{alignat*}
It is clear that $\mathcal E _{0,0}(\rr d)$ contains all constant functions
on $\rr d$, and that $\mathcal E _{0}(\rr d)\setminus \mathcal E _{0,0}(\rr d)$
contains all non-constant trigonometric polynomials.

\par

\subsection{Ordered, dual and phase split bases}\label{subsec1.2}

\par

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

\par

\begin{defn}\label{Def:OrdBasis}
Let $E = \{ e_1,\dots,e_d \}$ be an \emph{ordered} basis of
$\rr {d}$. Then $E'$ denotes the basis of $e_1',\dots,e_d'$ 
in $\rr {d}$ which satisfies
$$
\scal {e_j} {e'_k}=2\pi \delta_{jk}
\quad \text{for every}\quad
j,k =1,\dots, d.
$$
The corresponding lattices are given by
\begin{align*}
\Lambda _E
&=
\sets{n_1e_1+\cdots+n_de_d}{(n_1,\dots,n_d)\in \zz d},
\intertext{and}
\Lambda'_E
&=
\Lambda_{E'}=\sets{\nu _1e'_1+\cdots+\nu _de'_d}{(\nu _1,\dots,\nu _d)
\in \zz d}.
\end{align*}
The sets $E'$ and $\Lambda '_E$ are
called the dual basis and dual lattice of $E$ and $\Lambda _E$, respectively.
If $E_1,E_2$ are ordered bases of $\rr d$ such that a permutation of
$E_2$ is the dual basis for $E_1$, then the pair $(E_1,E_2)$ are called
\emph{permuted dual bases} (to each others on $\rr d$).
\end{defn}

\par

\begin{rem}\label{Rem:TEMap}
Evidently, if $E$ is the same as in Definition \ref{Def:OrdBasis},
then there is a matrix $T_E$ with $E$ as the image of
the standard basis in $\rr d$. Then $E'$ is the image of the standard basis
under the map $T_{E'}= 2\pi(T^{-1}_E)^t$.
\end{rem}

\par

Two ordered bases on $\rr d$ can be used to construct a uniquely defined
ordered basis for $\rr {2d}$ as in the following definition.

\par

\begin{defn}\label{Def:FromTwoToOneBasis}
Let $E_1,E_2$ be ordered bases of $\rr d$,
$$
V_1=\sets {(x,0)\in \rr {2d}}{x\in \rr d},
\quad
V_2=\sets {(0,\xi )\in \rr {2d}}{\xi \in \rr d}
$$
and let $\pi _j$ from $\rr {2d}$ to $\rr d$, $j=1,2$, be the projections
$$
\pi _1(x,\xi ) = x
\quad \text{and}\quad
\pi _2(x,\xi ) = \xi .
$$
Then $E_1\times E_2$ is the ordered basis $\{ e_1,\dots ,e_{2d}\}$
of $\rr {2d}$ such that
\begin{alignat*}{2}
\{ e_1,\dots ,e_d\} &\subseteq V_1,
&\qquad
E_1 &= \{ \pi _1(e_1),\dots ,\pi _1(e_d)\} ,
\\[1ex]
\{ e_{d+1},\dots ,e_{2d}\}
&\subseteq V_2 &
\quad \text{and}\quad
E_2 &= \{ \pi _2(e_{d+1}),\dots ,\pi _2(e_{2d})\} .
\end{alignat*}
\end{defn}

\par

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

\par

\begin{defn}\label{Def:Phasesplit}
Let $V_1$, $V_2$, $\pi _1$ and $\pi _2$ be as in Definition
\ref{Def:FromTwoToOneBasis}, $E$ be an ordered basis of
the phase space $\rr {2d}$ and let
$E_0\subseteq E$. Then
$E$ is called \emph{phase split} (with respect to $E_0$),
if the following is true:
\begin{enumerate}
\item the span of $E_0$ and $E\setminus E_0$ equal
$V_1$ and $V_2$, respectively;

\vspace{0.1cm}

\item let $E_1=\pi _1(E_0)$ and $E_2=
\pi _2(E\setminus E_0)$ be the bases in $\rr d$ which preserves the
orders from $E_0$ and $E\setminus E_0$. Then $(E_1,E_2)$
are permuted dual bases.
\end{enumerate}
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 \emph{strongly phase split}
(with respect to $E_0$).
\end{defn}

\par

In Definition \ref{Def:Phasesplit} it is understood that the orderings of
$E_0$ and $E\setminus E_0$
are inherited from the ordering in $E$.

\par

\begin{rem}\label{Rem:CommentPhasSplit}
Let $E$ and $E_j$, $j=0,1,2$ be the same as in Definition \ref{Def:Phasesplit}.
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. The pair $(E_1,E_2)$ is called
\emph{the pair of permuted dual bases, induced by $E$ and $E_0$}.

\par

On the other hand, suppose that $(E_1,E_2)$ is a pair of permuted dual bases to each
others on $\rr d$. Then it is clear that for
$E_1\times E_2=\{ e_1,\dots ,e_{2d}\}$ in Definition \ref{Def:FromTwoToOneBasis}
and $E_0=\{ e_1,\dots ,e_d\}$, we have that
$E_0$ and $E$ fullfils all properties in Definition \ref{Def:Phasesplit}. In this case,
$E_1\times E_2$ is called the phase split basis (of $\rr {2d}$) induced by $(E_1,E_2)$.

\par

It follows that if $E'$, $E_1'$ and $E_2'$ are the dual bases of $E$, $E_1$ and $E_2$,
repsectively, then $E'=E_1'\times E_2'$.
\end{rem}

\par

\subsection{Invariant quasi-Banach spaces and spaces of
mixed quasi-normed spaces of Lebesgue types}\label{subsec1.3}

\par

We recall that a quasi-norm $\nm {\, \cdot \, }{\mathscr B}$ of order $r \in (0,1]$ on the
vector-space $\mathscr B$ over $\mathbf C$ is a nonnegative functional on
$\mathscr B$ which satisfies
\begin{alignat}{2}
 \nm {f+g}{\mathscr B} &\le 2^{\frac 1r-1}(\nm {f}{\mathscr B} + \nm {g}{\mathscr B}), &
\quad f,g &\in \mathscr B ,
\label{Eq:WeakTriangle1}
\\[1ex]
\nm {\alpha \cdot f}{\mathscr B} &= |\alpha| \cdot \nm f{\mathscr B},
& \quad \alpha &\in \mathbf{C},
\quad  f \in \mathscr B
\notag
\intertext{and}
   \nm f {\mathscr B} &= 0\quad  \Leftrightarrow \quad f=0. & &
\notag
\end{alignat}
The space $\mathscr B$ is then called a quasi-norm space. A complete
quasi-norm space is called a quasi-Banach space. If $\mathscr B$
is a quasi-Banach space with
quasi-norm satisfying \eqref{Eq:WeakTriangle1}
then by \cite{Aik,Rol} there is an equivalent quasi-norm to $\nm \, \cdot \,  {\mathscr B}$
which additionally satisfies
\begin{align}\label{Eq:WeakTriangle2}
\nm {f+g}{\mathscr B}^r \le \nm {f}{\mathscr B}^r + \nm {g}{\mathscr B}^r, 
\quad f,g \in \mathscr B .
\end{align}
From now on we always assume that the quasi-norm of the quasi-Banach space $\mathscr B$
is chosen in such way that both \eqref{Eq:WeakTriangle1} and \eqref{Eq:WeakTriangle2}
hold.

\par

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

\par

A \emph{weight} or \emph{weight function} on $\rr d$ is a
positive function $\omega
\in  L^\infty _{loc}(\rr d)$ such that $1/\omega \in  L^\infty _{loc}(\rr d)$.
The weight $\omega$ is called \emph{moderate},
if there is a positive weight $v$ on $\rr d$ such that
\begin{equation}\label{moderate}
\omega (x+y) \lesssim \omega (x)v(y),\qquad x,y\in \rr d.
\end{equation}
If $\omega$ and $v$ are weights on $\rr d$ such that
\eqref{moderate} holds, then $\omega$ is also called
\emph{$v$-moderate}.
We note that \eqref{moderate} implies that $\omega$ fulfills
the estimates
\begin{equation}\label{moderateconseq}
v(-x)^{-1}\lesssim \omega (x)\lesssim v(x),\quad x\in \rr d.
\end{equation}
We let $\mathscr P _E(\rr d)$ be the set of all moderate weights on $\rr d$.

\par

It can be proved that if $\omega \in \mathscr P _E(\rr d)$, then
$\omega$ is $v$-moderate for some $v(x) = e^{r|x|}$, provided the
positive constant $r$ is large enough (cf. \cite{Gc2.5}). In particular,
\eqref{moderateconseq} shows that for any $\omega \in \mathscr P
_E(\rr d)$, there is a constant $r>0$ such that
$$
e^{-r|x|}\lesssim \omega (x)\lesssim e^{r|x|},\quad x\in \rr d.
$$

\par

We say that $v$ is
\emph{submultiplicative} if $v$ is even and \eqref{moderate}
holds with $\omega =v$. In the sequel, $v$ and $v_j$ for
$j\ge 0$, always stand for submultiplicative weights if
nothing else is stated. The next definition is similar to \cite[Section 3]{FG1}
in the Banach space case.

\par

\begin{defn}\label{Def:InvSpaces}
Let $r\in (0,1]$, $v\in \mathscr P _E(\rr {d})$ and let $\mathscr B = \mathscr B (\rr d)
\subseteq L^r_{loc}(\rr {d})$ be a quasi-Banach space such that $\Sigma _1
(\rr d)\subseteq \mathscr B (\rr d)$. Then $\mathscr B$ is called $v$-invariant on
$\rr d$ if the following is true:
\begin{enumerate}
\item $x\mapsto f(x+y)$ belongs to $\mathscr B$ for every $f\in \mathscr B$
and $y\in \rr {d}$.

\vspace{0.1cm}

\item There is a constant $C>0$ such that $\nm {f_1}{\mathscr B}\le C\nm {f_2}{\mathscr B}$
when $f_1,f_2\in \mathscr B$ are such that $|f_1|\le |f_2|$. Moreover,
$$
\nm {f(\, \cdot \,  +y)}{\mathscr B}\lesssim \nm {f}{\mathscr B}v(y),\qquad
f\in \mathscr B ,\ y\in \rr {d}.
$$
\end{enumerate}
\end{defn}

\par

Let $\mathscr B$ be as in Definition \ref{Def:InvSpaces}, $E$ be a basis for
$\rr d$ and let $\kappa (E)$ be the closed parallelepiped spanned by $E$.
The \emph{discrete version, $\ell _{\mathscr B ,E} = \ell _{\mathscr B ,E}(\Lambda _E)$,
of $\mathscr B$} with respect to $E$
is the set of all $a\in \ell _0'(\Lambda _E)$ such that
$$
\nm a{\ell _{\mathscr B ,E}} \equiv \Nm {\sum _{j\in \Lambda _E}a(j)\chi _{j+\kappa (E)}}{\mathscr B}
$$
is finite.

\par

An important example on $v$-invariant spaces concerns mixed quasi-norm
spaces of Lebesgue type, given in the following definition.

\par

\begin{defn}\label{Def:DiscLebSpaces}
Let $E = \{  e_1,\dots ,e_d \}$ be an ordered basis of $\rr d$, $\kappa (E)$ be the
parallelepiped spanned by $E$, $\omega \in \mathscr P _E(\rr d)$
$\boldsymbol q =(q_1,\dots ,q_d)\in (0,\infty ]^{d}$ and $r=\min (1,\boldsymbol q )$.
If  $f\in L^r_{loc}(\rr d)$, then
$$
\nm f{L^{\boldsymbol q }_{E,(\omega )}}\equiv
\nm {g_{d-1}}{L^{q_{d}}(\mathbf R)}
$$
where  $g_k\, :\, \rr {d-k}\to \mathbf R$,
$k=0,\dots ,d-1$, are inductively defined as
\begin{align*}
g_0(x_1,\dots ,x_{d})
&\equiv
|f(x_1e_1+\cdots +x_{d}e_d)\omega (x_1e_1+\cdots +x_{d}e_d)|,
\\[1ex]
\intertext{and}
g_k(\boldsymbol z _k) &\equiv
\nm {g_{k-1}(\, \cdot \,  ,\boldsymbol z_k)}{L^{q_k}(\mathbf R)},
\quad \boldsymbol z _k\in \rr {d-k},\ k=1,\dots ,d-1.
\end{align*}
If $\Omega \subseteq \rr d$ is measurable,
then $L^{\boldsymbol q }_{E,(\omega )}(\Omega )$ consists
of all $f\in L^r_{loc}(\Omega )$ with finite quasi-norm
$$
\nm f{L^{\boldsymbol q}_{E,(\omega )}(\Omega )}
\equiv
\nm {f_\Omega }{L^{\boldsymbol q}_{E,(\omega )}(\rr d)},
\qquad
f_\Omega (x)
\equiv
\begin{cases}
f(x), &\text{when}\ x\in \Omega
\\[1ex]
0, &\text{when}\ x\notin \Omega .
\end{cases}
$$
The space $L^{\boldsymbol q }_{E,(\omega )}(\Omega )$ is called 
\emph{$E$-split Lebesgue space (with respect to $\omega$, $\boldsymbol q$
and $\Omega$)}.
\end{defn}

\par

We let $\ell ^{\boldsymbol p} _{E,(\omega )}(\Lambda _E)$ be the discrete
version of $\mathscr B = L^{\boldsymbol p} _{E,(\omega )}(\rr d)$ when
${\boldsymbol p} \in (0,\infty ]^d$.



\par

Suppose that $E$ and $\Lambda$ are the same as in Definition
\ref{Def:DiscLebSpaces}. Then we let $(\ell _E^0)'(\Lambda )$
be the set of
all formal sequences $\{ a(j)\} _{j\in \Lambda}$, and we let
$\ell _E^0(\Lambda )$
be the set of all such sequences such that at most finite
numbers of $a(j)$ are
non-zero.

\par

\begin{rem}\label{Rem:LebExpIdent}
Evidently, $L^{\boldsymbol q}_{E,(\omega )} (\Omega )$ and
$\ell ^{\boldsymbol q}_{E,(\omega )} (\Lambda )$
in Definition \ref{Def:DiscLebSpaces} are quasi-Banach spaces of order
$\min ({\boldsymbol p},1)$. We set
$$
L^{\boldsymbol q}_{E} = L^{\boldsymbol q}_{E,(\omega )}
\quad \text{and}\quad
\ell ^{\boldsymbol q}_{E} = \ell ^{\boldsymbol q}_{E,(\omega )}
$$
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,
\begin{alignat*}{5}
L^{q}_{E,(\omega )} &=  L^{\boldsymbol q}_{E,(\omega )},
&\quad
L^{q}_{E} &= L^{\boldsymbol q}_{E},
&\quad
\ell ^{q}_{E,(\omega )} &= \ell ^{\boldsymbol q}_{E,(\omega )}
&\quad &\text{and} &\quad 
\ell ^{q}_{E} &= \ell ^{\boldsymbol q}_{E}
\intertext{for such $\boldsymbol q$, and notice that these spaces agree with}
&{\phantom =}L^q_{(\omega )}, &\qquad
&{\phantom =}L^q, &\qquad
&{\phantom =}\ell ^{q}_{(\omega )}
&\quad &\text{and}&\quad
&{\phantom =}\ell ^{q},
\end{alignat*}
respectively, with equivalent quasi-norms.
\end{rem}

\par

\subsection{Modulation and Wiener spaces}\label{subsec1.4}

\par

We consider a general class of modulation spaces
given in the following definition (cf. \cite{Fei5}).

\par

\begin{defn}\label{Def:ModSpaces}
Let $\omega ,v\in \mathscr P _E(\rr {2d})$ be such that $\omega$ is $v$-moderate,
$\mathscr B$ be a $v$-invariant quasi-Banach space on
$\rr {2d}$, and let $\phi \in \mathcal S _{1/2}(\rr d)\setminus 0$.
Then the \emph{modulation space} $M(\omega ,\mathscr B)$ consists of all $f\in \mathcal S _{1/2}'
(\rr d)$ such that
\begin{equation}\label{Eq:ModSpNorm}
\nm f{M(\omega ,\mathscr B)} \equiv \nm {V_\phi f \cdot \omega}{\mathscr B}
\end{equation}
is finite.
\end{defn}

\par

An important family of modulation spaces which contains
the classical modulation spaces, introduced by Feichtinger in
\cite{F1}, is given next.

\par

\begin{defn}\label{Def:ExtClassModSpaces}
Let ${\boldsymbol p} ,\boldsymbol q \in (0,\infty ]^d$, $E_1$ and $E_2$ be ordered bases
of $\rr d$, $E=E_1\times E_2$,
$\phi \in \Sigma _1(\rr d)\setminus 0$ and let
$\omega \in \mathscr P _E(\rr {2d})$.
For any $f\in \Sigma _1'(\rr d)$ set
\begin{multline*}
\nm f{M^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}}
\equiv
\nm {H_{1,f,E_1,{\boldsymbol p} ,\omega }}{L^{\boldsymbol q}_{E_2}},
\\[1ex]
\quad \text{where}\quad
H_{1, f,E_1,{\boldsymbol p} ,\omega }(\xi )
\equiv
\nm {V_\phi f (\, \cdot \,  ,\xi )\omega (\, \cdot \,  ,\xi )}{L^{{\boldsymbol p}}_{E_1}}
\end{multline*}
and
\begin{multline*}
\nm f{W^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}}
\equiv
\nm {H_{2,f,E_2,\boldsymbol q ,\omega }}{L^{{\boldsymbol p}}_{E_1}},
\\[1ex]
\quad \text{where}\quad
H_{2, f,E_2,\boldsymbol q ,\omega }(x)
\equiv
\nm {V_\phi f (x,\, \cdot \,  )\omega (x,\, \cdot \,  )}{L^{\boldsymbol q}_{E_2}}
\end{multline*}
The \emph{modulation space}
$M^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}(\rr d)$
($W^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}(\rr d)$)
consist of all $f\in \Sigma _1'(\rr d)$ such that
$\nm f{M^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}}$ 
($\nm f{W^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}}$) is finite.
\end{defn}

\par

The theory of modulation spaces has developed in different ways since they
were introduced in \cite{F1} by Feichtinger. (Cf. e.{\,}g. \cite{Fei5,GaSa,Gc2,Toft15}.)
For example, let ${\boldsymbol p}$, $\boldsymbol q$, $E$, $\omega$ and $v$ be the same
as in Definition \ref{Def:ModSpaces} and \ref{Def:ExtClassModSpaces},
and let $\mathscr B = L^{{\boldsymbol p} ,\boldsymbol q}_E(\rr {2d})$ and $r=\min (1,{\boldsymbol p} ,\boldsymbol q)$.
Then $M(\omega ,\mathscr B)=M^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}(\rr d)$
is a quasi-Banach space. Moreover, $f\in M^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}(\rr d)$
if and only if $V_\phi f \cdot \omega \in L^{{\boldsymbol p} ,\boldsymbol q}_{E}(\rr {2d})$,
and different choices of $\phi$ give rise to equivalent quasi-norms in
Definition \ref{Def:ExtClassModSpaces}.
We also note that for any such $\mathscr B$, then
$$
\Sigma _1(\rr d) \subseteq M^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}(\rr d) \subseteq \Sigma _1'(\rr d).
$$
Similar facts hold for the space $W^{{\boldsymbol p} ,\boldsymbol q}_{E,(\omega )}(\rr d)$.
(Cf. \cite{GaSa,Toft15}.)

\par

We shall consider various kinds of Wiener spaces involved later on when
finding different characterizations of modulation spaces. The following
type of Wiener spaces can essentially be found in e.{\,}g.
\cite{FG1,GaSa,Gc2}, and is related to coorbit spaces of Lebesgue spaces. 

\par

\begin{defn}\label{Def:WienerSpace}
Let $\boldsymbol r \in (0,\infty ]^d$, 
$\omega _0\in \mathscr P _E(\rr d)$, $\omega \in \mathscr P _E(\rr {2d})$,
$\phi \in \Sigma _1(\rr d)\setminus 0$,
$E\subseteq \rr d$ be an ordered basis, and let $\kappa (E)$ be the closed parallelepiped
spanned by $E$. Also let $\mathscr B = \mathscr B (\rr d)$ and $\mathscr B _0 = \mathscr B _0(\rr d)$
be invariant QBF-spaces on $\rr d$, $f$ and $F$ be measurable on $\rr d$ respective
$\rr {2d}$, $F_\omega =F\cdot \omega$, and let
$\ell _{\mathscr B ,E}(\Lambda _E)$ be the discrete version of $\mathscr B$ with respect to $E$.
\begin{enumerate}
\item Then $\nm f{\mathsf{W} ^{\boldsymbol r}_{E}(\omega _0,\ell _{\mathscr B ,E} )}$ is given by
\begin{align*}
\nm f{\mathsf{W} ^{\boldsymbol r}_{E}(\omega _0,\ell _{\mathscr B ,E} )}
&\equiv
\nm {h_{E,\omega _0,\boldsymbol q ,f}}{\ell _{\mathscr B ,E}(\Lambda _E)},
\\[1ex]
h_{E,\omega _0,\boldsymbol q ,f} (j)
&=
\nm {f} {L_E^{\boldsymbol r}(j+\kappa (E))}\omega _0(j),
\quad
j\in \Lambda _E.
\end{align*}
The set $\mathsf{W} ^{\boldsymbol r}_{E}(\omega ,\ell _{\mathscr B ,E} )$ consists of all measurable
$f$ on $\rr d$ such that $\nm f{\mathsf{W} ^{\boldsymbol r}_{E}(\omega _0,\ell _{\mathscr B ,E} )}<\infty$;

\vspace{0.1cm}

\item Then $\nm F{\mathsf{W} ^{\boldsymbol r}_{k,E}(\omega ,\ell _{\mathscr B ,E}\,,\mathscr B _0 )}$, $k=1,2$,
are given by
\begin{alignat*}{2}
\nm F{\mathsf{W} ^{\boldsymbol r}_{1,E}(\omega ,\ell _{\mathscr B ,E}\,,\mathscr B _0 )}
&\equiv
\nm {\varphi _{F,\omega ,\boldsymbol r ,\mathscr B ,E}}{\mathscr B _0},&\quad
\varphi _{F,\omega ,\boldsymbol r ,\mathscr B ,E}(\xi )
&=
\nm {F_\omega (\, \cdot \,  ,\xi )}{\mathsf{W} ^{\boldsymbol r}_{E}(1,\ell _{\mathscr B ,E} )},
\intertext{and}
\nm F{\mathsf{W} ^{\boldsymbol r}_{2,E}(\omega ,\ell _{\mathscr B ,E}\, ,\mathscr B _0 )}
&\equiv
\nm {\psi _{F,\omega ,\mathscr B _0}}{\mathsf{W} ^{\boldsymbol r}_{E}
(1,\ell _{\mathscr B ,E} )},&
\qquad
\psi _{F,\omega ,\mathscr B _0}(x) &= \nm {F_\omega (x,\, \cdot \,  )}{\mathscr B _0}.
\end{alignat*}
The set $\mathsf{W} ^{\boldsymbol r}_{k,E}(\omega ,\ell _{\mathscr B ,E}\,,\mathscr B _0 )$ 
consists of all
measurable $F$ on $\rr {2d}$ such that
$\nm F{\mathsf{W} ^{\boldsymbol r}_{k,E}
(\omega ,\ell _{\mathscr B ,E}\,,\mathscr B _0 )}<\infty$, $k=1,2$.
\end{enumerate}
\end{defn}

\par

The space $\mathsf{W} ^{\boldsymbol r}_{E}(\omega _0,\ell _{\mathscr B ,E})$
in Definition
\ref{Def:WienerSpace} is essentially a Wiener amalgam space with
$L^{\boldsymbol r}_E$ as local (quasi-)norm and $\mathscr B$ or
$\ell _{\mathscr B ,E}(\Lambda _E)$
as global component. They are also related to
coorbit spaces. (See \cite{Fe0,FG1,FG2,FeLu,Rau1,Rau2}.)

\par

In fact, $\mathsf{W} ^\infty (\omega _0,\ell ^{{\boldsymbol p}})$ in Definition
\ref{Def:WienerSpace} (i.{\,}e. the case
$\boldsymbol r = (\infty ,\dots ,\infty )$ and $E$ is the standard basis)
is the \emph{coorbit space} of
$L^{{\boldsymbol p}}(\rr d)$ with weight $\omega _0$, and is
sometimes denoted by
$$
\mathsf{Co} (L^{{\boldsymbol p}}_{(\omega _0)}(\rr d))\quad \text{or}\quad 
W(L_{(\omega _0)} ^{{\boldsymbol p}}) = W(L_{(\omega)}
^{{\boldsymbol p}}(\rr d)),
$$
in the literature (cf. \cite{Gc2,Rau1,Rau2}).

\par

\begin{rem}\label{Rem:PerWienerSpace}
Let ${\boldsymbol p}$, $\omega _0$, $\omega$, $E$, $\mathscr B$,
$\mathscr B _0$, $f$ and $F$
be the same as in Definition \ref{Def:WienerSpace}.
Evidently, by using the fact that $\omega _0$ is
$v_0$-moderate for some $v_0$,
it follows that
\begin{align*}
\nm {f\cdot \omega _0}{\mathsf{W} ^{\boldsymbol q}_{E}(1,\ell _{\mathscr B ,E})}
&\asymp
\nm {f}{\mathsf{W} ^{\boldsymbol q}_{E}(\omega _0,\ell _{\mathscr B ,E})}
\intertext{and}
\nm {F\cdot \omega}
{\mathsf{W} ^{\boldsymbol q}_{k,E}(1,\ell _{\mathscr B ,E}\, ,\mathscr B _0 )}
&=
\nm {F}{\mathsf{W} ^{\boldsymbol q}_{k,E}(\omega ,\ell
_{\mathscr B ,E}\, ,\mathscr B _0)}
\end{align*}
for $k=1,2$. Furthermore,
\begin{alignat*}{1}
\mathsf{W} ^{\boldsymbol q}_{1,E}(\omega ,\ell _{\mathscr B ,E}\, ,\mathscr B _0)
&=
\omega ^{-1}
\cdot
\mathsf{W} ^{\boldsymbol q}_{E}(1,\ell _{\mathscr B ,E}\, ;\, \mathscr B _0)
\intertext{and}
\mathsf{W} ^{\boldsymbol q}_{2,E}(\omega ,\ell _{\mathscr B ,E}\, ,\mathscr B _0)
&=
\omega ^{-1}
\cdot
\mathscr B _0 (\rr d\, ;\, \mathsf{W} ^{\boldsymbol q}_{E}
(1,\ell _{\mathscr B ,E} ) ).
\end{alignat*}
Here and in what follows,
$\mathscr B (\rr {d}\, ;\, \mathscr B _0)
=
\mathscr B (\rr {d}\, ;\, \mathscr B _0(\rr {d_0}))$
is the set of all functions $g$ in $\mathscr B $ with values in
$\mathscr B _0$, which are equipped with the quasi-norm
$$
\nm g{\mathscr B (\rr {d};\mathscr B _0)}\equiv \nm {g_0}{\mathscr B},
\qquad g_0(x) \equiv \nm {g(x)}{\mathscr B _0},
$$
when $\mathscr B (\rr {d})$ and $\mathscr B _0(\rr {d_0})$ are
invariant QBF-spaces.
\end{rem}

\par

Later on we discuss periodicity in the framework of certain
modulation spaces which are related to spaces which are defined
by imposing $L^\infty$-conditions on the configuration
variable of corresponding short-time Fourier transforms.

\par

\begin{defn}\label{Def:AltModSpace}
Let $E$, $\boldsymbol r$, $\mathscr B _0$ and $\omega \in \mathscr P _E(\rr {2d})$
be the same as in Definition \ref{Def:WienerSpace}, and let $\phi \in
\Sigma _1(\rr d)\setminus 0$. Then
$\EuScript M ^{\boldsymbol r}_E(\omega ,\mathscr B _0)$ and
$\EuScript W ^{\boldsymbol r}_E(\omega ,\mathscr B _0)$)
are the sets of all $f\in \Sigma _1'(\rr d)$ such that
\begin{align*}
\nm f{\EuScript M ^{\boldsymbol r}_E(\omega ,\mathscr B _0)}
&\equiv
\nm {V_\phi f}{\mathsf{W} ^{\boldsymbol r}_{1,E}(\omega ,\ell ^{\infty}_E,\mathscr B _0)}
\quad
\intertext{respectively}
\nm f{\EuScript W ^{\boldsymbol r}_E(\omega ,\mathscr B _0)}
&\equiv
\nm {V_\phi f}{\mathsf{W} ^{\boldsymbol r}_{2,E}(\omega ,\ell ^{\infty}_E,\mathscr B _0)}
\end{align*}
are finite.
\end{defn}

\par

\begin{rem}\label{Rem:WienerSpace}
For the spaces in Definition \ref{Def:WienerSpace}
we set $\mathsf{W} ^{q_0,\boldsymbol r _0} =  \mathsf{W} ^{\boldsymbol r}$, when
\begin{alignat*}{2}
\boldsymbol r _0 &=(r_1,\dots ,r_d)\in (0,\infty ]^d,&
\quad \text{and}\quad
\boldsymbol r &= (q_0,\dots ,q_0,r_1,\dots ,r_d)\in (0,\infty ]^{2d},
\end{alignat*}
and similarly for other types of exponents and for the spaces
in Definitions \ref{Def:ExtClassModSpaces}
and \ref{Def:AltModSpace}. (See also Remark
\ref{Rem:LebExpIdent}.)
We also set
$$
M^{\infty ,\boldsymbol q}_{E,(\omega )}
=
M^{\infty ,\boldsymbol q}_{E_2,(\omega )}
\quad \text{and}\quad
W^{\infty,\boldsymbol q}_{E,(\omega )}
=
W^{\infty,\boldsymbol q}_{E_2,(\omega )}
$$
when $E_1,E_2$ are ordered bases of $\rr d$ and
$E=E_1\times E_2$, for spaces in Definition \ref{Def:ExtClassModSpaces},
since these spaces are independent of $E_1$.
\end{rem}

\par

In Section \ref{sec2} we prove that if $\mathscr B _0$ is an $E$-split
Lebesgue space on $\rr d$ and $\omega (x,\xi )\in \mathscr P _E(\rr {2d})$
which is constant with respect to the $x$ variable, then
$\EuScript M ^{\boldsymbol r}_E(\omega ,\mathscr B _0)$
and $\EuScript W ^{\boldsymbol r}_E(\omega ,\mathscr B _0)$ are independent of $\boldsymbol r$
and agree with modulation spaces of the form in
Definition \ref{Def:ModSpaces}
(cf. Proposition \ref{Prop:SpecCaseWienerRel1}).

\par

The next result is a reformulation of \cite[Proposition 3.4]{Toft15}, and
indicates how Wiener spaces are connected to modulation spaces. The proof
is therefore omitted. Here, let
\begin{equation}\label{Eq:SubMultMod}
(\Theta _\rho v)(x,\xi )=v(x,\xi )\eabs {x,\xi}^\rho ,
\quad \text{where}\quad
\rho \ge 2d\left ( \frac 1r-1 \right ),
\end{equation}
for any submultiplicative $v\in \mathscr P _E(\rr {2d})$ and $r\in (0,1]$.
It follows that $L^1_{(\Theta _\rho v)}(\rr {2d})$ is
continuously embedded in
$L^r_{(v)}(\rr {2d})$, giving that $M^1_{(\Theta _\rho v)}(\rr d)
\subseteq M^r_{(v)}(\rr d)$. Hence if $\phi \in M^1_{(\Theta _\rho v)}
\setminus 0$, $\varepsilon _0$ is chosen such that $S^\Lambda _{\phi ,\phi}$
is invertible on $M^1_{(\Theta _\rho v)}(\rr d)$ for every
$\Lambda =\varepsilon \Lambda _E$, $\varepsilon \in (0,\varepsilon _0]$, it follows that both
$\phi$ and its canonical dual with respect to $\Lambda$ belong to
$M^r_{(v)}(\rr d)$. Notice that such $\varepsilon _0>0$ exists in view of
\cite[Theorem S]{Gc1}.

\par

\begin{prop}\label{Prop:WienerEquiv}
Let $E$ be a phase split basis for $\rr {2d}$,
${\boldsymbol p} \in (0,\infty ]^{2d}$, $r=\min (1,{\boldsymbol p} )$,
$\omega ,v\in \mathscr P _E(\rr {2d})$
be such that $\omega$ is $v$-moderate, $\rho$ and $\Theta _\rho v$ be
as in \eqref{Eq:SubMultMod} with strict inequality when $r<1$,
and let $\phi _1,\phi _2\in M^1_{(\Theta _\rho v)} (\rr d)\setminus 0$.
Then
$$
\nm f{M^{{\boldsymbol p}}_{E,(\omega )}}
\asymp
\nm {V_{\phi _1} f}{L^{\boldsymbol p} _{E,(\omega )}}
\asymp 
\nm {V_{\phi _2} f}{\mathsf{W} ^\infty _{E}
(\omega ,\ell ^{\boldsymbol p} _{E})},\quad f\in
\mathcal S '_{1/2}(\rr d).
$$
In particular, if $f\in \mathcal S _{1/2}'(\rr d)$, then
$$
f\in M^{{\boldsymbol p}}_{E,(\omega )}(\rr {2d})
\quad \Leftrightarrow \quad
V_{\phi _1} f \in L^{\boldsymbol p} _{E,(\omega )}(\rr {2d})
\quad \Leftrightarrow \quad
V_{\phi _2} f \in \mathsf{W} ^\infty _{E}(\omega ,\ell ^{\boldsymbol p} _{E}
(\Lambda _E)).
$$
\end{prop}

\par

In Section \ref{sec2} we extend this result in such way
that we may replace
$\mathsf{W} ^\infty _{E}(\omega ,\ell ^{\boldsymbol p} _{E})$
by $\mathsf{W} ^r _{E}(\omega ,\ell ^{\boldsymbol p} _{E})$ for any $r>0$.

\par

\subsection{Classes of periodic elements}

\par

We consider
spaces of periodic Gevrey functions and their duals.

\par

Let $s,\sigma \in\mathbf{R}_{+}$ be such that $s+t\geq 1$,
$f\in (\mathcal S_{s}^{\sigma})'(\rr d)$, 
$E$ be a basis of $\rr d$ and let $E_0\subseteq E$.
Then $f$ is called \emph{$E_0$-periodic} if $f(x+y)=f(x)$ 
for every $x\in \rr d$ and $y\in E_0$.

\par

We note that for any $\Lambda _E$-periodic function
$f\in C^{\infty}(\rr d)$, we have 
\begin{align}
f &= \sum_{\alpha\in \Lambda ' _E} c(f,{\alpha})e^{i\scal \, \cdot \,  \alpha},
\label{Eq:Expan2}
\intertext{where $c(f,{\alpha})$ are the Fourier coefficients given by}
c(f,{\alpha}) &\equiv \vert \kappa (E) \vert ^{-1}
(f,e^{i\scal \, \cdot \,  \alpha})_{L^{2}(E)}.\notag
\end{align}

\par

For any $s\ge 0$ and basis $E\subseteq \rr d$ we let
$\mathcal E _{0,\sigma}^{E}(\rr d)$
and $\mathcal E _{\sigma}^{E}(\rr d)$ be the sets of all $E$-periodic 
elements
in $\mathcal E _{0,\sigma}(\rr d)$ and in $\mathcal E _{\sigma}(\rr d)$, respectively. Evidently,
$$
\mathcal E _\sigma^E(\rr d)\simeq \mathcal E _\sigma (\rr d/\Lambda _E)
\quad \text{and}\quad
\mathcal E _{0,\sigma}^E(\rr d)\simeq \mathcal E _{0,\sigma}(\rr d/\Lambda _E),
$$
which is a common approach in the literature.

\par

\begin{rem}
Let $E$ be an ordered basis on $\rr d$
and $V$ be a topological space of functions or (ultra-)distributions on
$\rr d$. Then we use the convention that $V^E$ ($E$ as upper case index) denotes
the $E$ periodic elements in $V$, while $V_E$ ($E$ as lower case index) is the
space analogous to $V$ when $E$ is used as basis.
\end{rem}

\par

Let $s,s_0,\sigma ,\sigma _0>0$ be such that $s+\sigma \ge 1$,
$s_0+\sigma _0\ge 1$ and $(s_0,\sigma _0)\neq (\frac 12,\frac 12)$. Then
we recall that the duals
$(\mathcal E _\sigma ^E)'(\rr d)$ and $(\mathcal E _{0,\sigma _0}^E)'(\rr d)$ of
$\mathcal E _\sigma ^E(\rr d)$ and $\mathcal E _{0,\sigma _0}^E(\rr d)$, respectively,
can be identified with the $E$-periodic elements in $(\mathcal S _s^\sigma)'(\rr d)$
and $(\Sigma _{s_0}^{\sigma _0})'(\rr d)$ respectively via unique extension of the form
\begin{align*}
(f,\phi)_{E} &= \sum_{\alpha \in \Lambda '_{E}} c(f,{\alpha})
\overline{c(\phi ,{\alpha})}
\end{align*}
on $\mathcal E _{0,\sigma _0}^E (\rr d)\times \mathcal E _{0,\sigma _0}^E (\rr d)$.
We also let $(\mathcal E _0^E)'(\rr d)$ be the set of all formal expansions
in \eqref{Eq:Expan2} and $\mathcal E _0^E(\rr d)$ be the set of
all formal expansions in \eqref{Eq:Expan2} such that at most finite
numbers of $c(f,{\alpha})$ are non-zero (cf. \cite{ToNa}). We refer to
\cite{Pil2,ToNa} for more characterizations of $\mathcal E _{\sigma}^E$,
$\mathcal E _{0,\sigma}^E$ and their duals.

\par


\par

The following definition takes care of spaces of formal
expansions \eqref{Eq:Expan2} with coefficients
obeying specific quasi-norm estimates.

\par

\begin{defn}\label{Def:PerQuasiBanachSpaces}
Let $E$ be a basis of $\rr d$,
$\mathscr B$ be a
quasi-Banach space  continuously embedded in $\ell _0' (\Lambda ' _E)$
and let $\omega _0$ be a weight on $\rr d$. Then
$\mathcal L ^E(\omega _0,\mathscr B )$ consists of all 
$f\in(\mathcal E_{0}^E)'(\rr d)$ such that 
\begin{equation*}
\nm f{\mathcal L ^E(\omega _0,\mathscr B)}
\equiv
\nm {\{ c(f,\alpha)\omega _0(\alpha) \}
_{\alpha \in \Lambda '_E}} {\mathscr B}
\end{equation*}
is finite.
\end{defn}

\par

If $\omega _0\in \mathscr P _E(\rr d)$ and $\omega (x,\xi )=\omega _0(\xi )$,
then
\begin{align}
\begin{aligned}\label{Eq:ModPerNormEquiv1}
\nm f{\EuScript M ^{\boldsymbol r}_E (\omega ,\mathscr B )}
&=
\nm {g\omega _0}{\mathscr B},
\\[1ex]
\text{when} \quad
g(\xi ) &= \nm {V_\phi f(\, \cdot \,  ,\xi)}{L^{\boldsymbol r}_E(\kappa (E))},
\quad
f\in (\mathcal E _0^E)'(\rr d),
\end{aligned}
\intertext{and}
\begin{aligned}\label{Eq:ModPerNormEquiv2}
\nm f{\EuScript W ^{\boldsymbol r}_E (\omega ,\mathscr B )}
&=
\nm h{L^{\boldsymbol r}_E(\kappa (E))},
\\[1ex]
\text{when} \quad
h(x) &= \nm {V_\phi f(x,\, \cdot \,  )\omega _0}{\mathscr B},
\quad
\phantom{{}_{L^{\boldsymbol r}_E(\kappa (E))}}
\quad f\in (\mathcal E _0^E)'(\rr d),
\end{aligned}
\end{align}
because the $E$-periodicity of $x\mapsto |V_\phi f(x,\xi )|$
when $f$ is $E$ periodic gives

\begin{equation}\label{Eq:AltFormPerNorms}
\begin{aligned}
g(\xi ) &= \nm {V_\phi f(\, \cdot \,  ,\xi)}{L^{\boldsymbol r}_E(\kappa (E))}
=
\nm {V_\phi f(\, \cdot \,  ,\xi)}{L^{\boldsymbol r}_E(x+\kappa (E))},
\\[1ex]
\nm h{L^{\boldsymbol r}_E(\kappa (E))}
&=
\nm h{L^{\boldsymbol r}_E(x+\kappa (E))},
\qquad \qquad \qquad \qquad \qquad x\in \rr d.
\end{aligned}
\end{equation}

\par

\begin{prop}\label{Prop:PerMod}
Let $E$ be a basis of $\rr d$, $r\in(0,1]$,
$\mathscr B \subseteq L^{r}_{loc}(\rr d)$ be an $E'$-split
Lebesgue space, $\ell _{\mathscr B ,E}(\Lambda _E)$ be its discrete version,
$\omega _0\in \mathscr P _E(\rr d)$ and let $\omega (x,\xi )=\omega _0(\xi)$
when $x,\xi \in \rr d$. Then
\begin{equation*}
\mathcal L ^E(\omega _0,\ell _{\mathscr B ,E})
=
\EuScript M^{\infty}_{E}(\omega ,\mathscr B )\bigcap (\mathcal E _0^E)'(\rr d)
=
\EuScript W^{\infty}_{E}(\omega ,\mathscr B )\bigcap (\mathcal E _0^E)'(\rr d).
\end{equation*}
\end{prop}

\par

When proving that $\mathsf{W} ^{\boldsymbol r} _{E}(\omega ,\ell ^{\boldsymbol p} _{E})$
is independent of $\boldsymbol r \in (0,\infty ]^d$ in Section \ref{sec2},
as announced earlier, it will at the same time follow that if
$\mathscr B$ is a suitable quasi-norm space of Lebesgue type, then
\begin{equation}\label{Eq:ModrumInvariants1}
\begin{alignedat}{3}
\EuScript M ^{\boldsymbol r _1}_E (\omega ,\mathscr B ) &= \EuScript W ^{\boldsymbol r _2}_E (\omega ,\mathscr B )
& \quad &\text{when} & \quad \omega &\in \mathscr P _E(\rr {2d})
\end{alignedat}
\end{equation}
for every $\boldsymbol r _1,\boldsymbol r _2\in (0,\infty ]^d$.

\par

\begin{rem}\label{Rem:PerDistr}
The link between periodic Gelfand-Shilov distributions and formal Fourier
series expansions is given by the formula
\begin{equation}\label{Eq:PerDistAction}
\scal f\phi = (2\pi )^{\frac d2}\sum _{\alpha \in \Lambda _E'}c(f,\alpha )
\widehat \phi (-\alpha ).
\end{equation}
\end{rem}

\par

\subsection{The Zak transform}

\par

For any ordered basis $E$ of $\rr d$ and $f\in \mathscr S (\rr d)$,
the \emph{Zak transform} is defined by
\begin{equation}\label{Eq:DefZak}
(Z_E f)(x,\xi ) \equiv \sum _{j\in \Lambda _E}
f(x-j)e^{i\scal j\xi}
\end{equation}

\par

Several properties for the Zak transform can be found in \cite{Gc2}. 
For example, by
the definition it follows that $Z_E$ is continuous from $\mathscr S 
(\rr d)$ to the set of all smooth functions
on $\rr {2d}$ 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
\begin{equation}\label{Eq:QuasiPer}
\begin{alignedat}{1}
F(x+k,\xi ) = e^{i\scal k\xi} F(x,\xi )
\quad \text{and}\quad
F(x,\xi +\kappa ) &= F(x,\xi ),
\\[1ex]
k &\in \Lambda _E,\ 
\kappa \in \Lambda _E'.
\end{alignedat}
\end{equation}
It follows by similar arguments as in Section 7.2 in \cite{Ho1}
that if $F$ above
is a distribution (Gevrey distribution), then $F$ is a tempered
distribution (Gelfand-Shilov) distribution. For conveniency we
set $Z_1=Z_E$ when $E$ is the standard basis for $\rr d$.

\par

For the Zak transform we recall the following important mapping
properties on $L^2(\rr d)$ and $\mathscr S (\rr d)$. 

\par

\begin{prop}\label{Prop:ZakTransfBasicMaps}
Let $E$ be an ordered basis of $\rr d$. Then the following is true:
\begin{enumerate}
\item The operator $Z_E$ is homeomorphic from $L^2(\rr d)$
to the set of all quasi-periodic elements of order $E$ in
$L^2_{loc}(\rr {2d})$.
Furthermore, 
\begin{equation}\tag*{(\ref{Eq:ZakonL2})$'$}
\nm {Z_E f}{L^2(\kappa (E\times E'))} = |\kappa (E')|^{\frac d2}
\nm f{L^2},
\qquad f\in L^2(\rr d).
\end{equation}
holds;

\vspace{0.1cm}

\item The operator $Z_E$ restricts to a homeomorphism from
$\mathscr S (\rr d)$ to the set of all quasi-periodic elements of
order $E$ in $C^\infty (\rr {2d})$.
\end{enumerate}
\end{prop}

\par

\begin{proof}
Let $T_E$ be as in Remark \ref{Rem:TEMap}.
By straight-forward computations it follows that
\begin{equation}\label{Eq:ZakTransfers}
Z_Ef(x,\xi ) = (Z_1f_E)(T_E^{-1}x,T_E^*\xi ),\quad
f_E=f\circ T_E.
\end{equation}
The assertion (1) now follows from \eqref{Eq:ZakonL2},
\eqref{Eq:ZakTransfers} and
suitable changes of variables in the involved integrals.
The details are left for the reader.

\par

The assertion (2) follows from \eqref{Eq:ZakTransfers}
and \cite[Theorem 8.2.5]{Gc2}.
\end{proof}

\par


\section{Estimates on Wiener spaces and periodic elements
in modulation spaces}
\label{sec2}

\par

In this section we deduce equivalences between
various Wiener (quasi-)norm estimates on short-time
Fourier transforms. Especially we prove that \eqref{Eq:ModrumInvariants1}
holds for every $\boldsymbol r _1,\boldsymbol r _2\in (0,\infty ]^d$.



\par


\par

\subsection{Estimates of Wiener spaces}

\par

We begin with the following inclusions between
the different Wiener spaces in the previous section.

\par

\begin{prop}\label{Prop:WienerRel1}
Let $(E_1,E_2)$ be permuted dual bases of $\rr d$,
$E=E_1\times E_2$,
${\boldsymbol p} ,\boldsymbol q ,\boldsymbol r \in (0,\infty ]^d$ $r_1\in (0,\min ({\boldsymbol p} ,\boldsymbol q ,\boldsymbol r )]$,
$r_2\in (0,\min (\boldsymbol q )]$, and let $\omega _1,\omega _2
\in \mathscr P _E(\rr {2d})$ be such that
$$
\omega _1(x,\xi ) =
\omega _2(\xi ,x),\qquad x,\xi \in \rr d.
$$
Then
\begin{multline}\label{Eq:WienerRel1}
\mathsf{W} ^{\boldsymbol r ,\infty}_E(\omega ,\ell
^{{\boldsymbol p} ,\boldsymbol q}_E(\Lambda _E))
\hookrightarrow
\mathsf{W} ^{\boldsymbol r}_{1,E_1}(\omega
,\ell _{E_1}^{{\boldsymbol p}}(\Lambda _{E_1}),L^{\boldsymbol q}_{E_2}(\rr d))
\\[1ex]
\hookrightarrow
\mathsf{W} ^{r_1}_E(\omega ,\ell ^{{\boldsymbol p} ,\boldsymbol q}_E
(\Lambda _E))
\end{multline}
and
\begin{multline}\label{Eq:WienerRel2}
\mathsf{W} ^{\infty}_{E'}(\omega ,\ell
^{\boldsymbol q ,{\boldsymbol p}}_{E'}(\Lambda _{E}'))
\hookrightarrow
\mathsf{W} ^{r_2}_{2,E_2'}(\omega ,\ell ^{{\boldsymbol p}}_{E_2'}(\Lambda _{E_2}'), L^{\boldsymbol q}_{E_1'}(\rr d))
\\[1ex]
\hookrightarrow
\mathsf{W} ^{r_2}_{E'}(\omega ,\ell
^{\boldsymbol q ,{\boldsymbol p}}_{E'}(\Lambda _{E}')).
\end{multline}
\end{prop}



\par

\begin{rem}\label{Rem:IncreasingWienerSpaces}
For the involved spaces in Proposition \ref{Prop:WienerRel1} it follows from
H{\"o}lder's inequality that
$$
\mathsf{W} ^{\boldsymbol r}_{1,E_1}(\omega ,\ell ^{{\boldsymbol p}}_{E_1}(\Lambda _{E_k}),L^{\boldsymbol q}_{E_2}(\rr d)),
\quad
\mathsf{W} ^{\boldsymbol r}_{2,E_2'}(\omega ,\ell ^{{\boldsymbol p}}_{E_2'}(\Lambda _{E_2}'),L^{\boldsymbol q}_{E_1'}(\rr d))
$$
and
$$
\mathsf{W} ^{\boldsymbol r}_E(\omega ,\ell ^{{\boldsymbol p}}_{E}(\Lambda _{E}))
$$
increase with respect to ${\boldsymbol p}$ and decrease with respect to $\boldsymbol r$.
\end{rem}

\par


\par

We need the following lemma for the proof of Proposition \ref{Prop:WienerRel1}.

\par

\begin{lemma}\label{Lemma:WienerNormEst2}
Let $\omega \in \mathscr P _E(\rr d)$, $E$ be an ordered
basis of $\rr d$,
$\kappa (E)$ the parallelepiped spanned
by $E$, ${\boldsymbol p} \in (0,\infty ]^d$,
$r\in (0,\min ({\boldsymbol p} )]$ and
let $f$ be measurable on $\rr d$. Then
\begin{equation}\label{Eq:WienerNormEst2}
\nm a{\ell ^{{\boldsymbol p}}_E(\Lambda _E)}
\lesssim
\nm f{L^{{\boldsymbol p}}_{E,(\omega )}},
\quad
a(j) = \nm f{L^r_E(j+\kappa (E))}\omega (j).
\end{equation}
\end{lemma}

\par

\begin{proof}
Let $f$ be measurable on $\rr d$, $g_k$ be the same as in
Definition \ref{Def:DiscLebSpaces},
$T_E$ be the linear map which maps the standard basis
into $E$,
$Q_k=[0,1]^k$, and let ${\boldsymbol p} _k=(p_{k+1},\dots ,p_d)$,
when $k\ge 1$. Then
$$
\nm f{L^r_E(T_E(j)+\kappa (E))}\omega (j)
\asymp
\nm {f\cdot \omega }{L^r_E(T_E(j)+\kappa (E))}
\asymp
\nm {g_0}{L^r_E(j+Q_d)},\quad j\in \zz d.
$$
This reduce the situation to the case that $E$ is the
standard basis, $\kappa (E)=Q_d$ and $\omega =1$.
Moreover, by replacing $|f|^r$ with $f$ and $p_jr$
by $p_j$, $j=1,\dots ,d$, we may assume that $r=1$
(and that each $p_j\ge 1$).

\par

By induction it suffices to prove that if
\begin{align}
a_k(l) &= \nm {g_k}{L^1(l+Q_{d-k})},\quad l\in \zz {d-k},
\notag
\intertext{then}
\nm {a_k}{\ell ^{{\boldsymbol p} _k}(\zz {d-k})}
&\lesssim
\nm {a_{k+1}}{\ell ^{{\boldsymbol p} _{k+1}}(\zz {d-k-1})},\quad
k=0,\dots ,d-1,
\label{Eq:WienerNormEst2Ess}
\end{align}
since $\nm {a_0}{\ell ^{{\boldsymbol p} _0}(\zz {d})}$ is equal to the
left-hand side of \eqref{Eq:WienerNormEst2}, and
$a_d=\nm {a_d}{\ell ^\infty (\zz 0)}$ is equal to
the right-hand side of \eqref{Eq:WienerNormEst2}.

\par

Let $m\in \zz {d-k-1}$ be fixed. We only prove \eqref{Eq:WienerNormEst2Ess}
in the case $p_{k+1}<\infty$. The case $p_{k+1}=\infty$ will follow by similar
arguments and is left for the reader. By first using Minkowski's inequality
and then H{\"o}lder's inequality we get
\begin{multline*}
\nm {a_k(\, \cdot \,  ,m)}{\ell ^{p_{k+1}}(\mathbf Z)}
=
\left ( \sum _{l_1\in \mathbf Z}\nm {g_k}{L^1((l_1,m)+Q_{d-k})}^{p_{k+1}}
\right )^{\frac 1{p_{k+1}}}
\\[1ex]
=
\left ( \sum _{l_1\in \mathbf Z}
\left (
\int _{m+Q_{d-k-1}}
\left (
\int _{l_1+Q_1}
g_k(t,y)\, dt
\right )\, dy
\right )^{p_{k+1}}
\right )^{\frac 1{p_{k+1}}}
\\[1ex]
\le
\int _{m+Q_{d-k-1}}
\left (
\sum _{l_1\in \mathbf Z}
\left (
\int _{l_1+Q_1}
g_k(t,y)\, dt
\right )^{p_{k+1}}
\right )^{\frac 1{p_{k+1}}}
\, dy
\\[1ex]
\lesssim
\int _{m+Q_{d-k-1}}
\left (
\sum _{l_1\in \mathbf Z}
\int _{l_1+Q_1}
g_k(t,y)^{p_{k+1}}\, dt
\right )^{\frac 1{p_{k+1}}}
\, dy
\\[1ex]
=
\int _{m+Q_{d-k-1}}
\left (
\int _{\mathbf R}
g_k(t,y)^{p_{k+1}}\, dt
\right )^{\frac 1{p_{k+1}}}
\, dy
=
\int _{m+Q_{d-k-1}}
g_{k+1}(y)\, dy.
\end{multline*}
Hence,
\begin{equation}\label{Eq:WienerNormEst2Ess2}
\nm {a_k(\, \cdot \,  ,m)}{\ell ^{p_{k+1}}(\mathbf Z)}
\lesssim
a_{k+1}(m),\qquad m\in \zz {d-k-1}.
\end{equation}
By applying the $\ell ^{{\boldsymbol p} _{k+1}}(\zz {d-k-1})$-norm on
\eqref{Eq:WienerNormEst2Ess2} we get
\eqref{Eq:WienerNormEst2Ess}, and thereby
\eqref{Eq:WienerNormEst2}.
\end{proof}

\par

\begin{proof}[Proof of Proposition \ref{Prop:WienerRel1}]
Since the map $F\mapsto F\cdot \omega$ is homeomorphic between
the involved spaces and their corresponding non-weighted versions, we may assume
that $\omega _1=\omega _2=1$. Furthermore, by a linear change of variables,
we may assume that $E_1$ is the standard basis and $E_2=2\pi E_1$. Then
$\kappa (E_1)=Q_d$, $E_1'=E_2$ and $E_2'=E_1$.

\par

Let $F$ be measurable on $\rr {2d}$,
\begin{align*}
f_{1,\boldsymbol r}(\xi ,j) &= \nm {F(\, \cdot \,  ,\xi )}{L^{\boldsymbol r}(j+\kappa (E_1))},
\quad
g_1(\xi ) = \nm {f_{1,\boldsymbol r}(\xi ,\, \cdot \,  )}{\ell ^{{\boldsymbol p}}}
\intertext{and}
G_1(j,\iota ) &= \nm F{L^{\boldsymbol r ,\infty}_{(j,\iota )+\kappa (E_1\times E_2)}}.
\end{align*}
Then
$$
g_1 \le g\equiv
\sum _{\iota +\Lambda _{E_2}}\big ( \nm g{L^\infty (\iota +2\pi Q)}\big ) \cdot
\chi _{\iota +2\pi Q},
$$
and
\begin{equation*}
\nm F{\mathsf{W} ^{\boldsymbol r}_{1,E_1}(1,\ell ^{{\boldsymbol p}},L^{\boldsymbol q})}
=
\nm {g_1}{L^{\boldsymbol q}}
\le
\nm g{L^{\boldsymbol q}}
=
\nm {G_1}{\ell ^{{\boldsymbol p} ,\boldsymbol q}}
\asymp
\nm F{\mathsf{W} ^{\boldsymbol r ,\infty}_E (1,\ell ^{{\boldsymbol p} ,\boldsymbol q})}.
\end{equation*}
This implies that
$\mathsf{W} ^{\boldsymbol r ,\infty}_E (1,\ell ^{{\boldsymbol p} ,\boldsymbol q})
\hookrightarrow
\mathsf{W} ^{\boldsymbol r}_{1,E_1}(1,\ell ^{{\boldsymbol p}},L^{\boldsymbol q})$,
and the first inclusion in \eqref{Eq:WienerRel1} follows.

\par

In order to prove the second inclusion in
\eqref{Eq:WienerRel1}, we may assume
that $r_0<\infty$, since otherwise the result is trivial. Let
\begin{align*}
\quad
\psi (\xi ) &= \nm {f_{1,r_0} (\xi ,\, \cdot \,  )}{\ell ^{{\boldsymbol p}}}, 
\quad
a(\iota ) = \nm \psi {L^{\boldsymbol q}(\iota +\kappa (E_2))}
\intertext{and}
H_1(j,\iota ) &= \nm {f_{1,r_0} (\, \cdot \,  ,j)}{L^{r_0}(\iota 
+\kappa (E_2))}.
\end{align*}
Then
$$
\nm \psi{L^{\boldsymbol q} (\rr d)} 
=
\nm F{\mathsf{W} _{1,E_1}^{r_0}(1,\ell ^{{\boldsymbol p}} ,L^{\boldsymbol q})}
\quad \text{and}\quad
\nm {H_1}{\ell ^{{\boldsymbol p} ,\boldsymbol q}}
=
\nm F{\mathsf{W} _{E}^{r_0}(1,\ell ^{{\boldsymbol p} ,\boldsymbol q})}.
$$

\par

By Minkowski's inequality and the fact that
$\min ({\boldsymbol p} )\ge r_0$ we get
\begin{multline*}
\nm {H_1(\, \cdot \,  ,\iota )}{\ell ^{{\boldsymbol p}}}
=
\Nm {\left ( \int _{\iota +\kappa (E_2)} f_{1,r_0}
(\xi ,\, \cdot \,  )^{r_0}\, d\xi
\right )^{\frac 1{r_0}}}{\ell ^{{\boldsymbol p}}}
\\[1ex]
=
\left (\Nm {\left ( \int _{\iota +\kappa (E_2)}
f_{1,r_0} (\xi ,\, \cdot \,  )^{r_0}\, d\xi
\right )}{\ell ^{{\boldsymbol p} /{r_0}}}\right ) ^{\frac 1{r_0}}
\\[1ex]
\le
\left (\int _{\iota +\kappa (E_2)}
\nm {f_{1,r_0} (\xi ,\, \cdot \,  )^{r_0}}{\ell ^{{\boldsymbol p} /{r_0}}}
\, d\xi \right ) ^{\frac 1{r_0}}
= a(\iota ).
\end{multline*}
Hence $\nm {H_1}{\ell ^{{\boldsymbol p} ,\boldsymbol q}}
\le \nm a{\ell ^{\boldsymbol q}}$.
By Lemma \ref{Lemma:WienerNormEst2} it follows that
$\nm a{\ell ^{\boldsymbol q}}\le \nm \psi{L^{\boldsymbol q}}$, and the 
second inclusion
of \eqref{Eq:WienerRel1} follows by combining these relations.

\par

It remains to prove \eqref{Eq:WienerRel2}. Again we may assume that
$r_2<\infty$, since otherwise the result is trivial. Let
\begin{align*}
f_{2,q}(x,\iota ) &= \nm {F(x,\, \cdot \,  )}{L^{q}(\iota +\kappa (E_2))},
\quad
f_3(x) = \nm {F(x,\, \cdot \,  )}{L^{\boldsymbol q}(\rr d)},
\\[1ex]
H_{2,q_1,q_2}(\iota ,j) &= \nm {f_{2,q_1}(\, \cdot \,  ,\iota )}{L^{q_2} (j+\kappa (E_1))},
\quad \text{and}\quad H_{2,q}=H_{2,q,q}
\end{align*}
when $q,q_1,q_2\in (0,\infty ]$. Then the fact that $r_2\le \min (\boldsymbol q)$, Minkowski's inequality and
Lemma \ref{Lemma:WienerNormEst2} give
\begin{multline*}
\nm {H_{2,r_2}(\, \cdot \,  ,j)}{\ell ^{\boldsymbol q}}
\le 
\left (
\int _{j+\kappa (E_1)}
\nm {f_{2,r_2}(x,\, \cdot \,  )}{\ell ^{\boldsymbol q}}^{r_2}\, dx
\right )^{\frac 1{r_2}}
\\[1ex]
\le 
\left (
\int _{j+\kappa (E_1)}
\nm {F(x,\, \cdot \,  )}{L^{\boldsymbol q}}^{r_2}\, dx
\right )^{\frac 1{r_2}}.
\end{multline*}
By applying the $\ell ^{{\boldsymbol p}}$ norm on the latter inequality we get
$$
\nm F{\mathsf{W} ^{r_2}_{E'}(1,\ell ^{\boldsymbol q ,{\boldsymbol p}})}
\le
\nm F{\mathsf{W} ^{r_2}_{2,E_2}(1,\ell ^{{\boldsymbol p}},L^{\boldsymbol q})},
$$
and the second relation in \eqref{Eq:WienerRel2} follows.

\par

On the other hand,  we have
\begin{multline*}
\left (
\int _{j+\kappa (E_1)}
\nm {F(x,\, \cdot \,  )}{L^{\boldsymbol q}(\rr d)}^{r_2}
\right )^{\frac 1{r_2}}\, dx
\lesssim
\left (
\int _{j+\kappa (E_1)}
\nm {f_{2,\infty}(x,\, \cdot \,  )}{\ell ^{\boldsymbol q}}^{r_2}\, dx
\right )^{\frac 1{r_2}}
\\[1ex]
\le \nm {H_{2,\infty}(\, \cdot \,  ,j)}{\ell ^{\boldsymbol q}}
\end{multline*}
Again, by applying the $\ell ^{{\boldsymbol p}}$ norm with respect to the $j$ variable,
we get
$$
\nm F{\mathsf{W} ^{r_2}_{2,E_2}(1,\ell ^{{\boldsymbol p}},L^{\boldsymbol q})}
\le
\nm F{\mathsf{W} ^{\infty}_{E'}(1,\ell ^{\boldsymbol q ,{\boldsymbol p}})},
$$
and the first relation in \eqref{Eq:WienerRel2} follows.
\end{proof}

\par

\subsection{Wiener estimates on short-time Fourier
transforms, and modulation spaces}

\par

Essential parts of our analysis
are based on Lebesgue estimates of the \emph{semi-discrete
convolution} with respect to (the ordered) basis $E$ in $\rr d$, given by
\begin{equation}\label{EqDistSemContConv}
(a*_{[E]}f)(x) \sum _{j\in \Lambda _E}a(j)f(x - j),
\end{equation}
when $f\in \mathcal S _{1/2}'(\rr d)$ and $a \in \ell _0 (\Lambda _E)$.

\par

The next result is an extension of
\cite[Proposition 2.1]{Toft15} and
\cite[Lemma 2.6]{GaSa}, but a special case of
\cite[Theorem 2.1]{Toft22}.
The proof is therefore omitted. An other special case of
\cite[Theorem 2.1]{Toft22} will be used in Section
\ref{sec3} when discussing
characterizations of the modulation spaces via
norm estimates of
Zak transforms. The domain of integration is of the form
\begin{equation}\label{Eq:IDomain1}
I=\sets {x_1e_1+\cdots +x_de_d}{x_k\in J_k},
\quad
J_k =
\begin{cases}
[0,1], & e_k\in E_0
\\[1ex]
\mathbf R, & e_k\notin E_0
\end{cases}
\end{equation}

\par

\begin{prop}\label{Prop:SemiContConvEstNonPer}
Let $E$ be an ordered basis
of $\rr d$, $E_0\subseteq E$, $I$ be given by
\eqref{Eq:IDomain1},
$\omega ,v\in \mathscr P _E(\rr d)$ be
such that $\omega$ is $v$-moderate, and let
${\boldsymbol p} ,\boldsymbol r \in (0,\infty ]^{d}$ be such that
$$
r_k\le \min _{m\le k}(1,p_m).
$$
Also let  $f$ be measurable on $\rr d$ such that $|f|$ is $E_0$-periodic
and $f\in L^{{\boldsymbol p}}_{E,(\omega )}(I)$.
Then the map $a\mapsto a*_{[E]}f$ from $\ell _0(\Lambda _E)$ to
$L^{{\boldsymbol p}}_{E,(\omega )}(I)$ extends uniquely to a
linear and continuous map from $\ell ^{\boldsymbol r}_{E ,(v)}(\Lambda _E)$ to
$L^{{\boldsymbol p}}_{E,(\omega )}(I)$, and
\begin{equation}\label{convest1}
\nm {a*_{[E]}f}{L^{{\boldsymbol p} }_{E,(\omega )}(I)}\le
C\nm {a}{\ell ^{\boldsymbol r}_{E ,(v)}(\Lambda _E)}
\nm {f}{L^{{\boldsymbol p} }_{E,(\omega )}(I)},
\end{equation}
for some constant $C>0$ which is independent of
$a\in \ell ^{\boldsymbol r}_{E ,(v)}(\Lambda _E)$ and
measurable $f$ on $\rr d$ such that $|f|$ is $E_0$-periodic.
\end{prop}

\par

We have now the following result, which agrees with
Proposition \ref{Prop:WienerEquiv} when
$\boldsymbol r =( \infty ,\dots ,\infty )$.

\par

\renewcommand{}{Proposition \ref{Prop:WienerEquiv}$'$}

\par

\begin{tom}
Let $E$ be a phase split basis for $\rr {2d}$, ${\boldsymbol p} ,\boldsymbol r \in (0,\infty ]^{2d}$,
$r\in (0,\min (1,{\boldsymbol p} )]$, $\omega ,v\in \mathscr P _E(\rr {2d})$
be such that $\omega$ is $v$-moderate, $\rho$ and $\Theta _\rho v$ $\rho$ be
as in \eqref{Eq:SubMultMod} with strict inequality when $r<1$, and let $\phi _1,\phi _2\in
M^1_{(\Theta _\rho v)} (\rr d)\setminus 0$.
Then
$$
\nm f{M^{{\boldsymbol p}}_{E,(\omega )}}
\asymp
\nm {V_{\phi _1} f}{L^{\boldsymbol p} _{E,(\omega )}}
\asymp 
\nm {V_{\phi _2} f}{\mathsf{W} ^{\boldsymbol r} _{E}(\omega ,\ell ^{\boldsymbol p} _{E})},\quad f\in
\mathcal S '_{1/2}(\rr d).
$$
In particular, if $f\in \mathcal S _{1/2}'(\rr d)$, then
$$
f\in M^{{\boldsymbol p}}_{E,(\omega )}(\rr {2d}))
\quad \Leftrightarrow \quad
V_{\phi _1} f \in L^{\boldsymbol p} _{E,(\omega )}(\rr {2d})
\quad \Leftrightarrow \quad
V_{\phi _2} f \in \mathsf{W} ^{\boldsymbol r} _{E}(\omega ,\ell ^{\boldsymbol p} _{E}
(\Lambda _E)).
$$
\end{tom}

\par

We need the following lemma for the proof.

\par

\begin{lemma}\label{SubharmonicEst}
Let $p\in (0,\infty ]$, $r>0$, $(x_0,\xi _0)\in \rr {2d}$ be fixed,
and let $\phi \in \mathcal S _{1/2}(\rr d)$ be a Gaussian. Then
$$
|V_\phi f(x_0,\xi _0)| \le C\nm {V_\phi f}{L^p(B_r(x_0,\xi _0))},\quad
f\in \mathcal S _{1/2}'(\rr d),
$$
where the constant $C$ is independent of $(x_0,\xi _0)$ and $f$.
\end{lemma}

\par

When proving Lemma \ref{SubharmonicEst} we may first reduce ourself
to the case that the Gaussian $\phi$ should be centered at origin, by
straight-forward arguments involving pullbacks with translations.
The result then follows by using the same arguments as in
\cite[Lemma 2.3]{GaSa} and its proof, based on the fact that
$$
z\mapsto F_w(z) \equiv e^{c_1|z|^2+c_2(z,w)+c_3|w|^3}V_\phi f(x,\xi ),
\quad z=x+i\xi
$$
is an entire function for some choice of the constant
$c_1$ (depending on $\phi$).

\par

\begin{proof}[Proof of Proposition
\ref{Prop:WienerEquiv}${}^{\, \prime}$]
Let
$F=V_{\phi} f$, $F_0=V_{\phi _0} f$, $\kappa (E)$
be the (closed) parallelepiped which is spanned by
$E=\{ e_1,\dots ,e_{2d}\}$, and let
$$
\kappa _M(E) = \sets {x_1e_1+\cdots +x_{2d}e_{2d}}{|x_k|\le 2,\ k=1,\dots ,2d}.
$$
Also choose $r_0>0$ small enough such that
$$
\kappa (E)+B_{r_0}(0,0) \subseteq \kappa _M(E)
$$
The result holds when $\boldsymbol r =(\infty ,\dots ,\infty )$, in view of Proposition
\ref{Prop:WienerEquiv}. By H{\"o}lder's inequality we also have
\begin{equation}\label{Eq:WienerSTFT1}
\nm {V_{\phi}f}{\mathsf{W} ^{\boldsymbol r} _{E}(\omega ,\ell ^{\boldsymbol p} _{\sigma})}
\lesssim
\nm {V_{\phi}f}{\mathsf{W} ^\infty _{E}(\omega ,\ell ^{\boldsymbol p} _{\sigma})}.
\end{equation}
We need to prove the reversed inequality
\begin{equation}\label{Eq:WienerSTFT2}
\nm {V_{\phi}f}{\mathsf{W} ^\infty _{E}(\omega ,\ell ^{\boldsymbol p} _{\sigma})}
\lesssim
\nm {V_{\phi}f}{\mathsf{W} ^{\boldsymbol r} _{E}(\omega ,\ell ^{\boldsymbol p} _{\sigma})},
\end{equation}
and it suffices to prove this for $\boldsymbol r = (r,\dots ,r)$ for some $r\in (0,1]$
in view of H{\"o}lder's inquality.

\par

First we consider the case when $\phi =\phi _0$. If $r>0$ is small enough
and $j\in \Lambda _E$, then
Lemma \ref{SubharmonicEst} gives
for some $(x_j,\xi _j)\in j+\kappa (E)$ that
\begin{equation*}
\nm {V_{\phi _0}f}{L^\infty (j+\kappa (E))}
=
|{V_{\phi _0}f}(x_j,\xi _j)|
\lesssim
\nm {V_{\phi _0}f}{L^r (B_r(x_j,\xi _j))}
\le
\nm {V_{\phi _0}f}{L^r (j+\kappa _M(E))}
\end{equation*}
Hence,
\begin{multline*}
\nm {V_{\phi _0}f}{\mathsf{W} ^\infty _{E}(\omega ,\ell ^{\boldsymbol p} _{E})}
=
\nm { \{ \nm {V_{\phi _0}f}{L^\infty (j+\kappa (E)}\omega (j)\}
_{j\in \Lambda _E}}{\ell ^{\boldsymbol p} _{E}(\Lambda _E)}
\\[1ex]
\lesssim
\nm { \{ \nm {V_{\phi _0}f}{L^r (j+\kappa _M(E)}\omega (j)\}
_{j\in \Lambda _E}}{\ell ^{\boldsymbol p} _{E}(\Lambda _E)}
\\[1ex]
\asymp
\nm { \{ \nm {V_{\phi _0}f}{L^r (j+\kappa (E)}\omega (j)\}
_{j\in \Lambda _E}}{\ell ^{\boldsymbol p} _{E}(\Lambda _E)}
=
\nm {V_{\phi _0}f}{\mathsf{W} ^r _{E}(\omega ,\ell ^{\boldsymbol p} _{E})},
\end{multline*}
and \eqref{Eq:WienerSTFT2} holds for $\phi = \phi _0$.

\par

Next suppose that $\phi$ is arbitrary and let $n\ge 1$ be
a large enough integer such that if
$$
E_n = \frac 1n \cdot E \equiv \left \{  \frac {e_1}n,\dots ,\frac {e_{2d}}n \right \}
\quad \text{and}\quad
\Lambda
= \frac 1n \Lambda _E = \Lambda _{E_n},
$$
then
$$
\{ \phi (\, \cdot \,  -k)e^{i\scal \, \cdot \,  \kappa}\} _{(k,\kappa )\in \Lambda}
$$
is a frame. Since $\phi \in M^1_{(\Theta _\rho v)}$, it follows
that its canonical dual $\psi$ also belongs to
$M^1_{(\Theta _\rho v)}$ (cf. \cite[Theorem S]{Gc1}).
Consequently, any $f$ possess the expansions
\begin{align}
f
&=
\sum _{(k,\kappa )\in \Lambda} V_\phi f(k,\kappa )
\psi (\, \cdot \,  -k)e^{i\scal \, \cdot \,  \kappa}
\notag
\\[1ex]
&=
\sum _{(k,\kappa )\in \Lambda} V_\psi f(k,\kappa )
\phi (\, \cdot \,  -k)e^{i\scal \, \cdot \,  \kappa}
\label{Eq:GaborExp}
\end{align}
with suitable interpretation of convergences.

\par

Let
$$
F_0=|V_{\phi _0}f|\cdot \omega ,
\quad
F=|V_{\phi }f|\cdot \omega ,
\quad \text{and}\quad
a(\boldsymbol k ) = |V_\psi \phi _0 (-\boldsymbol k )| .
$$
As in the proofs of \cite[Theorem 3.1]{GaSa} and
\cite[Proposition 3.1]{Toft15} we use the fact that
\begin{equation}\label{Eq:STFTWindTrans}
|V_{\phi _0}f| \le (2\pi )^{-\frac d2}
a *_{[E_n ]}|V_\phi f| ,
\end{equation}
which follows from
\begin{multline*}
|V_{\phi _0}f(x,\xi )| = (2\pi )^{-\frac d2}|(f,e^{i\scal \, \cdot \,  {\xi}}
\phi _0(\, \cdot \,  -x))|
\\[1ex]
\le (2\pi )^{-\frac d2}\sum _{(k,\kappa )\in \Lambda}
|(V_\psi \phi _0)(k,\kappa )|  |(f,e^{i\scal \, \cdot \,  {\xi +\kappa}}
\phi (\, \cdot \,  -x-k))|
\\[1ex]
=  (2\pi )^{-\frac d2}\sum _{(k,\kappa )\in \Lambda}
|(V_\psi \phi _0)(k,\kappa )|  |V_{\phi }f(x+k,\xi +\kappa )|
\\[1ex]
=
(a *_{[E_n ]} |V_{\phi }f|)(x,\xi  ).
\end{multline*}
Here we have used \eqref{Eq:GaborExp} with $\phi _0$ in place of $f$,
in the inequality. By using that
$$
\omega (x,\xi )\lesssim v(k,\kappa )\omega (x+k,\xi +\kappa ),
$$
\eqref{Eq:STFTWindTrans} gives
\begin{equation}\label{Eq:F0FConvEst}
F_0 \lesssim (a\cdot v)*_{[E_n]}F.
\end{equation}
If we set
$$
b_0(j) = \int _{j+\kappa (E)} |F_0(X)|^r\, dX
\quad \text{and}\quad
b(j) = \int _{j+\kappa (E)} |F(X)|^r\, dX,
\qquad j\in \Lambda ,
$$
integrate \eqref{Eq:F0FConvEst} and use the fact
that $r\le 1$, we get for $j\in \Lambda$ that
\begin{multline*}
b_0(j)
\lesssim
\int _{j+\kappa (E)}
\left (
\sum _{k\in \Lambda}a(k)v(k)|F(X-k)|
\right )^r
\, dX
\\[1ex]
\lesssim
\sum _{k\in \Lambda}(a(k)v(k))^r \int _{j+\kappa (E)}
|F(X-k)|^r\, dX 
= ((a\cdot v)^r * b ))(j),
\end{multline*}
where $*$ is the discrete convolution with respect to the
lattice $\Lambda$.

\par

Let $\boldsymbol q = {\boldsymbol p} /r$.
Then $\min (\boldsymbol q )\ge 1$,
and Young's inequality applied on the last inequality
gives
\begin{multline}\label{Eq:F0FComputations}
\nm {F_0}{\mathsf{W} ^r_E(1,\ell ^{{\boldsymbol p}}_E )}
=
\nm {b_0^{\frac 1r}}{\ell ^{{\boldsymbol p}} _E (\Lambda _E)}
\lesssim
\nm {(a\cdot v)^r*b}{\ell ^{\boldsymbol q} _E (\Lambda _E)}
^{\frac 1r}
\\[1ex]
\le
\nm {(a\cdot v)^r*b}{\ell ^{\boldsymbol q} _E (\Lambda )}^{\frac 1r}
\le
\left (
\nm {(a\cdot v)^r}{\ell ^1(\Lambda )}
\nm b{\ell ^{\boldsymbol q} _E (\Lambda )}
\right )^{\frac 1r}
\\[1ex]
\asymp
\nm {a}{\ell ^r_{(v)}(\Lambda )}
\nm b{\ell ^{\boldsymbol q} _E (\Lambda )}
^{\frac 1r}
\le
\nm {a}{\ell ^1_{(\Theta _\rho v)}(\Lambda )}
\nm {b^{\frac 1r}}{\ell ^{{\boldsymbol p}} _E (\Lambda )}
\\[1ex]
\lesssim
\nm {\phi}{M^1_{(\Theta _\rho v)}}
\nm {b^{\frac 1r}}{\ell ^{{\boldsymbol p}} _E (\Lambda )}.
\end{multline}
In the last steps we have used H{\"o}lder's inequality and
$$
\nm {V_{\phi _0}\phi}{L^1_{(\Theta _\rho v)}(\rr {2d})}
\asymp
\nm { \{ \nm {V_{\phi _0}\phi}{L^\infty (j+\kappa (E)}
(\Theta _\rho v)(j)\}
_{j\in \Lambda _E}}{\ell ^1(\Lambda _E)}
\asymp \nm \phi {M^1_{(\Theta _\rho v)}}.
$$

\par

We have
\begin{equation*}
\nm {b^{\frac 1r}}{\ell ^{{\boldsymbol p}} _E (\Lambda )}
\\[1ex]
=
\nm { \{ \nm F{L^r(j+\kappa (E))} \} _{j\in \Lambda }}
{\ell ^{{\boldsymbol p}}_E (\Lambda )},
\end{equation*}
$\bigcup _{j\in \Lambda _E}(j+\kappa (E))
=\rr d$, $\Lambda _E$ and $\Lambda$ are lattices
such that $\Lambda$ contains $\Lambda _E$, and
$\Lambda$ is $n$ times as dense as $\Lambda _E$. From
these facts it follows by straight-forward computations
that
\begin{multline*}
\nm { \{ \nm F{L^r(j+\kappa (E))} \} _{j\in \Lambda }}
{\ell ^{{\boldsymbol p}}_E (\Lambda )}
\asymp
\nm { \{ \nm F{L^r(j+\kappa (E))} \} _{j\in \Lambda _E}}
{\ell ^{{\boldsymbol p}}_E (\Lambda _E)}
\\[1ex]
\asymp
\nm { \{ \nm {V_\phi f}{L^r(j+\kappa (E))}\omega (j)
\} _{j\in \Lambda _E}}
{\ell ^{{\boldsymbol p}}_E (\Lambda _E)}
= \nm F{\mathsf{W} ^r_E(\omega ,\ell ^{{\boldsymbol p}}_E )}.
\end{multline*}
Here the second relation follows from the fact that
$\omega (x)\asymp \omega (j)$ when $j\in \Lambda _E$ and
$x\in j+\kappa (E)$, which follows from \eqref{moderate}.
By combining these relations with \eqref{Eq:F0FComputations}
we get
$$
\nm {F_0}{\mathsf{W} ^r_E(1,\ell ^{{\boldsymbol p}}_E )}
\lesssim
\nm {F}{\mathsf{W} ^r_E(1,\ell ^{{\boldsymbol p}}_E )}.
$$
Hence, Proposition \ref{Prop:WienerEquiv} and the fact that we have
already proved \eqref{Eq:WienerSTFT2} when $\phi$ equals $\phi _0$
gives
\begin{multline*}
\nm {V_{\phi}f}{\mathsf{W} ^\infty _{E}(\omega ,\ell ^{\boldsymbol p} _{\sigma})}
\asymp
\nm {V_{\phi _0}f}{\mathsf{W} ^\infty _{E}(\omega ,\ell ^{\boldsymbol p} _{\sigma})}
\asymp
\nm {F_0}{\mathsf{W} ^\infty _{E}(1,\ell ^{\boldsymbol p} _{\sigma})}
\lesssim
\nm {F_0}{\mathsf{W} ^{\boldsymbol r} _{E}(1,\ell ^{\boldsymbol p} _{\sigma})}
\\[1ex]
\lesssim
\nm {F}{\mathsf{W} ^{\boldsymbol r} _{E}(1,\ell ^{\boldsymbol p} _{\sigma})}
\asymp
\nm {V_{\phi}f}{\mathsf{W} ^{\boldsymbol r} _{E}(\omega ,\ell ^{\boldsymbol p} _{\sigma})}.
\qedhere
\end{multline*}
\end{proof}

\par

By combining Proposition \ref{Prop:WienerEquiv}$'$ with
Proposition \ref{Prop:WienerRel1}
and Remark \ref{Rem:IncreasingWienerSpaces}
we get the following.

\par

\begin{prop}\label{Prop:SpecCaseWienerRel1}
Let $E_0$ be a basis for $\rr d$, $E_0'$ be its dual basis,
$E=E_0\times E_0'$, $\boldsymbol q ,\boldsymbol r \in (0,\infty ]^{d}$,
$\omega _0,v_0\in \mathscr P _E(\rr {d})$
be such that $\omega _0$ is $v_0$-moderate,
$\omega (x,\xi )=\omega _0(\xi )$,
$v(x,\xi )=v_0(\xi )$, $\Theta _\rho v$ be
as in \eqref{Eq:SubMultMod} with strict inequality when $r<1$, and let $\phi \in
M^1_{(\Theta _\rho v)} (\rr d)\setminus 0$.
Then
\begin{alignat*}{2}
M^{\infty ,\boldsymbol q}_{E,(\omega )}(\rr d)
&=
\EuScript M ^{\boldsymbol r}_{E_0}(\omega _0,L^{\boldsymbol q} _{E_0'}(\rr d)),&
\quad
W^{\infty ,\boldsymbol q}_{E,(\omega )}(\rr d)
&=
W^{\boldsymbol r}_{E_0}(\omega _0,L^{\boldsymbol q} _{E_0'}(\rr d)),
\intertext{and}
\nm f{M^{\infty ,\boldsymbol q}_{E,(\omega )}}
&\asymp
\nm {V_\phi f}
{\mathsf{W} ^{\boldsymbol r}_{1,E_0}(\omega ,\ell ^\infty _E,L^{\boldsymbol q}_{E_0'})},&
\quad
\nm f{W^{\infty ,\boldsymbol q}_{E,(\omega )}}
&\asymp
\nm {V_\phi f}
{\mathsf{W} ^{\boldsymbol r}_{2,E_0}(\omega ,\ell ^\infty _E,L^{\boldsymbol q}_{E_0'})}.
\end{alignat*}
\end{prop}

\par

\subsection{Periodic elements in modulation spaces}

\par

By a straight-forward combination of
Propositions \ref{Prop:PerMod}  and
\ref{Prop:SpecCaseWienerRel1}
we get the following. The details are left for the reader.

\par

\begin{prop}\label{Prop:PeriodicMod}
Let $E_0$ be a basis for $\rr d$, $E_0'$ be its dual basis,
$E=E_0\times E_0'$, $\boldsymbol q ,\boldsymbol r \in (0,\infty ]^{d}$,
$\omega _0,v_0\in \mathscr P _E(\rr {d})$
be such that $\omega _0$ is $v_0$-moderate,
$\omega (x,\xi )=\omega _0(\xi )$,
$v(x,\xi )=v_0(\xi )$, $\Theta _\rho v$ be
as in \eqref{Eq:SubMultMod} with strict inequality
when $r<1$, and let $\phi \in
M^1_{(\Theta _\rho v)} (\rr d)\setminus 0$.
Then
\begin{multline}\label{Eq:NormsEquivPerElem}
\nm f{M^{\infty ,\boldsymbol q}_{E,(\omega )}}
\asymp 
\nm f{W^{\infty ,\boldsymbol q}_{E,(\omega )}}
\asymp
\nm f{\EuScript M ^{\boldsymbol r}_{E_0}(\omega ,L^{\boldsymbol q} _{E_0'})}
\\[1ex]
\asymp
\nm f{\mathsf{W} ^{\boldsymbol r}_{E_0}(\omega ,L^{\boldsymbol q} _{E_0'})}
\asymp
\nm f{\mathcal L ^E(\omega _0,\ell ^{\boldsymbol q}_{E_0'}(\Lambda _{E_0}'))},
\qquad f\in (\mathcal E _0^{E})'(\rr d).
\end{multline}
\end{prop}

\par

As an immediate consequence of the previous result we get the
following extension of Proposition \ref{Prop:PerMod}. The details
are left for the reader.

\par

\renewcommand{}{Proposition \ref{Prop:PerMod}$'$}

\par

\begin{tom}
Let $E$ be a basis of $\rr d$, $\boldsymbol r \in (0,\infty ]^d$, $r\in(0,1]$,
$\mathscr B \subseteq L^{r}_{loc}(\rr d)$ be an $E'$-split
Lebesgue space, $\ell _{\mathscr B ,E}(\Lambda _E)$ its discrete version,
and let $\omega\in \mathscr P _E(\rr d)$. Then
\begin{equation*}
\mathcal L ^{E}(\omega ,\ell _{\mathscr B ,E})
=
\EuScript M^{\boldsymbol r}_{E}(\omega ,\mathscr B )\bigcap (\mathcal E _0^E)'(\rr d)
=
\EuScript W^{\boldsymbol r}_{E}(\omega ,\mathscr B )\bigcap (\mathcal E _0^E)'(\rr d)
\end{equation*}
\end{tom}

\par

\begin{rem}
Let
$$
E_0= \{ e_1,\dots ,e_d\} ,\quad 
E_0'= \{ \varepsilon _1,\dots ,\varepsilon _d\} ,\quad 
\boldsymbol q = (q_1,\dots ,q_d),\quad
\boldsymbol r = (r_1,\dots ,r_d),
$$
$\omega$, $v$  and $\phi$ be the same as in Proposition
\ref{Prop:PeriodicMod},
and let $r_0\le \min (\boldsymbol r )$ and $f\in f\in (\mathcal E _0^{E})'(\rr d)$
with Fourier series expansion \eqref{Eq:Expan2}.
Then \eqref{Eq:ModPerNormEquiv1}--\eqref{Eq:AltFormPerNorms} and
\eqref{Eq:NormsEquivPerElem} imply that 
\begin{equation}\label{Eq:ModPerNormEquiv3}
\nm {V_\phi f \cdot \omega}{L^{\boldsymbol r ,\boldsymbol q}_E(\kappa (E_0)\times \rr d)}
\asymp
\nm {V_\phi f \cdot \omega}{L^{\boldsymbol q ,\boldsymbol r}_{E'}(\rr d \times \kappa (E_0))}
\asymp
\nm {c(f,\, \cdot \,  )}{\ell ^{\boldsymbol q}_{E_0',(\omega _0)}}.
\end{equation}

\par

Let $\nmm \, \cdot \, $ be the quasi-norm on the left-hand side of
\eqref{Eq:ModPerNormEquiv3}, after the orders of the
involved $L^{q_k}_{e_k'}(\mathbf R)$ and $L^{r_k}_{e_k}(\kappa (e_k))$
quasi-norms have been permuted in such way that the internal order of the hitting
$L^{q_k}_{e_k'}(\mathbf R)$ quasi-norms remains the same. Then
\begin{equation}\label{Eq:PermutNormEst}
\nm F{L^{r_0,\boldsymbol q}_E(\kappa (E_0)\times \rr d)}
\lesssim
\nmm F
\lesssim
\nm F{L^{\infty,\boldsymbol q}_E(\kappa (E_0)\times \rr d)},
\end{equation}
by repeated application of H{\"o}lder's inequality. A combination of 
\eqref{Eq:ModPerNormEquiv3} and \eqref{Eq:PermutNormEst} give
\begin{equation}\label{Eq:ModPerNormEquiv4}
\nmm {V_\phi f \cdot \omega}
\asymp
\nm {c(f,\, \cdot \,  )}{\ell ^{\boldsymbol q}_{E_0',(\omega _0)}}.
\end{equation}

\par

In particular, if $e_j$ are the same as in Remark
\ref{Rem:CommentPhasSplit}, $E_*$ is the ordered
basis $\{ e_1,e_{d+1}, \dots ,e_d,e_{2d}\}$ of $\rr {2d}$,
$$
\Omega
=
\sets {y_1e_1+\cdots +y_{2d}e_{2d}}
{0\le y_j\le 1\ \text{and}\ y_{d+j}\in \mathbf R,\, j=1,\dots ,d}
$$
and $\boldsymbol q _0=(q_1,q_1,q_2,q_2,\dots ,q_d,q_d)\in (0,\infty ]^{2d}$, then
\begin{equation}\label{Eq:ModPerNormEquiv5}
\nm {V_\phi f \cdot \omega}{L^{\boldsymbol q _0}_{E_*}(\Omega )}
\asymp
\nm {c(f,\, \cdot \,  )}{\ell ^{\boldsymbol q}_{E_0',(\omega _0)}}.
\end{equation}
\end{rem}

\par

\begin{rem}
With the same notation as in the previous remark, we note that
if $E_0'$ is the standar basis of $\rr d$, $X_j=(x_j,\xi _j)$, $j=1,\dots ,d$,
$I=\mathbf R\times [0,2\pi]$ and $\max (\boldsymbol q )<\infty$,
then \eqref{Eq:ModPerNormEquiv5} is the same as
\begin{multline}\tag*{(\ref{Eq:ModPerNormEquiv5})$'$}
\left (
\int _{I}
\left (
\cdots
\left (
\int _{I}
|V_\phi f(x,\xi )\omega _0(\xi )|^{q_1}\, dX_1
\right )^{\frac {q_2}{q_1}}
\cdots
\right )^{\frac {p_d}{p_{d-1}}}
\, dX_d\right )^{\frac 1{p_d}}
\\[1ex]
\asymp
\left (
\sum _{\alpha _d\in \mathbf Z}
\left (
\cdots
\left (
\sum _{\alpha _1\in \mathbf Z}
|c(f,\alpha )\omega _0(\alpha )|^{q_1}
\right )^{\frac {q_2}{q_1}}
\cdots
\right )^{\frac {p_d}{p_{d-1}}}
\right )^{\frac 1{p_d}}
\end{multline}
\end{rem}

\par

\section{Zak transform on Gelfand-Shilov spaces,
Lebesgue spaces and modulation spaces}
\label{sec3}

\par

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
\cite{Ti,TiHe}.

\par

\subsection{The Zak transform on test function spaces
and their distribution spaces}

\par

For the classical spaces $\mathscr S (\rr d)$
and its distribution space $\mathscr S '(\rr d)$ we
have the following

\par

\begin{thm}\label{Thm:ZakClassical}
Let $E$ be an ordered basis of $\rr d$. Then the
following is true:
\begin{enumerate}
\item The operator $Z_E$ is a homeomorphism from
$\mathscr S (\rr d)$
to the set of quasi-periodic elements of order
$E$ in $C^\infty (\rr {2d})$;

\vspace{0.1cm}

\item The operator $Z_E$ from $\mathscr S (\rr d)$
to $C^\infty (\rr {2d})$ is uniquely to a
homeomorphism from $\mathscr S '(\rr d)$ to the
set of quasi-periodic elements of order $E$
in $\mathcal S '(\rr {2d})$.
\end{enumerate}
\end{thm}

\par

The assertion (1) in Theorem \ref{Thm:ZakClassical}
is essentially the same as \cite[Theorem 8.2.5]{Gc2},
and (2) in the same theorem follows by similar arguments
as in the proof of Theorem \ref{Thm:ZakDist} below.
The verifications of Theorem \ref{Thm:ZakClassical}
are therefore left for the reader.

\par

The analogous result of the (1) previous theorem for 
Gelfand-Shilov functions is the following.

\par

\begin{thm}\label{Thm:ZakTestFunctions}
Let $s,\sigma >0$ and $E$ be an ordered basis.
Then the operator $Z_E$ from $\mathscr S (\rr d)$ to
$C^\infty (\rr {2d})$
restricts to a homeomorphism from $\mathcal S _s^\sigma (\rr d)$
to the set of quasi-periodic elements of order $\rho $ in
$\mathcal E _{\sigma ,s}(\rr {2d})$.

\par

The same holds true with $\Sigma _s^\sigma (\rr d)$
and $\Sigma _{s,\sigma}^{\sigma ,s}(\rr {2d})$
in place of $\mathcal S _s^\sigma (\rr d)$
and $\mathcal S _{s,\sigma}^{\sigma ,s}(\rr {2d})$
at each occurrence.
\end{thm}

\par

Before the proof of the previous result we
present the analogous result for
corresponding distribution spaces.





\par

\begin{thm}\label{Thm:ZakDist}
Let $s,\sigma >0$ and
$E$ be an ordered basis of $\rr d$. Then the
operator $Z_E$ from $\mathscr S (\rr d)$ to
$C^\infty (\rr {2d})$
extends uniquely to a homeomorphism from $(\mathcal S
_s^\sigma)'(\rr d)$
to the set of quasi-periodic elements of order $E$
in $(\mathcal S _{s,\sigma}^{\sigma ,s})'(\rr {2d})$.

\par

The same holds true with $(\Sigma _s^\sigma)'(\rr d)$
and $(\Sigma _{s,\sigma}^{\sigma ,s})'(\rr {2d})$
in place of $(\mathcal S _s^\sigma)'(\rr d)$
and $(\mathcal S _{s,\sigma}^{\sigma ,s})'(\rr {2d})$
at each occurrence.
\end{thm}








\par

For the proof of Theorem \ref{Thm:ZakDist}
we need the following lemma on tensor product
of Gelfand-Shilov distributions.

\par

\begin{lemma}\label{Lemma:GSTensorProducts}
Let $s_j,\sigma _j>0$ and $f_j\in (\mathcal S _{s_j}^
{\sigma _j})'(\rr {d_j})$, $j=1,2$. Then there
is a unique $f\in
(\mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})$
such that
$$
\scal f{\varphi _1\otimes \varphi _2}
=\scal {f_1}{\varphi _1}\scal {f_2}{\varphi _2}
,\qquad \varphi _j\in \mathcal S _{s_j}^{\sigma _j}(\rr {d_j}),\ j=1,2.
$$
Moreover, if $\varphi \in \mathcal S _{s_1,s_2}
^{\sigma _1,\sigma _2}(\rr {d_1+d_2})$,
$$
\varphi _1 (x_1)= \scal {f_2}{\varphi (x_1,\, \cdot \,  )}
\quad \text{and}\quad
\varphi _2 (x_2)= \scal {f_1}{\varphi (\, \cdot \,  ,x_2)},
$$
then
$$
\scal f{\varphi} = \scal {f_1}{\varphi _1} = \scal {f_2}{\varphi _2}.
$$

\par

The same holds true with $\Sigma _{s_j}^{\sigma _j}$,
$\Sigma _{s_1,s_2}
^{\sigma _1,\sigma _2}$, $(\Sigma _{s_j}^{\sigma _j})'$ and
$(\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2})'$ in place of
$\mathcal S _{s_j}^{\sigma _j}$, $\mathcal S _{s_1,s_2}^{\sigma 
_1,\sigma _2}$,
$(\mathcal S _{s_j}^{\sigma _j})'$ and $(\mathcal S 
_{s_1,s_2}^{\sigma _1,\sigma _2})'$,
respectively, $j=1,2$.
\end{lemma}

Lemma \ref{Lemma:GSTensorProducts}
is essentially a restatement of Theorem 2.4
in \cite{Toft24}. The proof is therefore omitted.

\par

\begin{rem}\label{Rem:TensorUniqueness}
We notice that the uniqueness assertions in
Lemma \ref{Lemma:GSTensorProducts} is an
immediate consequence of \cite[Lemma 2.3]{Toft24}
which asserts that if
$f\in
(\mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})$
($f\in
(\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})$)
satisfies
$$
\scal f{\varphi _1\otimes \varphi _2}=0
$$
for every $\varphi _j\in \mathcal S _{s_j}^{\sigma _j}(\rr {d_j})$
($\varphi _j\in \Sigma _{s_j}^{\sigma _j}(\rr {d_j})$),
then $f=0$ (as an element in
$(\mathcal S _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})$
($(\Sigma _{s_1,s_2}^{\sigma _1,\sigma _2})'(\rr {d_1+d_2})$)).
\end{rem}

\par

\begin{proof}[Proof of Theorem \ref{Thm:ZakDist}]
Let $T_E$ be the same as in \eqref{Eq:ZakTransfers}.
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
\eqref{Eq:ZakTransfers}
that it suffices to prove the result when $E$
is the standard basis.

\par

We begin to prove (2). Let $\Phi \in \mathcal S 
_{s,\sigma}^{\sigma ,s}(\rr {2d})$. Then
\begin{gather*}
\mathscr F _2^{-1}\Phi \in \mathcal S _s^\sigma (\rr {2d}),\quad 
\mathscr F _1 (\mathscr F _2^{-1}\Phi ) \in \mathcal S _{\sigma ,s}^{s,\sigma}(\rr {2d}),
\\[1ex]
|\mathscr F _2^{-1}\Phi (x,y)|\lesssim e^{-r_0(|x|^{\frac 1s}+|y|^{\frac 1s})}
\quad \text{and}\quad
|\mathscr F _1\Phi (\xi ,\eta)|\lesssim e^{-r_0(|\xi |^{\frac 1\sigma }+|\eta |^{\frac 1\sigma })}
\end{gather*}

\par

If $f\in \mathscr S (\rr d)$, then
\begin{multline}\label{Eq:ZakDistFormula}
\scal {Z_1 f}\Phi = \iint _{\rr {2d}} \left (\sum _{j\in \zz d}f(x-j)e^{i\scal j\xi}
\right )\Phi (x,\xi )\, dxd\xi
\\[1ex]
=
\sum _{j\in \zz d} \left (
\int _{\rr d} f(x) (\mathscr F _2^{-1}\Phi )(x+j, j) 
\, dx
\right )
\\[1ex]
=
\sum _{j\in \zz d} (\check f *\Phi _2(\, \cdot \,  ,j))(j),
\end{multline}
where $\Phi _2= \mathscr F _2^{-1}\Phi $ and $\check f(x)=f(-x)$.

\par

Assume instead that $f\in (\mathcal S _s^\sigma)'(\rr d)$ is 
arbitrary. We claim that the series
on the right-hand side of \eqref{Eq:ZakDistFormula} converges 
absolutely for every $\Phi$ as above.

\par

In fact, since $f\in (\mathcal S _s^\sigma)'(\rr d)$, we have
$$
|\scal f\phi |
\lesssim
\nm {e^{r|\, \cdot \,  |^{\frac 1t}} \phi}{L^\infty}
+
\nm {e^{r|\, \cdot \,  |^{\frac 1\sigma }} \widehat \phi}{L^\infty}
$$
for every $r>0$, giving that for some $r_0>0$ and $c\ge 1$
we have
\begin{multline*}
| (\check f *\Phi _2(\, \cdot \,  ,y))(x) |
\lesssim
\nm {e^{r|\, \cdot \,  |^{\frac 1s}} \Phi _2(x-\, \cdot \,  ,y) }{L^\infty}
+
\nm {e^{r|\, \cdot \,  |^{\frac 1\sigma }}
\widehat \Phi (\, \cdot \,  ,y ) }{L^\infty}
\\[1ex]
\lesssim e^{rc|x|^{\frac 1s}-2r_0|y|^{\frac 1s}/c}.
\end{multline*}
Hence, if $r$ is chosen small enough and $x=y=j$, then
\begin{equation}\label{Eq:ConvGSEst}
| (\check f *\Phi _2(\, \cdot \,  ,2\pi \rho j ))(\rho j) | \lesssim e^{-r_0| j |^{\frac 1s}/c},
\end{equation}
when $(\mathcal S _s^\sigma)'(\rr d)$. The absolutely
convergence of the series
of the right-hand side of \eqref{Eq:ZakDistFormula}
now follows from \eqref{Eq:ConvGSEst}.

\par

If $f\in (\mathcal S _s^\sigma)'(\rr d)$, then $Z_1 f$
is defined as the element in
$(\mathcal S _{s,\sigma}^{\sigma ,s})'(\rr {2d})$, given
by the right-hand side of
\eqref{Eq:ZakDistFormula}. The previous estimates
show that this
definition makes sense, and that the map
$f\mapsto Z_1 f$ is continuous
from from $(\mathcal S _s^\sigma)'(\rr d)$
to the set of all quasi-periodic elements of order $1$
in $(\mathcal S _{s,\sigma}^{\sigma ,s})'(\rr {2d})$.
By approximating elements in $(\mathcal S _s^\sigma )'(\rr d)$
by sequences of elements in $\mathscr S (\rr d)$, it also
follows that the continuous extension of $Z_1$
to such distribution is unique.

\par

We need to prove that any quasi-periodic element
of order $1$ in
$(\mathcal S _{s,\sigma}^{\sigma ,s}) '(\rr {2d})$ is the Zak transform of an element in
$(\mathcal S _s^\sigma)'(\rr d)$. Therefore, let
$\varphi \in \mathcal S _s^\sigma (\rr d)$, $F$ be a quasi-periodic elements of order $1$
in $(\mathcal S _{s,\sigma}^{\sigma ,s})'(\rr {2d})$, and let 
$g_\varphi\in (\mathcal S _\sigma ^s)'(\rr d)$ be defined
by
$$
\scal {g_\varphi}{\psi} = \scal F{\varphi \otimes \psi},
\qquad \psi \in \mathcal S _s^t(\rr d).
$$
Then $g_\varphi$ is $2\pi$-periodic, and it follows from
Remark 2.3 and Proposition 2.5
in \cite{ToNa} that if $\phi \in \mathcal S _\sigma ^s
(\rr d)\setminus 0$, then
\begin{align*}
g_\varphi &= \sum _{k\in \zz d}c(g_\varphi ,k)e^{i\scal k\xi},
\intertext{where the series converges in
$(\mathcal S _\sigma ^s)'(\rr d)$, and}
c(g_\varphi ,0)
&=
\frac {1}{(2\pi )^d\nm \phi{L^2}^2}\int _{Q_{d,2\pi}}
\left ( \int _{\rr d}
(V_\phi g_\varphi )(\eta ,y)\widehat \phi (-y)e^{i\scal y\eta}\, dy
\right ) \, d\eta
\intertext{and}
c(g_\varphi ,k)
&=
c(g_\varphi e^{-i\scal k{\, \cdot \, }},0)
\end{align*}

\par

By straight-forward computations we get
$$
c(g_\varphi ,0) = 
\frac {1}{(2\pi )^d\nm \phi{L^2}^2}
\int _{Q_{d,2\pi}} \left ( \int _{\rr d}
\scal F{\varphi \otimes (\overline{\phi (\, \cdot \,  -\eta )}
e^{-{i\scal y\, \cdot \, }})}
\widehat \phi (-y)e^{i\scal y\eta}\, dy
\right ) \, d\eta ,
$$
and it is clear that the map which takes $\varphi$ into the 
right-hand side
defines a continuous linear form on
$\mathcal S _s^\sigma (\rr d)$. Hence
$$
c(g_\varphi ,0) = \scal f\varphi ,\qquad \varphi \in
\mathcal S _s^\sigma (\rr d),
$$
for some $f\in (\mathcal S _s^\sigma)'(\rr d)$.

\par

It now follows from the quasi-periodicity of $F$
that
\begin{multline*}
c(g_\varphi ,k) = c(g_\varphi e^{-i\scal k{\, \cdot \, }},0)
\\[1ex]
c(g_{\varphi (\, \cdot \,  +k)},0) = \scal f{\varphi (\, \cdot \,  +k)}
= \scal {f(\, \cdot \,  - k)}\varphi .
\end{multline*}
Hence, if $F_0=F-Z_1f$, then $\scal {F_0}{\varphi \otimes \psi}=0$
when $\varphi \in \mathcal S _s^\sigma (\rr d)$ and $\psi \in \mathcal S _\sigma ^s(\rr d)$. By Remark \ref{Rem:TensorUniqueness}
it now follows that $F=Z_1f$, which gives the result.
\end{proof}

\par

We need the following lemma for the proof of Theorem
\ref{Thm:ZakTestFunctions}.

\par

\begin{lemma}\label{Lemma:ZakTestFunctions}
Let $r,s>0$. Then there is a constant $h>0$ such that
$$
|t|^\beta e^{-r\, |t|^{\frac 1s}} \lesssim h^\beta \beta !^s,
\qquad t\in \mathbf R,\ \beta \in \mathbf N.
$$
\end{lemma}

\par

\begin{proof}
By reasons of symmetry and Stirling's formula, the result follows if we prove
$$
t^\tau e^{-r\, t^{\frac 1s}} \lesssim h^\tau \tau ^{s\tau},
\qquad 0\le t,\tau \in \mathbf R
$$
for some $h>0$. By taking the logaritm it follows that we need to prove that
for some constant $C>0$,
$$
g(\tau ) = -rt^{\frac 1s} + \tau \log t - C\tau -s\tau \log \tau 
$$
is bounded from above by a constant which is independent of $t,\tau >0$.

\par

By differentiation and checking the sign of $g'(\tau)$, it follows that $g(\tau)$
has a global maximum for
$$
\tau _0=e^{-1-\frac Cs} t^{\frac 1s}
$$
with value
$$
g(\tau _0) = t^{\frac 1s}(-r +seh^{-\frac 1s})
$$
By choosing
$$
h>
\left (
\frac r{se}
\right )^s
$$
it follows that $g(\tau )$ is negative, giving the result.
\end{proof}

\par

\begin{proof}[Proof of Theorem \ref{Thm:ZakTestFunctions}]
By similar arguments as in the proof of Theorem
\ref{Thm:ZakDist} we may assume that $E$ is the
standard basis for $\rr d$.

\par

The assertion (1) is the same as Theorem 8.2.5 in \cite{Gc2}.

\par

In order to prove (2) we shall follow the proof of
Theorem 8.2.5 in \cite{Gc2}. In fact,
assume first that $f\in \mathcal S _s^\sigma (\rr d)$, $x\in  k_0+Q_{d,1}$ for some
fixed $k_0\in \zz d$, and let $F=Z_1 f$. Then
$$
|\partial _x^\alpha f(x-k)|\le C h^\alpha e^{-2r (|k_1|^{\frac 1s}+\cdots +|k_d|^{\frac 1s})}
\alpha !^\sigma , \quad x\in k_0+Q_{d,1},
$$
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 $\alpha$.

\par

The series in \eqref{Eq:DefZak} is absolutely convergent together with all its derivatives.
This gives
\begin{multline*}
|(\partial _x^\alpha \partial _\xi ^\beta F)(x,\xi )| 
\le
\sum _{k\in \zz d}|f^{(\alpha )}(x-k)|
k^{\beta}
\\[1ex]
\lesssim
h_1^{|\alpha +\beta |}\sum _{k\in \zz d}
e^{-2r (|k_1|^{\frac 1s}+\cdots +|k_d|^{\frac 1s})}|k^\beta |
\\[1ex]
=
h_1^{|\alpha +\beta |}
\prod _{j=1}^d
\left (
\sum _{k_j\in \mathbf Z}e^{-r|k_j|^{\frac 1s}} \left (e^{-r|k_j|^{\frac 1s}} |k_j|^{\beta _j}\right )
\right ) ,
\end{multline*}
for some constant $h_1>0$. By Lemma \ref{Lemma:ZakTestFunctions}
it follows that
$$
e^{-r|k_j|^{\frac 1s}} |k_j|^{\beta _j} \lesssim h^{\beta _j}\beta _j!^s
$$
for some constant $h>0$. A combination of these estimates give
$$
|(\partial _x^\alpha \partial _\xi ^\beta F)(x,\xi )| 
\lesssim
h^{|\alpha +\beta |}\alpha !^\sigma \beta !^s,
$$
and it follows that $F\in \mathcal E _{\sigma ,s}(\rr {2d})$.
This shows that
$Z_1$ is continuous from $\mathcal S _s^\sigma (\rr d)$
to the set of all quasi-periodic elements of order
$1$ in $\mathcal E _{\sigma ,s}(\rr {2d})$.

\par

Next we show that any quasi-periodic element $F$
of order $1$ in $\mathcal E _{\sigma ,s}(\rr {2d})$
is the Zak transform of an element in
$\mathcal S _s^\sigma (\rr d)$. By Theorem 8.2.5
in \cite{Gc2} it follows that $F=Z_1 f$ when
$$
f(x) = \int _{Q_{d,2\pi}}F(x,\xi )\, d\xi .
$$
We need to prove that $f\in \mathcal S _s^\sigma (\rr d)$.

\par

Since $k\mapsto f(x-k)$ is the Fourier coefficient
of order $k$ for the function
$\xi \mapsto F(x,\xi)$, we have
$$
f(x-k)= \int _{Q_{d,2\pi}} F(x,\xi )e^{-i\scal k\xi}
\, d\xi .
$$
By applying the operator $k^\alpha
\partial _x^\beta$ and integrating by parts we get
\begin{multline*}
k^\alpha (\partial _x^\beta f)(x-k) = 
\int _{Q_{d,2\pi}} (\partial _x^\beta F)(x,\xi )
k^\alpha e^{-i\scal k\xi}\, d\xi 
\\[1ex]
= (-1)^{|\beta |}i^{|\alpha |}
\int _{Q_{d,2\pi}} (\partial _x^\beta
\partial _\xi ^\alpha F)(x,\xi )
e^{-i\scal k\xi}\, d\xi .
\end{multline*}
This gives
\begin{multline*}
\sup _{x\in \rr d} |x^\alpha f^{(\beta )}(x)|
\asymp
\sup _{k\in \zz d} \sup _{x\in Q_{d,\rho}}
|k^\alpha f^{(\beta )}(x-k)|
\\[1ex]
\lesssim
\nm{\partial _x^\beta \partial _\xi ^\alpha F}
{L^\infty (Q_{d,\rho}\times Q_{d,2\pi})}
\lesssim
h^{|\alpha +\beta |}\alpha !^s\beta !^\sigma ,
\end{multline*}
which is the same as $f\in \mathcal S _s^\sigma (\rr d)$.
This gives (2). The assertion (3) follows by similar
arguments and is left for the reader.
\end{proof}

\par

For completeness we also show that all quasi-periodic
distribution are Gelfand-Shilov distributions. (Cf.
\cite[Section 7.2]{Ho1}.)

\par

\begin{prop}\label{Prop:QPerGSDist}
Let $s,\sigma >1$ and $E$ be an ordered basis of
$\rr d$. Then the following is true:
\begin{enumerate}
\item The set of all quasi-periodic elements of
order $E$ in $\mathscr D '(\rr {2d})$ are contained
in $\mathscr S '(\rr {2d})$;

\vspace{0.1cm}

\item The set of all quasi-periodic elements of
order $E$ in $\mathcal D _{\sigma ,s} '(\rr {2d})$ are contained
in $(\mathcal S _{s,\sigma}^{\sigma ,s})'(\rr {2d})$;

\vspace{0.1cm}

\item The set of all quasi-periodic elements of
order $E$ in $\mathcal D _{0,\sigma ,s} '(\rr {2d})$
are contained in
$(\Sigma _{s,\sigma}^{\sigma ,s} )'(\rr {2d})$.
\end{enumerate}
\end{prop}

\par

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

\par

Let $F\in C^\infty (\rr {2d})$ be quasi-periodic
of order $E$,
$\Phi \in \mathcal S _{s,\sigma}^{\sigma ,s}(\rr {2d})$
and let $\chi \in C_0^\infty (\rr {2d})
\cap \mathcal E _{\sigma ,s}(\rr {2d})$ be such that
$$
\sum _{k\in \Lambda _E}\sum _{\kappa\in \Lambda _E'}
\chi (x+k,\xi +\kappa ) =1.
$$
If $\Phi \in \mathcal S _{s,\sigma}^{\sigma ,s}(\rr {2d})$,
then it follows by the quasi-periodisity of $F$
and some straight-forward computations that
\begin{align}
\scal F\Phi &= \scal F{T_\chi \Phi},
\label{Eq:QuasiPerAction1}
\intertext{where}
(T_\chi \Phi)(x,\xi )
&=
\sum _{k\in \Lambda _E}\sum _{\kappa \in \Lambda _E'}
e^{-i\scal k\xi}\Phi (x-k,\xi -\kappa )\chi (x,\xi ),
\label{Eq:QuasiPerAction2}
\end{align}
and that $T_\chi$ in \eqref{Eq:QuasiPerAction2}
is continuous from
$\mathcal S _{s,\sigma}^{\sigma ,s}(\rr {2d})$
to $C_0^\infty (\rr {2d})
\cap \mathcal E _{\sigma ,s}(\rr {2d})$.

\par

Hence
by letting $\scal F\Phi$ be defined by the
right-hand side of \eqref{Eq:QuasiPerAction1}
when $F\in \mathcal D _{\sigma ,s} '(\rr {2d})$
and $\Phi \in \mathcal S _{s,\sigma}^{\sigma ,s}(\rr {2d})$,
it follows that $\Phi \mapsto \scal F{T_\chi \Phi}$
in \eqref{Eq:QuasiPerAction1}
defines a linear and continuous form on
$\mathcal S _{s,\sigma}^{\sigma ,s}(\rr {2d})$
which agree with the usual distribution action,
$\Phi \mapsto \scal F\Phi$ when
$\Phi \in C_0^\infty (\rr {2d})
\cap \mathcal E _{\sigma ,s}(\rr {2d})$.
\end{proof}

\par

\subsection{The Zak transform on Lebesgue and
modulation spaces}

\par

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
\begin{align}
(\mathsf{ZV}_{E ,\phi _1}^{(1)}f)(x,\xi ,\eta ) &= (V_{\phi _1}
(Z_E f(\, \cdot \,  ,\xi )))(x ,\eta )
\label{Eq:STFTZak1}
\intertext{and}
(\mathsf{ZV}_{E ,\phi _2}^{(2)} f)(x,\xi ,y) &= (V_{\phi _2} (Z_E f(x ,\, \cdot \,  )))(\xi ,y ),
\label{Eq:STFTZak2}
\end{align}
which are compositions of the Zak transform and the partial short-time
Fourier transforms with respect to the first and second variable, respectively.

\par

From the previous section 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 $\rr {2d}$ which satisfies \eqref{Eq:QuasiPer}, and a suitable function or
distribution $\Phi$ on $\rr {2d}$, we have
\begin{equation}\label{Eq:STFTQuasiPer}
\begin{alignedat}{2}
(V_\Phi F)(x+ k,\xi ,\eta ,y) &= e^{-i\scal k\eta} 
(V_\Phi F)(x,\xi ,\eta ,y-k),
& \quad k &\in \Lambda _E,
\\[1ex]
(V_\Phi F)(x,\xi +\kappa ,\eta ,y) &=
e^{-i\scal y\kappa}(V_\Phi F)(x,\xi ,\eta ,y),
&\quad \kappa &\in \Lambda _E',
\end{alignedat}
\end{equation}
which follows by straight-forward computations.
We remark that functions
and distributions which satisfy conditions given
in \eqref{Eq:STFTQuasiPer}
are special cases of so-called echo-periodic
functions and distributions, given in
\cite{Toft22}.

\par

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

\par

\begin{thm}\label{Thm:ZakLebesgue}
Let $E$ be an ordered basis of $\rr d$,
$p,r\in (0,\infty ]$, $\omega \in
\mathscr P _E(\rr d)$, $\phi \in \Sigma _1(\rr d)
\setminus 0$,
and let $f$ be a Gelfand-Shilov distribution on
$\rr d$. Then
\begin{align}
\nm f{L^p_{(\omega )}}
&\asymp
\nm
{G_{E,r,\omega ,f}}{L^{p}(\kappa (E)\times \rr d)},
\label{Eq:ZakLebesgue}
\intertext{where}
G_{E,r,\omega ,f}(x,y)
&\equiv
\nm{(\mathsf{ZV}_{E,\phi}^{(2)} f)(x,\, \cdot \,  ,y)\omega (-y)}{L^r(\kappa (E'))}.
\intertext{In particular,}
\nm f{L^p}
&\asymp
\nm{\mathsf{ZV}_{E ,\phi}^{(2)} f}
{L^p(\kappa (E\times E')\times \rr d)}.
\end{align}
\end{thm}

\par

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

\par

The distribution $\xi \mapsto Z_Ef(x,\xi )$ is
$E'$-periodic, and it follows from
\eqref{Eq:ModPerNormEquiv3} that
\begin{multline*}
\left ( \sum _{j\in \Lambda _E} |f(x-j)
\omega (x-j)|^p \right )^{\frac 1p}
\asymp
\left ( \sum _{j\in \Lambda _E}
|f(x-j)\omega (-j)|^p \right )^{\frac 1p}
\\[1ex]
\asymp \nm {G_{E,r,\omega ,f}(x,\, \cdot \,  )}{L^p(\rr d)},
\quad x\in \kappa (E).
\end{multline*}
The result now follows by apply the
$L^p(\kappa (E))$ quasi-norm with respect to
the $x$-variable.
\end{proof}

\par

In the same way we may identify modulation spaces by 
using the Zak transform as
in the next result.
Here we let
\begin{align}
\mathcal R\{  e_1,\dots e_d,e_{d+1},\dots ,e_{2d}\}
&=
\{ e_{d+1},\dots ,e_{2d},e_1,\dots ,e_d\} ,
\label{Eq:B0SpaceDef1}
\end{align}
We also recall Definition \ref{Def:ExtClassModSpaces}
and Remark \ref{Rem:WienerSpace} for definitions
and notions concerning the Wiener amalgam space
$W^{{\boldsymbol p}}_{E,(\omega )}(\rr d)$.

\par

\begin{thm}\label{Thm:ZakModulation}
Let $E,E_0$ be an ordered bases of $\rr d$, $\mathcal R$
be as in \eqref{Eq:B0SpaceDef1},
$E_1=E\times E_0'$, $E_2=\mathcal RE_1$,
${\boldsymbol p} \in (0,\infty ]^{2d}$,
$\omega _0\in \mathscr P _E(\rr {2d})$ and
$\omega \in \mathscr P _E(\rr {4d})$
be such that
$$
\omega (x,\xi ,\eta ,y)=\omega _0(x-y,\eta ).
$$
Then $Z_{E}$ from $\Sigma _1(\rr d)$ to
$C^\infty (\rr {2d})$ is uniquely
extendable to a homeomorphism from
$M^{{\boldsymbol p}}_{E_1,(\omega _0)}(\rr d)$
to the set of quasi-periodic elements of order
$E$ in $W^{\infty ,{\boldsymbol p}}_{E_2,(\omega )}(\rr {2d})$,
and
\begin{equation}\label{Eq:ZakModulation}
\nm {f}{M^{{\boldsymbol p}}_{E_1,(\omega _0)}}
\asymp
\nm {Z_{E} f}{W^{\infty ,{\boldsymbol p}}_{E_2,(\omega )}},
\qquad f\in \Sigma _1'(\rr d).
\end{equation}
\end{thm}

\par

\begin{proof}
Since $\omega (x,\xi ,\eta ,y)$ is constant with 
respect to the $\xi$-variable,
we identify $\omega (x,\xi ,\eta ,y)$ with
$\omega (x,\eta ,y)$.

\par

Let $\Phi =\phi _1\otimes \phi _2$ with
$\phi _1,\phi _2\in \Sigma _1(\rr d)\setminus 0$.
By straight-forward computations we get
\begin{equation}\label{Eq:PerPartZakSTFT}
(\mathsf{ZV}_{1,\phi _1}^{(1)}f)(x,\xi ,\eta )
=
\sum _{j\in \Lambda _E}\big ( (V_{\phi _1}f)(x-j,\eta ) e^{-i\scal j\eta}\big )e^{i\scal j\xi }.
\end{equation}
Let
$$
{\boldsymbol p} _1=(p_1,\dots ,p_d)
\quad \text{and}\quad
{\boldsymbol p} _2=(p_{d+1},\dots ,p_{2d})
$$
when
$$
{\boldsymbol p} = (p_1,\dots ,p_{2d}),
$$
and consider the functions
\begin{align*}
F(x,\eta )
&=
\nm {(V_{\phi _1}f)(x-\, \cdot \,  ,\eta )
\omega (x,\, \cdot \,  ,\eta )}
{\ell ^{{\boldsymbol p} _1}_{E}(\Lambda _{E})}.
\\[1ex]
g_0 (x)&= \nm {F(x,\, \cdot \,  )}{L^{{\boldsymbol p} _2}_{E_0'}(\rr d)}
\\[1ex]
G_0(x,\xi ,\eta ,y)
&\equiv
|V_\Phi(Z_E f)(x,\xi ,\eta ,y)\omega (x,\, \cdot \,  ,\eta )|,
\\[1ex]
G(x,\xi ,\eta )
&\equiv
\nm {G_0(x,\xi ,\eta ,\, \cdot \,  )}
{L^{{\boldsymbol p} _1}_{E}(\rr d)},
\\[1ex]
H(x,\xi )
&=
\nm {G(x,\xi ,\, \cdot \,  )}{L^{{\boldsymbol p} _2}_{E_0'}(\rr d)},
\intertext{and}
h_0(x) &= h_{0,r_0}(x) = \nm {H(x,\, \cdot \,  )}{L^{r_0}_{E_0'}(\kappa (E_0''))},
\quad r_0\in (0,\infty ].
\end{align*}
Since $\xi
\mapsto
(\mathsf{ZV}_{1,\phi _1}^{(1)}f)(x,\xi ,\eta )$
is $E'$-periodic
with Fourier coefficients
$$
j\mapsto (V_{\phi _1}f)(x-j,\eta )
e^{-i\scal j\eta}
$$
(cf. \eqref{Eq:PerPartZakSTFT}),
and the (partial) short-time Fourier transform of that distribution equals
$V_\Phi (Z_1 f)$, it
follows from \eqref{Eq:ModPerNormEquiv3} that
\begin{equation}\label{Eq:PerCoeffEquivHere1}
F(x,\eta ) \asymp \nm {G(x,\, \cdot \,  ,\eta )}{L^{r}_{E'}(\kappa (E'))},
\quad r\in (0,\infty ].
\end{equation}

\par

First let $r_0\le \min ({\boldsymbol p} )$. If we apply the 
$L^{{\boldsymbol p} _2}_{E_0'}$ norm on
\eqref{Eq:PerCoeffEquivHere1} with respect to the
$\eta$ variable
and using H{\"o}lder's inequality we get
\begin{multline}\label{Eq:f0h0Est1}
g_0(x)
=
\nm {F(x,\, \cdot \,  )}{L^{{\boldsymbol p} _2}_{E_0'}(\rr d)}
\asymp
\nm {G(x,\, \cdot \,  )}{L^{r_0,{\boldsymbol p} _2}_{E'\times E_0'}(\kappa (E') \times \rr d)}
\\[1ex]
\lesssim
\nm {H(x,\, \cdot \,  )}{L^{r_0}_{E'}(\kappa (E'))} = h_{0,r_0}(x).
\end{multline}
If
$$
g_1(x) = \nm {F_1(x,\, \cdot \,  )}{\ell ^{{\boldsymbol p} _2}_{E_0'}}
\quad \text{with}\quad
F_1(x,\iota ) = \nm {F(x,\iota )}{L^{r_0}_{E_0'}(\iota +\kappa (E_0'))},
$$
then the fact that $r_0\le \min ({\boldsymbol p} )$ and Jensen's inequality give
$g_1 \lesssim g_0$.

\par

By applying the $L^{r_0}_{E}(\kappa (E))$ norm on the latter inequality,
using the fact that
$$
\omega _0(x-y,\eta )
\asymp \omega _0(-y,\eta ),
\qquad x\in \kappa (E)
$$
and Jensen's inequality again we obtain
$$
\nm {a_{1,f}}{\ell ^{{\boldsymbol p}}_{E_1}}\lesssim \nm {g_0}{L^{r_0}(\kappa (E))},
\quad \text{where}\quad
a_{1,f}(j,\iota )
=
\nm {V_{\phi _1}f\cdot \omega _0}{L^{r_0}_{E_1}((j,\iota )+\kappa (E_1))}.
$$
That is,
$$
\nm {V_{\phi _1}f}{\mathsf{W} ^{r_0}_{E_1}
(\omega _0,\ell ^{{\boldsymbol p}}_{E_1}
(\Lambda _{E_1}))}
\lesssim
\nm {g_0}{L^{r_0}(\kappa (E))},
$$
which is the same as
\begin{equation}\label{Eq:f0fEst1}
\nm f{M^{{\boldsymbol p}}_{E_1,(\omega _0)}}
\lesssim
\nm {g_0}{L^{r_0}(\kappa (E))}
\end{equation}
in view of Proposition \ref{Prop:WienerEquiv}$'$.

\par

In order to estimate $h_{0,r_0}$ we apply 
\eqref{Eq:STFTQuasiPer}
to get
\begin{multline*}
|V_\Phi (Z_{E} f)(x+j,\xi +\iota ,\eta ,y)
\omega _0(x+j-y,\eta )|
\\[1ex]
=
|V_\Phi (Z_{E}f)(x,\xi ,\eta ,y-j)
\omega _0(x-y+ j,\eta )|,\quad
(j,\iota )\in \Lambda _{E\times E'}.
\end{multline*}
By first applying the $L^{{\boldsymbol p}_1}_{E}(\rr {d})$
norm with respect to the $y$ variable and
then the $L^{{\boldsymbol p}_2}_{E_0'}(\rr {d})$
norm with respect to the $\eta$ variable we get
$$
H(x+j,\xi +\iota )= H(x,\xi ),\qquad (j,\iota )\in \Lambda _{E\times E'}.
$$

\par


Hence, by applying the $L^{r_0}_{E}(\kappa (E))$ norm
on $h_{0,r_0}$ and using
H{\"o}lder's and Jensen's inequalities we get
\begin{multline}\label{Eq:h0Est1}
\nm {h_{0,r_0}}{L^{r_0}_{E}(\kappa (E))}
= \nm H{L^{r_0}_{E\times E'}(\kappa (E\times E'))}
\lesssim
\nm H{L^\infty_{E\times E'}(\kappa (E\times E'))}
\\[1ex]
=
\sup _{(j,\iota )\in \Lambda _{E\times E'}}
\left (
\nm H{L^\infty_{E\times E'}((j,\iota )+\kappa (E\times E'))}
\right ),
\end{multline}
A combination of
\eqref{Eq:f0h0Est1}--\eqref{Eq:h0Est1}
and Proposition \ref{Prop:WienerEquiv}$'$
now gives
\begin{equation}\label{Eq:ZakModulation2}
\nm {f}{M^{{\boldsymbol p}}_{E_1,(\omega _0)}}
\lesssim
\nm {Z_{E} f}{W^{\infty ,{\boldsymbol p}}_{E_2,(\omega )}},
\qquad f\in \Sigma _1'(\rr d).
\end{equation}

\par

In order to get the reversed estimate we again
apply the $L^{{\boldsymbol p} _2}_{E_0'}$ norm on
\eqref{Eq:PerCoeffEquivHere1} with respect to
the $\eta$ variable
and use H{\"o}lder's inequality to get
\begin{multline}\label{Eq:f0h0Est2}
g_0(x) = \nm {F(x,\, \cdot \,  )}{L^{{\boldsymbol p} _2}_{E_0'}(\rr d)}
\asymp
\nm {G(x,\, \cdot \,  )}
{L^{\infty ,{\boldsymbol p} _2}
_{E'\times E_0'}(\kappa (E')
\times \rr d)}
\\[1ex]
\gtrsim
\nm {H(x,\, \cdot \,  )}
{L^{\infty}_{E'}(\kappa (E'))}
=
h_{0,\infty}(x).
\end{multline}
If
$$
g_2(x) = \nm {F_2(x,\, \cdot \,  )}{\ell ^{{\boldsymbol p} _2}_{E_0'}}
\quad \text{with}\quad
F_2(x,\iota ) = \nm {F(x,\, \cdot \,  )}
{L^{\infty}_{E_0'}(\iota +\kappa (E_0'))},
$$
then Jensen's inequality give
$g_0 \lesssim g_2$.

\par

By applying the $L^{\infty}_{E}
(\kappa (E))$ norm on the latter inequality
and using Jensen's inequality again we obtain
$$
\nm {a_{2,f}}{\ell ^{{\boldsymbol p}}_{E_1}}\gtrsim \nm {g_0}{L^{\infty}(\kappa (E))},
\quad \text{where}\quad
a_{2,f}(j,\iota )
=
\nm {V_{\phi _1}f\cdot \omega _0}{L^{\infty}((j,\iota )+\kappa (E_1))}.
$$
That is,
$$
\nm {V_{\phi _1}f}
{\mathsf{W} ^{\infty}_{E_1}(\omega _0,
\ell ^{{\boldsymbol p}}_{E_1}(\Lambda _{E_1}))}
\gtrsim
\nm {g_0}{L^{r_0}(\kappa (E))},
$$
which is the same as
\begin{equation}\label{Eq:f0fEst2}
\nm f{M^{{\boldsymbol p}}_{E_1},(\omega _0)}
\gtrsim
\nm {g_0}{L^{\infty}(\kappa (E))}
\end{equation}
in view of Proposition \ref{Prop:WienerEquiv}$'$.

\par

By applying the
$L^{\infty}_{E}(\kappa (E))$ norm on
$h_{0,\infty}$ and using \eqref{Eq:h0Est1}
we get
\begin{equation}\label{Eq:h0Est2}
\nm {h_{0,\infty}}{L^{\infty}_{E}(\kappa (E))}
=
\nm H{L^{\infty}_{E\times E'}(\kappa (E\times E'))}
\asymp
\nm {Z_{E} f}{W^{\infty ,{\boldsymbol p}}_{E_2,(\omega )}},
\end{equation}
where the last relation follows
from Proposition \ref{Prop:WienerEquiv}$'$.
A combination of \eqref{Eq:f0h0Est2}, \eqref{Eq:f0fEst2}
and \eqref{Eq:h0Est2} now gives
\begin{equation}\label{Eq:ZakModulation3}
\nm {f}{M^{{\boldsymbol p}}_{E_1,(\omega _0)}}
\lesssim
\nm {Z_{E} f}{W^{\infty ,{\boldsymbol p}}_{E_2,(\omega )}},
\qquad f\in \Sigma _1'(\rr d),
\end{equation}
and the result follows by combining
\eqref{Eq:ZakModulation3} with
\eqref{Eq:ZakModulation2}.
\end{proof}

\par

If ${\boldsymbol p}$, $E$, $E_0$, $E_1$, $E_2$ and $\omega$ are
the same as in Theorem \ref{Thm:ZakModulation}, and
$f\in \Sigma _1'(\rr d)$ and $\Phi \in \Sigma _1(\rr {2d})\setminus 0$,
then it follows from the echo-periodicity
\eqref{Eq:STFTQuasiPer} that
\begin{equation}\label{Eq:STFTZakPartNorm}
H_{f,\omega ,E,E_0}(x,\xi )
\equiv
\nm {(V_\Phi (Z_Ef))(x,\xi ,\, \cdot \,  )\omega (x,\xi ,\, \cdot \,  )}{L^{{\boldsymbol p}}_{E_2}},
\end{equation}
is $E\times E'$ periodic. The following result now follows from Proposition
\ref{Prop:WienerEquiv}$'$, Theorem  \ref{Thm:ZakModulation}
and the previous observation. The details are left for the reader.

\par

\begin{thm}\label{Thm:ZakModulation2}
Let ${\boldsymbol p}$, $E$, $E_0$, $E_1$, $E_2$, $\omega$, $\omega _0$
and $H_{f,\omega ,E,E_0}$ be
the same as in Theorem \ref{Thm:ZakModulation} and
\eqref{Eq:STFTZakPartNorm}, and let
$f\in \Sigma _1'(\rr d)$, $\Phi \in \Sigma _1(\rr {2d})\setminus 0$
and $\boldsymbol r \in (0,\infty ]^{2d}$. Then
$$
f\in M^{{\boldsymbol p}}_{E_1,(\omega _0)}(\rr d)
\quad \Leftrightarrow \quad
H_{f,\omega ,E,E_0} \in L^{\boldsymbol r}_{E\times E'}(\kappa (E\times E')),
$$
and
$$
\nm f{M^{{\boldsymbol p}}_{E_1,(\omega _0)}}
\asymp
\nm {H_{f,\omega ,E,E_0}}{L^{\boldsymbol r}_{E\times E'}(\kappa (E\times E'))}.
$$
\end{thm}

\par


\par

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

\par

\begin{cor}\label{Cor:ZakModulation}
Let $E$ and $\Phi$ be the same as in
Theorem \ref{Thm:ZakModulation2}, and let
$p\in (0,\infty ]$. Then
\begin{align*}
\nm {f}{M^{p}}
&\asymp
\nm{(V_\Phi (Z_{E} f))}{L^p(\kappa (E\times E')
\times \rr {2d})},
\qquad f\in \Sigma _1'(\rr d).
\end{align*}
\end{cor}



\par

