Gelfand–Shilov Spaces for Extended Gevrey Regularity

Abstract

We consider spaces of smooth functions obtained by relaxing Gevrey-type regularity and decay conditions. It is shown that these classes can be introduced by using the general framework of the weighted matrices approach to ultradifferentiable functions. We examine alternative descriptions of Gelfand–Shilov spaces related to the extended Gevrey regularity and derive their nuclearity. In addition to the Fourier transform invariance property, we present their corresponding symmetric characterizations. Finally, we consider some time–frequency representations of the introduced classes of ultradifferentiable functions.

1. Introduction

Traditionally, Gelfand–Shilov spaces of ultradifferentiable functions are introduced as sets of all $f\in C^{\infty }({\mathbb{R}}^n)$ such that for given $t,s>0$,

$$\underset{x\in {\mathbb{R}}^n}{sup}|x^{\alpha }{\partial }^{\beta }f(x)|\le C{h}^{|\alpha +\beta |}\alpha {!}^t\beta {!}^s,\alpha ,\beta \in {\mathbb{N}}_0^n,$$

(1)

for some constants $C,h>0$; see [1]. In other words, the regularity and decay properties of f are controlled by Gevrey sequences ${(\beta {!}^s)}_{\beta \in {\mathbb{N}}_0^n},$ and ${(\alpha {!}^t)}_{\alpha \in {\mathbb{N}}_0^n}$, respectively.

These spaces are widely used in regularity theory for partial differential equations, e.g., to describe exponential decay and the holomorphic extension of solutions to globally elliptic equations [2,3] and in the regularizing properties of the Boltzmann equation [4]. For additional examples, we refer to [5,6,7].

More general Gelfand–Shilov-type spaces are obtained by using various types of defining sequences apart from the Gevrey sequences.

In this paper, we introduce Gelfand–Shilov-type spaces through regularity and decay conditions that go beyond the Gevrey sequences by considering two-parameter dependent sequences of the form ${(p^{\tau p^{\sigma }})}_{p\in {\mathbb{N}}_0}$, $\tau >0$, $\sigma >1$. Such sequences have been recently used to describe a convenient extension of Gevrey spaces [8,9,10]. For an overview and some applications of the extended Gevrey classes, we refer to [11]. We note that the extended Gevrey classes are defined by regularity conditions, without reference to decay, whereas the elements of the spaces considered here, in addition to regularity, also enjoy suitable decay properties.

A commonly used approach to ultradifferentiable functions, based on defining sequences, was introduced in [12]. Another widely used approach is the one based on the so-called weight functions, as considered in [13]. Recently, Rainer and Schindl proposed an approach that unifies both classical methods [14]. It is based on the concept of weight matrices, or multi-indexed weight sequences in the terminology of [15]. A general framework that provides a unified treatment of Gelfand–Shilov spaces defined via weight sequences or weight functions was very recently presented in [16,17]. There, one can find general results on the inclusion relation between the considered global ultradifferentiable classes.

Our aim is to study a new type of Gelfand–Shilov spaces that combines the general approach based on weight matrices with the framework of extended Gevrey regularity. This is achieved by considering the so-called limit classes, i.e., suitable unions and intersections with respect to the parameter $\tau >0$; see Definition 2.

In this way, we obtain prominent examples of global ultradifferentiable classes defined by the general weight matrix approach which go beyond the classical theory. At the same time, by considering specific sequences, we are able to derive results that cannot be extracted from the general theory while also extending those that are well known in the context of the Gevrey-type regularity.

To be more precise, we perform a thorough examination of general conditions from [15] that provide nuclearity of the considered spaces. Thereafter, we show the Fourier transform invariance and symmetric characterization of Gelfand–Shilov spaces for extended Gevrey regularity à la Chung–Chung–Kim [18]. We emphasize that the sequence ${(p^{\tau p^{\sigma }})}_{p\in {\mathbb{N}}_0}$, $\tau >0$, $\sigma >1$ does not satisfy Komatsu’s $(M.2)$ condition:

$$(M.2)(\exists A,B>0)M_{p+q}\le A{B}^{p+q}{M}_p{M}_q,p,q\in {\mathbb{N}}_0.$$

This invokes nontrivial modifications to the existing proofs for classical Gelfand–Shilov-type spaces.

Furthermore, we study the properties of several time–frequency representations acting on Gelfand–Shilov spaces via extended Gevrey sequences. This leads to characterizations that extend those given in, e.g., [19,20,21], in the context of classical Gelfand–Shilov spaces.

2. Weight Matrices and Weight Functions

A general approach to ultradifferentiable functions based on the concept of weight matrices is proposed in [14]. In addition, Gelfand–Shilov-type spaces of ultradifferentiable functions introduced by the means of weight matrices and weight functions are recently considered in [17]. In this section we briefly recall the notions of weight matrices and weight functions which will be used in our approach to Gelfand–Shilov spaces for extended Gevrey regularity.

2.1. Weight Sequences and Matrices

We first recall the notion of a weight matrix [14] or multi-indexed weight sequence system in terms of [15,17].

Definition 1. 

Let $M={(M_{\alpha })}_{\alpha \in {\mathbb{N}}_0^n}$ be a weight sequence, i.e. $M_{\alpha }>0$, $\alpha \in {\mathbb{N}}_0^n$, $M_0=1$, and ${lim}_{|\alpha |\to \infty }{(M_{\alpha })}^{1/|\alpha |}=\infty$.

A family $\mathcal{M}=\{M^{(\lambda )}:\lambda >0\}$ of weight sequences is a weight matrix if $M_{\alpha }^{(\lambda )}\le M_{\alpha }^{(\mu )}$ for all $\alpha \in {\mathbb{N}}_0^n$ and $\lambda \le \mu$.

We say that the sequence $M={(M_{\alpha })}_{\alpha \in {\mathbb{N}}_0^n}$ is an isotropic sequence if for all $\alpha \in {\mathbb{N}}_0^n$ with $|\alpha |=p$, $p\in {\mathbb{N}}_0$ one has $M_{|\alpha |}=N_p$ for some sequence ${(N_p)}_{p\in {\mathbb{N}}_0}$. In this case we simply write ${(M_p)}_{p\in {\mathbb{N}}_0}$ for this sequence.

Next, we introduce isotropic sequences related to extended Gevrey regularity.

Let $\tau >0$ and $\sigma >1.$ By $M^{(\tau )}={(M_p^{\tau ,\sigma })}_{p\in {\mathbb{N}}_0}$, we denote the sequence of positive numbers given by

$$M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }},p\in \mathbb{N},M_0^{\tau ,\sigma }=1.$$

(2)

We are interested in the weight matrix ${\mathcal{M}}_{\sigma },$$\sigma >1$ fixed, given by

$${\mathcal{M}}_{\sigma }=\left\{M^{(\tau )}={(M_p^{\tau ,\sigma })}_{p\in {\mathbb{N}}_0}:\tau >0\right\}.$$

(3)

The main properties of ${(M_p^{\tau ,\sigma })}_{p\in {\mathbb{N}}_0}$, $\tau >0$, $\sigma >1$, related to Komatsu’s theory of ultradistributions, are given in the next lemma (cf. [9] (Lemmas 2.2 and 3.1), [11] (Lemma 2)).

Lemma 1. 

Let $\tau >0$, $\sigma >1$, $M_0^{\tau ,\sigma }=1$, and $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$. Then, the following properties hold:

$(M.1)$

${(M_p^{\tau ,\sigma })}^2\le M_{p-1}^{\tau ,\sigma }{M}_{p+1}^{\tau ,\sigma },$$p\in \mathbb{N},$

$\widetilde{(M.2)}$

$M_{p+q}^{\tau ,\sigma }\le C^{p^{\sigma }+q^{\sigma }}{M}_p^{\tau 2^{\sigma -1},\sigma }{M}_q^{\tau 2^{\sigma -1},\sigma },$$p,q\in {\mathbb{N}}_0,$ for some constant $C\ge 1$,

$\widetilde{{(M.2)}^{\prime }}$

$M_{p+1}^{\tau ,\sigma }\le C^{p^{\sigma }}{M}_p^{\tau ,\sigma },$$p\in {\mathbb{N}}_0,$ for some constant $C\ge 1$,

${(M.3)}^{\prime }$

${\displaystyle \sum _{p=1}^{\infty }\frac{M_{p-1}^{\tau ,\sigma }}{M_p^{\tau ,\sigma }}<\infty .}$

In addition, ${\left({\left({\displaystyle \frac{M_p^{\tau ,\sigma }}{p!}}\right)}^{1/p}\right)}_{p\in {\mathbb{N}}_0}$ is an almost-increasing sequence, i.e.,

$${\left({\displaystyle \frac{M_p^{\tau ,\sigma }}{p!}}\right)}^{1/p}\le C{\left({\displaystyle \frac{M_q^{\tau ,\sigma }}{q!}}\right)}^{1/q},p\le q,$$

for some constant $C>0$, which implies ${lim}_{p\to \infty }{(M_p)}^{1/p}=\infty$ (cf. [11]).

We note that $(M.1)$ implies

$$M_p^{\tau ,\sigma }{M}_q^{\tau ,\sigma }\le M_{p+q}^{\tau ,\sigma },p,q\in {\mathbb{N}}_0,$$

(4)

and that ${(M.3)}^{\prime }$ (the so-called non-quasianaliticity condition) follows from the estimate

$$\begin{array}{c}\hfill {\displaystyle \frac{M_{p-1}^{\tau ,\sigma }}{M_p^{\tau ,\sigma }}}\le {\displaystyle \frac{1}{{(2p)}^{\tau {(p-1)}^{\sigma -1}}}},p\in \mathbb{N}.\end{array}$$

(5)

The proof of (5) can be found in, e.g., [11] (Appendix A).

More properties of ${(M_p^{\tau ,\sigma })}_{p\in {\mathbb{N}}_0}$ in the context of the weight matrix approach to ultradifferentiable functions (in the sense of A. Rainer and G. Schindl) can be found in, e.g., [8].

The sequence ${(M_p^{\tau ,\sigma })}_{p\in {\mathbb{N}}_0}$ gives rise to extended Gevrey classes ${\mathcal{E}}_{\tau ,\sigma }({\mathbb{R}}^n)$, also called Pilipović–Teofanov–Tomić classes (PTT-classes) in [8,22]. We refer to [11] for a recent survey on PTT-classes and their relation to classes of ultradifferentiable functions given by the weight matrix approach.

In fact, we use the weight matrix approach and consider PTT-limit classes. As it is noted in [8], the main reason to consider these classes is their stability under the action of ultradifferential operators.

2.2. Weight Functions and the Lambert Function

A continuous and increasing function $\omega :[0,\infty )$$\mapsto [0,\infty )$, $\omega (0)=0$ is called a BMT (Braun–Meise–Taylor) weight function or simply a weight function if it satisfies the following conditions:

($\alpha$)

$\omega (2t)=O(\omega (t)),t\to \infty ,$

($\beta$)

$\omega (t)=O(t),t\to \infty ,$

($\gamma$)

$o(\omega (t))=logt,t\to \infty ,$

($\delta$)

$\phi (t)=w(e^t),$ is convex.

Then, we define $\omega (t)=\omega (|t|)$ if $t\in {\mathbb{R}}^n$. Some classical examples of BMT weight functions are

$$\omega (t)={ln}_{+}^s|t|,\omega (t)={\displaystyle \frac{|t|}{{ln}^{s-1}(e+|t|)}},s>1,t\in {\mathbb{R}}^n,$$

where ${ln}_{+}k=max\{0,lnk\}$. Also, $\omega (t)={|t|}^s$ is a weight function if and only if $0<s\le 1$.

Next, we introduce the Lambert W function which describes the precise asymptotic behavior of the associated function related to the sequence ${(M_p^{\tau ,\sigma })}_{p\in {\mathbb{N}}_0}$.

The Lambert W function is defined as the inverse of $z{e}^z$, $z\in \mathbb{C}$. By $W(x)$, we denote the restriction of its principal branch to $[0,\infty )$. It is used as a convenient tool to describe asymptotic behavior in different contexts. We refer to [23,24] for a review of some applications of the Lambert W function in pure and applied mathematics.

Some basic properties of the Lambert function W are given below:

$(W1)$

$W(0)=0$, $W(e)=1$, $W(x)$ is continuous, increasing and concave on $[0,\infty )$.

$(W2)$

$W(x{e}^x)=x$ and $x=W(x)e^{W(x)}$, $x\ge 0$.

$(W3)$

W can be represented in the form of the absolutely convergent series

$$W(x)=lnx-ln(lnx)+\sum _{k=0}^{\infty }\sum _{m=1}^{\infty }{c}_{km}{\displaystyle \frac{{(ln(lnx))}^m}{{(lnx)}^{k+m}}},x\ge x_0>e,$$

with suitable constants $c_{km}$ and $x_0$, wherefrom the following estimates hold:

$$lnx-ln(lnx)\le W(x)\le lnx-{\displaystyle \frac{1}{2}}ln(lnx),x\ge e.$$

(6)

The equality in (6) holds if and only if $x=e$.

Note that $(W2)$ implies

$$W(xlnx)=lnx,x>1.$$

By using $(W3)$ we obtain

$$W(x)\asymp lnx,x\to \infty ,$$

and therefore

$$W(Cx)\asymp W(x),x\to \infty ,$$

for any $C>0$. Here, and in what follows, $f\asymp g$ means that functions f and g are equivalent, i.e., $f=O(g)$ and $g=O(f)$.

Remark 1. 

By [25] (Theorem 1) it follows that ${\displaystyle \omega (t)=\frac{{ln}^s(1+|t|)}{W^{s-1}(ln(1+|t|))}}$, $s>1$, $t\in {\mathbb{R}}^n$, is equivalent to a BMT weight function, and therefore it satisfies $(\alpha )-(\gamma )$. Actually, in [25], the function ${ln}_{+}|·|$ is considered instead of $ln(1+|·|)$, which does not change the conclusion.

3. Gelfand–Shilov-Type Classes

In this section, we introduce Gelfand–Shilov spaces related to extended Gevrey regularity and provide equivalent descriptions that will be used in the sequel. Additionally, we prove their nuclearity.

We use the common notation for multi-indices $\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}_0^n$: Let $\alpha ,\beta ,\gamma \in {\mathbb{N}}_0^n$ and $x\in {\mathbb{R}}^n.$ Then, $|\alpha |={\alpha }_1+\cdots +{\alpha }_n$; $\alpha !={\alpha }_1!\cdots {\alpha }_n!$; $\alpha \le \beta$⇔${\alpha }_j\le {\beta }_j$, $j=1,\dots ,n$; $\gamma \le min\{\alpha ,\beta \}$ means $\gamma \le \alpha$ and $\gamma \le \beta$; $\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\beta }}\right)={\prod }_{j=1}^n\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_j}{{\beta }_j}}\right)$, $\beta \le \alpha$; $x^{\alpha }={\prod }_{j=1}^n{x}_j^{{\alpha }_j}$. With ${\langle ·,·\rangle }_{L^2}$ we denote the standard scalar product in $L^2({\mathbb{R}}^n)$.

3.1. Definition and Its Relatives

Isotropic Gelfand–Shilov spaces related to the weight matrix ${\mathcal{M}}_{\sigma }$, where $\sigma >1$ is fixed, can be introduced as follows.

Definition 2. 

Let $\sigma >1$, the weight matrix ${\mathcal{M}}_{\sigma }$ be given by (3), and let $\phi \in C^{\infty }({\mathbb{R}}^n)$. Then, the Gelfand–Shilov space of the Roumieu type related to ${\mathcal{M}}_{\sigma }$ is given by

$$\begin{array}{cc}\hfill \phi \in {\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)\iff & (\exists \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)\hfill \\ \hfill & \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\hfill \end{array}$$

and the Gelfand–Shilov space of Beurling type related to ${\mathcal{M}}_{\sigma }$ is given by

$$\begin{array}{cc}\hfill \phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)\iff & (\forall \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)\hfill \\ \hfill & \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma }.\hfill \end{array}$$

It immediately follows that

$${\mathit{S}}_t^s({\mathbb{R}}^n)\subset {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)\subset {\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)\subset \mathcal{S}({\mathbb{R}}^n),s,t>0,\sigma >1,$$

where ${\mathit{S}}_t^s({\mathbb{R}}^n)$ is the Gelfand–Shilov space of smooth functions which satisfy (1) for some constants $C,h>0$, and $\mathcal{S}({\mathbb{R}}^n)$ is the Schwartz space of rapidly decreasing functions.

Let $\sigma >1$ be given, and let $M^{(\tau )}={(M_p^{\tau ,\sigma })}_{p\in {\mathbb{N}}_0}$, $\tau >0$; see (2). We denote by ${\mathit{S}}_{M^{(\tau )}}^{M^{(\tau )}}({\mathbb{R}}^n)$ the Banach space of all functions $\phi \in C^{\infty }({\mathbb{R}}^n)$ such that

$${\parallel \phi \parallel }_{{\mathit{S}}_{M^{(\tau )}}^{M^{(\tau )}}}=\underset{\alpha ,\beta \in {\mathbb{N}}_0^n}{sup}{\displaystyle \frac{{sup}_x|x^{\alpha }{\partial }^{\beta }\phi (x)|}{M_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma }}}<\infty .$$

Then,

$${\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)={ind\; lim}_{\tau >0}{\mathit{S}}_{M^{(\tau )}}^{M^{(\tau )}}({\mathbb{R}}^n)\mathrm{and}{\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)={proj\; lim}_{\tau >0}{\mathit{S}}_{M^{(\tau )}}^{M^{(\tau )}}({\mathbb{R}}^n).$$

Thus, ${\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$ is an $(LB)$-space and ${\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ is a Fréchet space.

We will use $[{\mathcal{M}}_{\sigma }]$ as a common notation for $\{{\mathcal{M}}_{\sigma }\}$ or $({\mathcal{M}}_{\sigma })$; i.e., ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}$ is either ${\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}$ or ${\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}$.

Notice that Gelfand–Shilov spaces related to a weight matrix $\mathcal{M}=\{M^{(\lambda )}:\lambda >0\}$, when the weight sequence $M={(M_{\alpha })}_{\alpha \in {\mathbb{N}}_0^n}$ satisfies suitable conditions, are usually given by

$$\begin{array}{c}{\mathcal{S}}_{\{\mathcal{M}\}}^{\{\mathcal{M}\}}({\mathbb{R}}^n)=\{\phi \in C^{\infty }({\mathbb{R}}^n):(\exists \lambda >0)(\exists h>0)(\exists C>0)\hfill \\ \hfill \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{h}^{|\alpha +\beta |}{M}_{\alpha }^{\lambda }{M}_{\beta }^{\lambda },\forall \alpha ,\beta \in {\mathbb{N}}_0^n\},\end{array}$$

(7)

and

$$\begin{array}{c}{\mathcal{S}}_{(\mathcal{M})}^{(\mathcal{M})}({\mathbb{R}}^n)=\{\phi \in C^{\infty }({\mathbb{R}}^n):(\forall \lambda >0)(\forall h>0)(\exists C>0)\hfill \\ \hfill \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{h}^{|\alpha +\beta |}{M}_{\alpha }^{\lambda }{M}_{\beta }^{\lambda },\forall \alpha ,\beta \in {\mathbb{N}}_0^n\},\end{array}$$

(8)

see, e.g., [26] (2.16). However, the ”geometric growth factor” $h^{|\alpha +\beta |}$ in (7) and (8) is excluded in the definition of corresponding Gelfand–Shilov spaces in [15] (3.1).

Let us show that for Gelfand–Shilov spaces in Definition 2 the expected presence of “non-standard growth factor” $h^{{|\alpha |}^{\sigma }+{|\beta |}^{\sigma }}$ related to the weight matrix ${\mathcal{M}}_{\sigma }$ given by (3) is irrelevant.

In the proof of Theorem 1, we will use the following result [9] (Lemma 2.3).

Lemma 2. 

$$\underset{\rho >0}{sup}{\displaystyle \frac{h^{{\rho }^{\sigma }}}{{\rho }^{\tau {\rho }^{\sigma }}}}=e^{{\displaystyle \frac{\tau }{\sigma e}}{h}^{{\displaystyle \frac{\sigma }{\tau }}}}$$

(9)

for any $h>0$, $\tau >0$ and $\sigma >1$.

Proof. 

We provide the proof, correcting a typographical error from [9]. Define $f(\rho )={\displaystyle \frac{h^{{\rho }^{\sigma }}}{{\rho }^{\tau {\rho }^{\sigma }}}}$, $\rho >0$. Differentiating $lnf(\rho ),$ we obtain ${(lnf(\rho ))}^{\prime }={\rho }^{\sigma -1}(\sigma lnh-\tau \sigma ln\rho -\tau )$, $\rho >0.$ For ${\rho }_0:=h^{{\displaystyle \frac{1}{\tau }}}{e}^{-1/\sigma },$ we have ${max}_{\rho >0}lnf(\rho )=lnf({\rho }_0)={\displaystyle \frac{\tau }{e\sigma }}{h}^{{\displaystyle \frac{\sigma }{\tau }}}$, which proves the claim. □

Lemma 2 implies that for any given $h>0$, we have

$$h^{{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{{\displaystyle \frac{\tau }{2}},\sigma }\le C{M}_{|\alpha |}^{\tau ,\sigma },\forall \alpha \in {\mathbb{N}}_0^n,$$

(10)

where the constant $C>0$ depends on $\tau$, $\sigma$, and h.

We also introduce the notation:

$$\begin{array}{c}{\mathcal{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)=\{\phi \in C^{\infty }({\mathbb{R}}^n):(\forall \tau >0)(\forall h>0)(\exists C>0)\hfill \\ \hfill \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{h}^{{|\alpha |}^{\sigma }+{|\beta |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\forall \alpha ,\beta \in {\mathbb{N}}_0^n\},\end{array}$$

(11)

and

$$\begin{array}{c}{\mathcal{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)=\{\phi \in C^{\infty }({\mathbb{R}}^n):(\exists \tau >0)(\exists h>0)(\exists C>0)\hfill \\ \hfill \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{h}^{{|\alpha |}^{\sigma }+{|\beta |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\forall \alpha ,\beta \in {\mathbb{N}}_0^n\}.\end{array}$$

The spaces ${\mathcal{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$ and ${\mathcal{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ are an $(LB)$- space and a Fréchet space, respectively, with topologies introduced analogously to those of ${\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$ and ${\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$.

Theorem 1. 

Let $\sigma >1$, and let the weight matrix ${\mathcal{M}}_{\sigma }$ be given by (3), $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $M_0^{\tau ,\sigma }=1$. Then,

$${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)={\mathcal{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n),$$

as locally convex vector spaces.

Proof. 

We prove ${\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)={\mathcal{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$, and leave the Roumieu case to the reader.

Let us first show that $\phi \in {\mathcal{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ if and only if

$$(\forall \tau >0)(\exists h_0>0)(\exists C>0)\underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{h}_0^{{|\alpha |}^{\sigma }+{|\beta |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },$$

(12)

for all $\alpha ,\beta \in {\mathbb{N}}_0^n$. The implication (11) ⇒ (12) trivially holds.

For the opposite inclusion, we invoke (9), wherefrom for any given $\tau >0$ and $h>0$ we have

$$\begin{array}{ccc}\hfill h_0^{{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{{\displaystyle \frac{\tau }{2}},\sigma }& =& {\left({\displaystyle \frac{h_0}{h}}\right)}^{{|\alpha |}^{\sigma }}{h}^{{|\alpha |}^{\sigma }}{\displaystyle \frac{M_{|\alpha |}^{\tau ,\sigma }}{M_{|\alpha |}^{\frac{\tau }{2},\sigma }}}\le \left(\underset{\alpha >0}{sup}{\displaystyle \frac{{(h_0/h)}^{{|\alpha |}^{\sigma }}}{M_{|\alpha |}^{\frac{\tau }{2},\sigma }}}\right)·h^{{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma }\hfill \\ & =& e^{{\displaystyle \frac{\tau }{2\sigma e}}{(h_0/h)}^{{\displaystyle \frac{2\sigma }{\tau }}}}{h}^{{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma }=C{h}^{{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma },\hfill \end{array}$$

and similarly

$$h_0^{{|\beta |}^{\sigma }}{M}_{|\beta |}^{{\displaystyle \frac{\tau }{2}},\sigma }\le C{h}^{{|\beta |}^{\sigma }}{M}_{|\beta |}^{\tau ,\sigma }.$$

Consequently,

$$\underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{h}^{{|\alpha |}^{\sigma }+{|\beta |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },$$

i.e., (12) ⇒ (11). Thus, it remains to prove that $\phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ if and only if (12) holds. Of course, $\phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ implies (12) (take $h_0\ge 1$ in (12)). Next, we assume that (12) holds and use (10) to conclude that $\phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$. □

Remark 2. 

By $\widetilde{(M.2)}$, (4), and Theorem 1, it follows that the product $M_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma }$ in the definition of ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$ can be replaced by $M_{|\alpha +\beta |}^{\tau ,\sigma }$ yielding the same locally convex vector spaces.

Next we show that instead of the sup norm one can use the $L^2$-norm in Definition 2. This result will be used in Section 4. We refer to [27] (Theorem 1) or [7] (Theorem 6.1.6) for Gelfand–Shilov spaces related to Gevrey sequences $M_{\alpha }=\alpha {!}^t$, $\alpha \in {\mathbb{N}}_0^n$, $t>0$.

We introduce the following notation.

$$\begin{array}{cc}\hfill \phi \in {\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\},2}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)\iff & (\exists \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)\hfill \\ \hfill & \parallel x^{\alpha }{\partial }^{\beta }{\phi (x)\parallel }_{L^2}\le C{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma };\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma }),2}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)\iff & (\forall \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)\hfill \\ \hfill & \parallel x^{\alpha }{\partial }^{\beta }{\phi (x)\parallel }_{L^2}\le C{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\hfill \end{array}$$

(14)

where ${\parallel ·\parallel }_{L^2}$ denotes the usual $L^2$-norm, and ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }],2}^{[{\mathcal{M}}_{\sigma }]}$ is either ${\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\},2}^{\{{\mathcal{M}}_{\sigma }\}}$ or ${\mathit{S}}_{({\mathcal{M}}_{\sigma }),2}^{({\mathcal{M}}_{\sigma })}$.

The spaces ${\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\},2}^{\{{\mathcal{M}}_{\sigma }\}}$ and ${\mathit{S}}_{({\mathcal{M}}_{\sigma }),2}^{({\mathcal{M}}_{\sigma })}$ are an $(LB)$- space and a Fréchet space, respectively, with topologies introduced analogously to those of ${\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$ and ${\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$.

Theorem 2. 

Let $\sigma >1$, and let the weight matrix ${\mathcal{M}}_{\sigma }$ be given by (3), $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $M_0^{\tau ,\sigma }=1$. Then,

$${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)={\mathit{S}}_{[{\mathcal{M}}_{\sigma }],2}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n),$$

(15)

as locally convex vector spaces.

Proof. 

We prove the Beurling case, i.e., ${\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)={\mathit{S}}_{({\mathcal{M}}_{\sigma }),2}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$, and leave the Roumieu case to the reader.

For the inclusion ${\mathit{S}}_{({\mathcal{M}}_{\sigma }),2}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)\subset {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$, we invoke the Sobolev embedding $H^s({\mathbb{R}}^n)\hookrightarrow L^{\infty }({\mathbb{R}}^n)$, with integer $s>n/2$, which implies

$$\underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C_1\parallel x^{\alpha }{\partial }^{\beta }{\phi (x)\parallel }_{H^s}\le C\sum _{|\gamma |\le s}{\parallel {\partial }^{\gamma }(x^{\alpha }{\partial }^{\beta }\phi (x))\parallel }_{L^2},$$

(16)

for some $C_1,C>0$, since the Sobolev norm ${\parallel f(x)\parallel }_{H^s}=\parallel (1+{|\omega |}^2{{)}^{s/2}\widehat{f}(\omega )\parallel }_{H^s}$ is equivalent to ${\sum }_{|\gamma |\le s}{\parallel {\partial }^{\gamma }f(x)\parallel }_{L^2}$.

By the Leibniz formula

$$\sum _{|\gamma |\le s}\parallel {\partial }^{\gamma }(x^{\alpha }{\partial }^{\beta }\phi (x)){\parallel }_{L^2}\le \sum _{|\gamma |\le s}\sum _{\delta \le min\{\gamma ,\alpha \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\gamma }{\delta }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\delta }}\right)\delta !{\parallel x^{\alpha -\delta }{\partial }^{\beta +\gamma -\delta }\phi (x)\parallel }_{L^2}.$$

(17)

Now, $\phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma }),2}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ implies

$$\begin{array}{cc}\hfill \parallel x^{\alpha -\delta }{\partial }^{\beta +\gamma -\delta }{\phi (x)\parallel }_{L^2}& \le C_1{M}_{|\alpha -\delta |}^{\tau ,\sigma }{M}_{|\beta +\gamma -\delta |}^{\tau ,\sigma }\hfill \\ \hfill & \le C_1{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta +s|}^{\tau ,\sigma }\hfill \\ \hfill & \le C_1{M}_{|\alpha |}^{\tau ,\sigma }{C}_2^{{s|\beta +s|}^{\sigma }}{M}_{|\beta |}^{\tau ,\sigma }\hfill \\ \hfill & \le C_1{M}_{|\alpha |}^{\tau ,\sigma }{C}_2^{s{2}^{\sigma -1}{(|\beta |}^{\sigma }+s^{\sigma })}{M}_{|\beta |}^{\tau ,\sigma }\hfill \\ \hfill & \le C_3{M}_{|\alpha |}^{\tau ,\sigma }{h}^{{|\beta |}^{\sigma }}{M}_{|\beta |}^{\tau ,\sigma },\hfill \end{array}$$

(18)

for some $h>0$, $C_j>0$, $j=1,2,3$, which depend on $\tau ,\sigma$ and the dimension n (via s), where we used the inequality

$${(p+q)}^{\sigma }\le 2^{\sigma -1}(p^{\sigma }+q^{\sigma }),p,q>0,\sigma >1.$$

Since the number of terms in the sums in (17) can be estimated by an integer independent on $\alpha$, by (16)–(18), and Theorem 1, it follows that for any $\tau >0$ there is a constant $C>0$ such that

$$\underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },$$

i.e., $\phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$.

It remains to prove the opposite inclusion, ${\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)\subset {\mathit{S}}_{({\mathcal{M}}_{\sigma }),2}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n).$

Let there be given $s>n/4$, so that $A_s=\parallel (1+{|x|}^2{{)}^{-s}\parallel }_{L^2}<\infty$. By using

$${(1+|x|}^2{)}^s=\sum _{|\gamma |\le s}{C}_{\gamma }|x^{2\gamma }|,x\in {\mathbb{R}}^n,$$

where the constants $C_{\gamma }$ can be estimated in terms of s, we obtain

$$\begin{array}{cc}\hfill \parallel x^{\alpha }{\partial }^{\beta }{\phi (x)\parallel }_{L^2}& \le A_s\underset{x}{sup}{(1+|x|}^2{)}^s|x^{\alpha }{\partial }^{\beta }\phi (x)|\hfill \\ \hfill & =A_s\sum _{|\gamma |\le s}{C}_{\gamma }\underset{x}{sup}|x^{\alpha +2\gamma }{\partial }^{\beta }\phi (x)|\hfill \\ \hfill & \le C_1\sum _{|\gamma |\le s}{M}_{|\alpha +2\gamma |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma }\hfill \\ \hfill & \le C_1\sum _{|\gamma |\le s}{C}_2^{|2\gamma |2^{\sigma -1}{(|\alpha |}^{\sigma }+{|2\gamma |}^{\sigma })}{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma }\hfill \\ \hfill & \le C_1{C}_2^{2^{\sigma -1}{(2s)}^{\sigma +1}}\sum _{|\gamma |\le s}{h}^{{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma }\hfill \\ \hfill & \le C{h}^{{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\hfill \end{array}$$

(19)

for some $h>0$ and $C,C_1,C_2>0$ which depend on the dimension n and $\sigma >1$.

Note that Theorem 1 holds true if the $L^{\infty }$-norm in ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$ and ${\mathcal{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$ is replaced by any $L^p$-norm, $1\le p\le \infty$, since in its proof we use only properties of the weight matrix ${\mathcal{M}}_{\sigma }$. This, together with (19), implies that $\phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma }),2}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$. □

Remark 3. 

If the $L^2$-norm in (13) and (14) is replaced by any $L^p$-norm, $1\le p<\infty$, and we denote the corresponding spaces by ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }],p}^{[{\mathcal{M}}_{\sigma }]}$, then, (15) in Theorem 2 extends to

$${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)={\mathit{S}}_{[{\mathcal{M}}_{\sigma }],p}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n),1\le p<\infty .$$

(20)

This can be proved by using the $L^p$-Sobolev embedding, the Hölder inequality, and a modification of the proof of Theorem 2. We refer to [27] where Gelfand–Shilov spaces related to Gevrey sequences are considered.

It is easy to verify that ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$ are closed under differentiation and multiplication by polynomials.

3.2. Nuclearity

In this subsection we prove that the spaces ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$ are nuclear. This is performed by using the general theory given in [15]. We perform nontrivial calculations to show that the general result [15] (Theorem 5.1) can be utilized to prove the nuclearity of ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$.

Recall that the associated function to the sequence $M_p^{\tau ,\sigma }$ is given by

$$T_{\tau ,\sigma }(x)=\underset{p\in {\mathbb{N}}_0}{sup}ln{\displaystyle \frac{x^p}{M_p^{\tau ,\sigma }}},x\ge 0.$$

(21)

Then, by [25] (Proposition 2) it follows that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{A}}{\tau }^{-{\displaystyle \frac{1}{\sigma -1}}}{\displaystyle \frac{{ln}^{\frac{\sigma }{\sigma -1}}(|t|)}{W^{\frac{1}{\sigma -1}}(ln(|t|))}}-B\le T_{\tau ,\sigma }(|t|)\le A{\tau }^{-{\displaystyle \frac{1}{\sigma -1}}}{\displaystyle \frac{{ln}^{\frac{\sigma }{\sigma -1}}(|t|)}{W^{\frac{1}{\sigma -1}}(ln(|t|))}}+B,|t|>1,\end{array}$$

(22)

for suitable constants $A>1$ and $B>0$.

Let there be given $\sigma >1$, and let $w^{\tau }(x):=e^{T_{\tau ,\sigma }(|x|)}$, $x\in {\mathbb{R}}^n$, $\tau >0$. We define the family of functions ${\mathcal{W}}_{\sigma }$ as follows:

$${\mathcal{W}}_{\sigma }=\left\{w^{\tau }:\tau >0\right\}.$$

By ${\mathit{S}}_{\{{\mathcal{W}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$ we denote the set of all $\phi \in C^{\infty }({\mathbb{R}}^n)$ such that

$$(\exists \tau >0)(\exists C>0)(\forall \beta \in {\mathbb{N}}_0^n)\underset{x}{sup}\left|w^{\tau }(x){\partial }^{\beta }\phi (x)\right|\le C{M}_{|\beta |}^{\tau ,\sigma };$$

(23)

and ${\mathit{S}}_{({\mathcal{W}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ consists of $\phi \in C^{\infty }({\mathbb{R}}^n)$ such that

$$(\forall \tau >0)(\exists C>0)(\forall \beta \in {\mathbb{N}}_0^n)\underset{x}{sup}\left|w^{\tau }(x){\partial }^{\beta }\phi (x)\right|\le C{M}_{|\beta |}^{\tau ,\sigma }.$$

Again, we use ${\mathit{S}}_{[{\mathcal{W}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}$ to denote ${\mathit{S}}_{\{{\mathcal{W}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$ or ${\mathit{S}}_{({\mathcal{W}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n).$

The spaces ${\mathit{S}}_{\{{\mathcal{W}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$ and ${\mathit{S}}_{({\mathcal{W}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n).$ are an $(LB)$- space and a Fréchet space, respectively, with topologies introduced analogously to those of ${\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$ and ${\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$.

Now, we can prove the main result of this section.

Theorem 3. 

Fix $\sigma >1$ and let ${\mathcal{M}}_{\sigma }$ be the weight matrix given by (3). Then:

1.

${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)={\mathit{S}}_{[{\mathcal{W}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$, as locally convex vector spaces;

2.

${\mathit{S}}_{[{\mathcal{W}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$ is a nuclear space.

Proof. 

We prove the Roumieu case here, and the Beurling case follows using similar arguments.

1. Since the sequence ${(M_p^{\tau ,\sigma })}_{p\in {\mathbb{N}}_0}$ given by (2) satisfies $(M.1)$, by [12] (Proposition 3.2) it follows that

$$M_p^{\tau ,\sigma }=\underset{|x|>0}{sup}{\displaystyle \frac{{|x|}^p}{e^{T_{\tau ,\sigma }(|x|)}}},p\in \mathbb{N}$$

(24)

for all $\tau >0$ and $\sigma >1$.

Let $\phi \in {\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$. Then, there exist ${\tau }_0,C_1>0$ such that

$$\underset{x\in {\mathbb{R}}^n}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C_1{M}_{|\alpha |}^{{\tau }_0,\sigma }{M}_{|\beta |}^{{\tau }_0,\sigma },$$

and by the simple inequality ${\displaystyle {|x|}^{|\alpha |}\le n^{|\alpha |}\sum _{|\gamma |=|\alpha |}|x^{\gamma }|}$, we get

$${|x|}^{|\alpha |}|{\partial }^{\beta }\phi (x)|\le n^{|\alpha |}\sum _{|\gamma |=|\alpha |}|x^{\gamma }{\partial }^{\beta }\phi (x)|\le 2^{n-1}{C}_1{\displaystyle \frac{{(2n)}^{|\alpha |}}{M_{|\alpha |}^{{\tau }_0,\sigma }}}{M}_{|\alpha |}^{2{\tau }_0,\sigma }{M}_{|\beta |}^{2{\tau }_0,\sigma }\le C{M}_{|\alpha |}^{2{\tau }_0,\sigma }{M}_{|\beta |}^{2{\tau }_0,\sigma },$$

for $\alpha ,\beta \in {\mathbb{N}}_0^n,$ $x\in {\mathbb{R}}^n$, where $C=2^{n-1}{C}_1{e}^{T_{{\tau }_0,\sigma }(2n)}$ and for the second inequality we used that the number of terms in sum does not exceed $2^{n+|\alpha |-1}$. Hence, we obtain

$$|{\partial }^{\beta }\phi (x)|\le C{M}_{|\beta |}^{2{\tau }_0,\sigma }\underset{\alpha }{inf}{\displaystyle \frac{M_{|\alpha |}^{2{\tau }_0,\sigma }}{{|x|}^{|\alpha |}}}=C{M}_{|\beta |}^{2{\tau }_0,\sigma }{e}^{-T_{2{\tau }_0,\sigma }(|x|)},\beta \in {\mathbb{N}}_0^n,x\in {\mathbb{R}}^n,$$

(25)

i.e., (23) holds with $\tau =2{\tau }_0$.

For the opposite direction, we note that $\phi \in {\mathit{S}}_{\{{\mathcal{W}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$, i.e., (23), together with (24) and $|x^{\alpha }{|\le |x|}^{|\alpha |}$, implies

$$|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C{M}_{|\beta |}^{{\tau }_0,\sigma }{|x|}^{|\alpha |}{e}^{-T_{{\tau }_0,\sigma }(|x|)}\le C{M}_{|\beta |}^{{\tau }_0,\sigma }{M}_{|\alpha |}^{{\tau }_0,\sigma },x\in {\mathbb{R}}^n,\alpha ,\beta \in {\mathbb{N}}_0^n,$$

for some $C,{\tau }_0>0$. Thus, $\phi \in {\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$, which proves 1.

2. By [15] (Theorem 5.1), it follows that ${\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)={\mathit{S}}_{\{{\mathcal{W}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)$ is a nuclear space if ${\mathcal{M}}_{\sigma }$ and ${\mathcal{W}}_{\sigma }$ satisfy the following conditions:

$$\begin{array}{cc}\hfill \{L\}& (\forall h>0)(\forall \tau >0)(\exists {\tau }_0,C>0)(\forall \alpha \in {\mathbb{N}}^n)\hfill \\ & h^{|\alpha |}{M}_{|\alpha |}^{\tau ,\sigma }\le C{M}_{|\alpha |}^{{\tau }_0,\sigma },\hfill \\ \hfill {\{\mathcal{M}.2\}}^{\prime }& (\forall \tau >0)(\exists {\tau }_0>0)(\exists h,C>0)(\forall \alpha \in {\mathbb{N}}^n)\hfill \\ & M_{|\alpha |+1}^{\tau ,\sigma }\le C{h}^{|\alpha |}{M}_{|\alpha |}^{{\tau }_0,\sigma },\hfill \\ \hfill \{M\}& (\forall {\tau }_1,{\tau }_2>0)(\exists {\tau }_0>0)(\exists C>0)(\forall x,y\in {\mathbb{R}}^n)\hfill \\ & w^{{\tau }_0}(x+y)\le C{w}^{{\tau }_1}(x)w^{{\tau }_2}(y),\hfill \\ \hfill \{N\}& (\forall \tau >0)(\exists {\tau }_0>0)\hfill \\ & w^{{\tau }_0}/w^{\tau }\in L^1,\hfill \end{array}$$

where $w^{\tau }(x)=e^{T_{\tau ,\sigma }(|x|)}$. In fact, $\{M\}$ is a stronger condition than $\{wM\}$, which is assumed in [15] (Theorem 5.1).

We note that $\{L\}$ follows from Lemma 2, i.e., from (10).

For ${\{\mathcal{M}.2\}}^{\prime }$, we take arbitrary $\tau >0$. Then, by ${\widetilde{(M.2)}}^{\prime }$ we have

$$M_{|\alpha |+1}^{\tau ,\sigma }\le A^{{|\alpha |}^{\sigma }+1}{M}_{|\alpha |}^{\tau ,\sigma }=A{\displaystyle \frac{A^{{|\alpha |}^{\sigma }}}{M_{|\alpha |}^{\tau ,\sigma }}}{M}_{|\alpha |}^{2\tau ,\sigma }\le C{M}_{|\alpha |}^{{\tau }_0,\sigma },$$

for ${\tau }_0=2\tau$ and some $C>0$, where we used (9).

To verify conditions $\{M\}$ and $\{N\}$, put

$${\displaystyle {\omega }_{\sigma }(x)=\frac{{ln}^{\frac{\sigma }{\sigma -1}}(1+|x|)}{W^{\frac{1}{\sigma -1}}(ln(1+|x|))},x\in {\mathbb{R}}^n\setminus \{0\},{\omega }_{\sigma }(0)=0,}$$

and consider

$${\mathcal{W}}_{\sigma }^{\prime }=\left\{{\omega }^{\lambda }:=e^{{\displaystyle \frac{1}{\lambda }}{\omega }_{\sigma }}:\lambda >0\right\}.$$

By (22), we obtain

$$(\forall \tau >0)(\exists {\lambda }_1,{\lambda }_2>0)(\exists C_1,C_2>0)(\forall x\in {\mathbb{R}}^n)C_1{\omega }^{{\lambda }_1}(x)\le w^{\tau }(x)\le C_2{\omega }^{{\lambda }_2}(x),$$

and conversely

$$(\forall \lambda >0)(\exists {\tau }_1,{\tau }_2>0)(\exists C_1,C_2>0)(\forall x\in {\mathbb{R}}^n)C_1{w}^{{\tau }_1}(x)\le {\omega }^{\lambda }(x)\le C_2{w}^{{\tau }_2}(x).$$

Thus, it is sufficient to check that ${\mathcal{W}}_{\sigma }^{\prime }$ satisfies $\{M\}$ and $\{N\}$.

Indeed, since ${\omega }_{\sigma }(x)$ is equivalent to a BMT weight function (see Remark 1), it satisfies BMT conditions $(\alpha )-(\gamma )$. Then, ref. [15] (Lemma 3.5) implies that ${\mathcal{W}}_{\sigma }^{\prime }$ satisfies conditions $\{M\}$ and $\{N\}$, and the theorem is proved. Alternatively, condition $\{M\}$ can be proved using (4) and [17] (Lemma 2.12). □

Remark 4. 

Note that Theorem 2 and its extension (20) follow from Theorem 3 (1.) and [15] (Theorem 5.7). However, the proof of Theorem 2 given here employs techniques that differ from those used in [15].

4. Fourier Transform Representation

The Gelfand–Shilov spaces given by (1) with $t=s>0$ are Fourier transform invariant, which is a convenient property in different applications. In this section, we address the Fourier transform invariance of ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$, and their characterizations in terms of the Fourier transform.

The Fourier transform of $f\in {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$ is given by

$$\begin{array}{c}\hfill (\mathcal{F}f)(\xi )=\widehat{f}(\xi )={\int }_{{\mathbb{R}}^n}f(x)e^{-2\pi ix\xi }dx,\xi \in {\mathbb{R}}^n,\end{array}$$

and the corresponding inverse Fourier transform is defined as

$$\begin{array}{c}\hfill ({\mathcal{F}}^{-1}f)(x)={\int }_{{\mathbb{R}}^n}f(\xi )e^{2\pi ix\xi }d\xi ,x\in {\mathbb{R}}^n.\end{array}$$

The invariance of the classical Gelfand–Shilov spaces under the Fourier transform is already given in [1]; see also a more recent source [7] (Chapter 6). The Fourier transform invariance for the Gelfand–Shilov type spaces related to the extended Gevrey regularity can be stated as follows.

Theorem 4. 

Let $\sigma >1$, and let the weight matrix ${\mathcal{M}}_{\sigma }$ be given by (3), $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $M_0^{\tau ,\sigma }=1$. Then,

$$\mathcal{F}({\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]})({\mathbb{R}}^n)={\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n).$$

Proof. 

We give the proof for the Beurling case, since the Roumieu case can be proved in a similar way.

Let $\phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ and let there be given $\tau >0$.

From the Leibniz formula and (4) we obtain

$$\begin{array}{cc}\hfill |{\partial }^{\beta }(x^{\alpha }\phi (x))|& =\left|\sum _{\delta \le min\{\alpha ,\beta \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\delta }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\delta }}\right)\delta !x^{\alpha -\delta }{\partial }^{\beta -\delta }\phi (x)\right|\hfill \\ \hfill & \le C\sum _{\delta \le min\{\alpha ,\beta \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\delta }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\delta }}\right)\delta !M_{|\alpha -\delta |}^{{\displaystyle \frac{\tau }{2}},\sigma }{M}_{|\beta -\delta |}^{{\displaystyle \frac{\tau }{2}},\sigma }\hfill \\ \hfill & \le C{M}_{|\alpha |}^{{\displaystyle \frac{\tau }{2}},\sigma }{M}_{|\beta |}^{{\displaystyle \frac{\tau }{2}},\sigma }\sum _{\delta \le min\{\alpha ,\beta \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\delta }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\delta }}\right){\displaystyle \frac{\delta !}{{(M_{|\delta |}^{\frac{\tau }{2},\sigma })}^2}}.\hfill \end{array}$$

Next, by using the fact that for any $\tau >0$ there exists $C_1>0$ such that

$$\delta !\le C_1{M}_{|\delta |}^{\tau ,\sigma },\delta \in {\mathbb{N}}_0^n,$$

(26)

(which follows from, e.g., [11] (4)), we obtain

$$\begin{array}{cc}\hfill |{\partial }^{\beta }(x^{\alpha }\phi (x))|& \le C{M}_{|\alpha |}^{{\displaystyle \frac{\tau }{2}},\sigma }{M}_{|\beta |}^{{\displaystyle \frac{\tau }{2}},\sigma }{C}_1\sum _{\delta \le min\{\alpha ,\beta \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\delta }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\delta }}\right)\hfill \\ \hfill & \le C_2{2}^{|\alpha |+|\beta |}{M}_{|\alpha |}^{{\displaystyle \frac{\tau }{2}},\sigma }{M}_{|\beta |}^{{\displaystyle \frac{\tau }{2}},\sigma }\le C_3{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\alpha ,\beta \in {\mathbb{N}}_0^n,\hfill \end{array}$$

for some $C_2,C_3>0$, where in the last inequality we applied (10). Similarly,

$$\begin{array}{c}\hfill {|x|}^2|{\partial }^{\beta }(x^{\alpha }\phi (x))|\le C{2}^{|\alpha |+2+|\beta |}{M}_{|\alpha |+2}^{{\displaystyle \frac{\tau }{2}},\sigma }{M}_{|\beta |}^{{\displaystyle \frac{\tau }{2}},\sigma },\alpha ,\beta \in {\mathbb{N}}_0^n,\end{array}$$

for some $C>0$. Applying $\widetilde{{(M.2)}^{\prime }}$ twice and then using (10) again, we obtain

$$\begin{array}{cc}\hfill {|x|}^2|{\partial }^{\beta }(x^{\alpha }\phi (x))|& \le C{2}^{|\alpha |+2+|\beta |}{A}^{{(|\alpha |+1)}^{\sigma }+{|\alpha |}^{\sigma }}{M}_{|\alpha |}^{{\displaystyle \frac{\tau }{2}},\sigma }{M}_{|\beta |}^{{\displaystyle \frac{\tau }{2}},\sigma }\hfill \\ \hfill & \le C_4{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\alpha ,\beta \in {\mathbb{N}}_0^n,\hfill \end{array}$$

for some $C_4>0$. Finally,

$$\begin{array}{cc}\hfill |{\xi }^{\beta }\mathcal{F}{(\phi )}^{(\alpha )}(\xi )|& \le \left|\mathcal{F}\left({\partial }^{\beta }(x^{\alpha }\phi (x))\right)(\xi )\right|=\left|{\int }_{{\mathbb{R}}^n}{\partial }^{\beta }(x^{\alpha }\phi (x))e^{-2\pi ix\xi }dx\right|\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^n}|{\partial }^{\beta }(x^{\alpha }\phi (x))|dx={\int }_{{\mathbb{R}}^n}{\displaystyle \frac{|(1+{|x|}^2){\partial }^{\beta }(x^{\alpha }\phi (x))|}{1+{|x|}^2}}dx\hfill \\ \hfill & \le C{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\alpha ,\beta \in {\mathbb{N}}_0^n,\hfill \end{array}$$

for some $C>0$. So, $\mathcal{F}({\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n))\subseteq {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n).$ The other inclusion can be obtained in the same way since $\mathcal{F}(\mathcal{F}\phi )(x)=\phi (-x).$ □

Remark 5. 

As in [1], we may also consider separate conditions on decay and regularity, and introduce the following Gelfand–Shilov spaces related to ${\mathcal{M}}_{\sigma }$, $\sigma >1$, given by (3). Let $\phi \in C^{\infty }({\mathbb{R}}^n)$. Then,

$$\begin{array}{cc}\hfill \phi \in {\mathit{S}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)\iff & (\exists \tau >0)(\forall \alpha \in {\mathbb{N}}_0^n)(\exists C_{\alpha }>0)(\forall \beta \in {\mathbb{N}}_0^n)\hfill \\ \hfill & \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C_{\alpha }{M}_{|\beta |}^{\tau ,\sigma };\hfill \\ \hfill \phi \in {\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n)\iff & (\exists \tau >0)(\forall \beta \in {\mathbb{N}}_0^n)(\exists C_{\beta }>0)(\forall \alpha \in {\mathbb{N}}_0^n)\hfill \\ \hfill & \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C_{\beta }{M}_{|\alpha |}^{\tau ,\sigma },\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \phi \in {\mathit{S}}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)\iff & (\forall \tau >0)(\forall \alpha \in {\mathbb{N}}_0^n)(\exists C_{\alpha }>0)(\forall \beta \in {\mathbb{N}}_0^n)\hfill \\ \hfill & \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C_{\alpha }{M}_{|\beta |}^{\tau ,\sigma };\hfill \\ \hfill \phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)\iff & (\forall \tau >0)(\forall \beta \in {\mathbb{N}}_0^n)(\exists C_{\beta }>0)(\forall \alpha \in {\mathbb{N}}_0^n)\hfill \\ \hfill & \underset{x}{sup}|x^{\alpha }{\partial }^{\beta }\phi (x)|\le C_{\beta }{M}_{|\alpha |}^{\tau ,\sigma }.\hfill \end{array}$$

From the proof of Theorem 4, it follows that $\mathcal{F}({\mathit{S}}_{[{\mathcal{M}}_{\sigma }]})({\mathbb{R}}^n)={\mathit{S}}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$, and $\mathcal{F}({\mathit{S}}^{[{\mathcal{M}}_{\sigma }]})({\mathbb{R}}^n)={\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n).$

The beautiful symmetric characterizations of Gelfand–Shilov spaces given in [18] can be formulated in terms of the extended Gevrey regularity as follows.

Theorem 5. 

Let $\sigma >1$, and let the weight matrix ${\mathcal{M}}_{\sigma }$ be given by (3), $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $M_0^{\tau ,\sigma }=1$. If $\phi \in C^{\infty }({\mathbb{R}}^n)$, then the following conditions are equivalent:

1.

$\phi \in {\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n);$

2.

$(\exists \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)$

$$\underset{x}{sup}|x^{\alpha }\phi (x)|\le C{M}_{|\alpha |}^{\tau ,\sigma }\mathrm{and}\underset{x}{sup}|{\partial }^{\beta }\phi (x)|\le C{M}_{|\beta |}^{\tau ,\sigma };$$

3.

$(\exists \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)$

$$\underset{x}{sup}|x^{\alpha }\phi (x)|\le C{M}_{|\alpha |}^{\tau ,\sigma }\mathrm{and}\underset{\xi }{sup}|{\xi }^{\beta }\widehat{\phi }(\xi )|\le C{M}_{|\beta |}^{\tau ,\sigma };$$

4.

$(\exists \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)$

$$\underset{\xi }{sup}|{\partial }^{\alpha }\widehat{\phi }(\xi )|\le C{M}_{|\alpha |}^{\tau ,\sigma }\mathrm{and}\underset{\xi }{sup}|{\xi }^{\beta }\widehat{\phi }(\xi )|\le C{M}_{|\beta |}^{\tau ,\sigma }.$$

Proof. 

1. ⇒ 2., 1. ⇒ 3., and 1. ⇒ 4. follow immediately from Definition 2 and Theorem 4.

2. ⇒ 1. By Theorem 2, we can use the $L^2$-norm instead of the $L^{\infty }$-norm, cf. (13). Using integration by parts, the Leibniz formula and the Schwarz inequality we obtain that there exist $\tau >0$ and $C_1>0$ such that

$$\begin{array}{cc}\hfill \parallel x^{\alpha }{\partial }^{\beta }{\phi (x)\parallel }_{L^2}^2& =|{({\partial }^{\beta }(x^{2\alpha }{\partial }^{\beta }\phi ),\phi )}_{L_2}|\hfill \\ \hfill & \le \sum _{\gamma \le min\{2\alpha ,\beta \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2\alpha }{\gamma }}\right)\gamma !\parallel {\partial }^{2\beta -\gamma }{\phi (x)\parallel }_{L^2}{\parallel x^{2\alpha -\gamma }\phi (x)\parallel }_{L^2}\hfill \\ \hfill & \le C_1\sum _{\gamma \le min\{2\alpha ,\beta \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2\alpha }{\gamma }}\right)\gamma !M_{|2\beta -\gamma |}^{\tau ,\sigma }{M}_{|2\alpha -\gamma |}^{\tau ,\sigma },\hfill \end{array}$$

and, by using (4) we obtain

$$\begin{array}{cc}\hfill \parallel x^{\alpha }{\partial }^{\beta }{\phi (x)\parallel }_{L^2}^2& \le C_1{M}_{|2\alpha |}^{\tau ,\sigma }{M}_{|2\beta |}^{\tau ,\sigma }\sum _{\gamma \le min\{2\alpha ,\beta \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2\alpha }{\gamma }}\right){\displaystyle \frac{\gamma !}{{(M_{|\gamma |}^{\tau ,\sigma })}^2}},\alpha ,\beta \in {\mathbb{N}}_0^n.\hfill \end{array}$$

Next, by (26), it follows that

$$\begin{array}{cc}\hfill \sum _{\gamma \le min\{2\alpha ,\beta \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2\alpha }{\gamma }}\right){\displaystyle \frac{\gamma !}{{(M_{|\gamma |}^{\tau ,\sigma })}^2}}& \le C_2\sum _{\gamma \le min\{2\alpha ,\beta \}}\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2\alpha }{\gamma }}\right)\hfill \\ \hfill & \le C_2{2}^{|2\alpha |+|\beta |}\le C_2{2}^{{|2\alpha |}^{\sigma }+{|2\beta |}^{\sigma }},\hfill \end{array}$$

for some $C_2>0$, so that

$$\begin{array}{cc}\hfill \parallel x^{\alpha }{\partial }^{\beta }{\phi (x)\parallel }_{L^2}^2& \le C_1{C}_2{2}^{{|2\alpha |}^{\sigma }+{|2\beta |}^{\sigma }}{M}_{|2\alpha |}^{\tau ,\sigma }{M}_{|2\beta |}^{\tau ,\sigma },\alpha ,\beta \in {\mathbb{N}}_0^n.\hfill \end{array}$$

Now, using $\widetilde{(M.2)},$ we obtain that there exists $A>0$ such that

$$\begin{array}{cc}\hfill \parallel x^{\alpha }{\partial }^{\beta }{\phi (x)\parallel }_{L^2}^2& \le C_1{C}_2{2}^{{|2\alpha |}^{\sigma }+{|2\beta |}^{\sigma }}{A}^{{2|\alpha |}^{\sigma }}{(M_{|\alpha |}^{\tau 2^{\sigma -1},\sigma })}^2{A}^{{2|\beta |}^{\sigma }}{(M_{|\beta |}^{\tau 2^{\sigma -1},\sigma })}^2\hfill \\ \hfill & \le C_1{C}_2{(2A)}^{{|2\alpha |}^{\sigma }+{|2\beta |}^{\sigma }}{(M_{|\alpha |}^{\tau 2^{\sigma -1},\sigma })}^2{(M_{|\beta |}^{\tau 2^{\sigma -1},\sigma })}^2,\alpha ,\beta \in {\mathbb{N}}_0^n.\hfill \end{array}$$

Finally, by (10) there exists $C_3>0$ such that

$$\begin{array}{cc}\hfill \parallel x^{\alpha }{\partial }^{\beta }{\phi (x)\parallel }_{L^2}& \le \sqrt{C_1{C}_2}{({(2A)}^{2^{\sigma -1}})}^{{|\alpha |}^{\sigma }+{|\beta |}^{\sigma }}{M}_{|\alpha |}^{\tau 2^{\sigma -1},\sigma }{M}_{|\beta |}^{\tau 2^{\sigma -1},\sigma }\hfill \\ \hfill & \le C_3{M}_{|\alpha |}^{\tau 2^{\sigma },\sigma }{M}_{|\beta |}^{\tau 2^{\sigma },\sigma },\alpha ,\beta \in {\mathbb{N}}_0^n,\hfill \end{array}$$

which implies (13).

Now, we prove that 3. ⇒ 2. Let $\tau >0$ and $C>0$ be as in 3. Reasoning in the similar way as in (25), note that the inequality $|{\xi }^{\beta }\widehat{\phi }(\xi )|\le C{M}_{|\beta |}^{\tau ,\sigma },$ $\xi \in {\mathbb{R}}^n,$$\beta \in {\mathbb{N}}_0^n,$ implies

$$|\widehat{\phi }(\xi )|\le C_1{e}^{-T_{2\tau ,\sigma }(|\xi |)},\xi \in {\mathbb{R}}^n,$$

(27)

for some $C_1>0$, where $T_{\tau ,\sigma }$ is the associated function to ${(M_{\alpha }^{\tau ,\sigma })}_{\alpha \in {\mathbb{N}}_0^n}$; see (21).

By using (27) and the fact that ${\displaystyle {\int }_{{\mathbb{R}}^n}{e}^{-\frac{1}{2}{T}_{2\tau ,\sigma }(|\xi |)}d\xi <\infty }$ (the integrand decreases faster than ${|\xi |}^{-p}$ for any $p\in {\mathbb{N}}_0$), we obtain

$$\begin{array}{cc}\hfill |{\partial }^{\beta }\phi (x)|& \le {(2\pi )}^{|\beta |}{\int }_{{\mathbb{R}}^n}|e^{2\pi ix\xi }{\xi }^{\beta }\widehat{\phi }(\xi )|d\xi \hfill \\ \hfill & \le C_1{(2\pi )}^{|\beta |}{\int }_{{\mathbb{R}}^n}{|\xi |}^{|\beta |}{e}^{-T_{2\tau ,\sigma }(|\xi |)}d\xi \hfill \\ \hfill & \le C_1{(2\pi )}^{|\beta |}\underset{\xi }{sup}{[|\xi |}^{2|\beta |}{e}^{-T_{2\tau ,\sigma }(|\xi |)}{]}^{1/2}{\int }_{{\mathbb{R}}^n}{e}^{-{\displaystyle \frac{1}{2}}{T}_{2\tau ,\sigma }(|\xi |)}d\xi \hfill \\ \hfill & \le C_2{(2\pi )}^{|\beta |}\underset{\xi }{sup}{[|\xi |}^{2|\beta |}{e}^{-T_{2\tau ,\sigma }(|\xi |)}{]}^{1/2}\hfill \\ \hfill & \le C_2{(2\pi )}^{|\beta |}{[M_{|2\beta |}^{2\tau ,\sigma }]}^{1/2},\beta \in {\mathbb{N}}_0^n,\hfill \end{array}$$

for some $C_2>0$.

Finally, $\widetilde{(M.2)}$ and (10) yield

$$\begin{array}{cc}\hfill |{\partial }^{\beta }\phi (x)|& \le C_2{(A^{{2|\beta |}^{\sigma }}{(M_{|\beta |}^{\tau 2^{\sigma },\sigma })}^2)}^{1/2}=C_2{A}^{{|\beta |}^{\sigma }}{M}_{|\beta |}^{\tau 2^{\sigma },\sigma }\hfill \\ \hfill & \le C_3{M}_{|\beta |}^{\tau 2^{\sigma +1},\sigma },\beta \in {\mathbb{N}}_0^n,\hfill \end{array}$$

for some $A,C_3>0$, which implies 2.

The remaining part 4. ⇒ 1. follows from 1.⇔ 2. and Theorem 4. □

Theorem 5 implies the following characterization in terms of the associated functions.

Corollary 1. 

Let $\sigma >1$, and let the weight matrix ${\mathcal{M}}_{\sigma }$ be given by (3), $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $M_0^{\tau ,\sigma }=1$. If $\phi \in C^{\infty }({\mathbb{R}}^n)$, then the following conditions are equivalent:

1.

$\phi \in {\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^n);$

2.

there exist $\tau >0$ such that

$$\underset{x}{sup}|\phi (x)|exp{T}_{\tau ,\sigma }(|x|)<\infty \mathrm{and}\underset{\xi }{sup}|\widehat{\phi }(\xi )|exp{T}_{\tau ,\sigma }(|\xi |)<\infty ,$$

where $T_{\tau ,\sigma }$ denotes the associated function to ${(M_{\alpha }^{\tau ,\sigma })}_{\alpha \in {\mathbb{N}}_0^n}$ given in (21).

A similar theorem can be shown for the Beurling case.

Theorem 6. 

Let $\sigma >1,$ and let the weight matrix ${\mathcal{M}}_{\sigma }$ be given by (3), $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $M_0^{\tau ,\sigma }=1$. Furthermore, let $T_{\tau ,\sigma }$ be the associated function to ${(M_{\alpha }^{\tau ,\sigma })}_{\alpha \in {\mathbb{N}}_0^n}$; see (21). If $\phi \in C^{\infty }({\mathbb{R}}^n)$, then the following conditions are equivalent:

1.

$\phi \in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n);$

2.

$(\forall \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)$

$$\underset{x}{sup}|x^{\alpha }\phi (x)|\le C{M}_{|\alpha |}^{\tau ,\sigma }\mathrm{and}\underset{x}{sup}|{\partial }^{\beta }\phi (x)|\le C{M}_{|\beta |}^{\tau ,\sigma };$$

3.

$(\forall \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)$

$$\underset{x}{sup}|x^{\alpha }\phi (x)|\le C{M}_{|\alpha |}^{\tau ,\sigma }\mathrm{and}\underset{\xi }{sup}|{\xi }^{\beta }\widehat{\phi }(\xi )|\le C{M}_{|\beta |}^{\tau ,\sigma };$$

4.

$(\forall \tau >0)(\exists C>0)(\forall \alpha ,\beta \in {\mathbb{N}}_0^n)$

$$\underset{\xi }{sup}|{\partial }^{\alpha }\widehat{\phi }(\xi )|\le C{M}_{|\alpha |}^{\tau ,\sigma }\mathrm{and}\underset{\xi }{sup}|{\xi }^{\beta }\widehat{\phi }(\xi )|\le C{M}_{|\beta |}^{\tau ,\sigma };$$

5.

$(\forall \tau >0)$

$$\underset{x}{sup}|\phi (x)|exp{T}_{\tau ,\sigma }(|x|)<\infty \mathrm{and}\underset{\xi }{sup}|\widehat{\phi }(\xi )|exp{T}_{\tau ,\sigma }(|\xi |)<\infty .$$

Remark 6. 

Part of the proof of Theorems 5 and 6 (when showing that 2. implies 1.) establishes the inclusion ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)\cap {\mathit{S}}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)\subseteq {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n).$ This, together with the obvious inclusion

$${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)\subseteq {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)\cap {\mathit{S}}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n),$$

implies

$${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)={\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)\cap {\mathit{S}}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n).$$

5. Time–Frequency Representations

The Fourier transform of $f\in L^2({\mathbb{R}}^n)$ provides information about its global frequency content. Different representations of f in phase space or time–frequency plane are used to obtain a localized version of the Fourier transform. Notable examples are the short-time Fourier transform, the Wigner transform, and Cohen’s class representations such as the Born–Jordan transform.

In this section we turn our attention to the Grossmann–Royer operators which originated from the problem of the physical interpretation of the Wigner distribution; see [28,29]. It tuned out that the Grossmann–Royer operators are closely related to the Heisenberg operators, well-known objects from quantum mechanics; see [30,31].

The Grossmann–Royer operator $R:L^2({\mathbb{R}}^n)\to L^2({\mathbb{R}}^{2n})$ is given by

$$Rf(x,\omega )=R(f(t))(x,\omega )=e^{4\pi i\omega (t-x)}f(2x-t),f\in L^2({\mathbb{R}}^n),x,\omega \in {\mathbb{R}}^n.$$

We define the Grossmann–Royer transform $R_{g}f$ as the weak sense interpretation of the Grossmann–Royer operator. The Grossmann–Royer transform is essentially the cross-Wigner distribution $W(f,g)$ (see Definition 3, Lemma 3 and [31] (Definition 12)) and is also closely related to $V_{g}f$ and $A(f,g),$ the short-time Fourier transform, and the cross-ambiguity transform, respectively; see Definition 3.

Definition 3. 

Let there be given $f,g\in L^2({\mathbb{R}}^n).$ The Grossmann–Royer transform of f and g is given by

$$R_{g}f(x,\omega )=R(f,g)(x,\omega )=\langle Rf,g\rangle =\int e^{4\pi i\omega (t-x)}f(2x-t)\overline{g(t)}dt,x,\omega \in {\mathbb{R}}^n.$$

The short-time Fourier transform of f with respect to the window g is given by

$$V_{g}f(x,\omega )=\int e^{-2\pi it\omega }f(t)\overline{g(t-x)}dt,x,\omega \in {\mathbb{R}}^n.$$

The cross-Wigner distribution of f and g is

$$W(f,g)(x,\omega )=\int e^{-2\pi i\omega t}f(x+{\displaystyle \frac{t}{2}})\overline{g(x-{\displaystyle \frac{t}{2}})}dt,x,\omega \in {\mathbb{R}}^n,$$

and the cross-ambiguity function of f and g is

$$A(f,g)(x,\omega )=\int e^{-2\pi i\omega t}f(t+{\displaystyle \frac{x}{2}})\overline{g(t-{\displaystyle \frac{x}{2}})}dt,x,\omega \in {\mathbb{R}}^n.$$

The transforms in Definition 3 are called time–frequency representations and we set

$$TFR(f,g)=\{R_{g}f,V_{g}f,W(f,g),A(f,g)\}.$$

Note that $R(f(t))(x,\omega )=e^{-4\pi i\omega x}{M}_{2\omega }{(T_{2x}f)}^{ˇ}(t)$ so that

$$R_{g}f(x,\omega )=e^{-4\pi i\omega x}\langle M_{2\omega }{(T_{2x}f)}^{ˇ},g\rangle ,$$

where translation and modulation operators are respectively given by

$$T_{x}f(·)=f(·-x)and{M}_{x}f(·)=e^{2\pi ix·}f(·),x\in {\mathbb{R}}^n,$$

and $\check{f}$ denotes the reflection $\check{f}(x)=f(-x)$.

By using an appropriate change in variables, the following relations between the time–frequency representations can be obtained (cf. [20]).

Lemma 3. 

Let $f,g\in L^2({\mathbb{R}}^n).$ Then, we have:

$$\begin{array}{ccc}\hfill W(f,g)(x,\omega )& =& 2^n{R}_{g}f(x,\omega ),\hfill \\ \hfill V_{g}f(x,\omega )& =& e^{-\pi ix\omega }{R}_{\check{g}}f\left({\displaystyle \frac{x}{2}},{\displaystyle \frac{\omega }{2}}\right),\hfill \\ \hfill A(f,g)(x,\omega )& =& R_{\check{g}}f\left({\displaystyle \frac{x}{2}},{\displaystyle \frac{\omega }{2}}\right),x,\omega \in {\mathbb{R}}^n.\hfill \end{array}$$

The Grossmann–Royer operator is self-adjoint, uniformly continuous on ${\mathbb{R}}^{2n}$, and the following properties hold:

• $\parallel R_g{f\parallel }_{L^{\infty }}\le {\parallel f\parallel }_{L^2}{\parallel g\parallel }_{L^2}$;
• $R_{g}f=\overline{R_{f}g}$;
• $R_{\widehat{g}}\widehat{f}(x,\omega )=R_{g}f(-\omega ,x),$$x,\omega \in {\mathbb{R}}^n$;
• For $f_1,f_2,g_1,g_2\in L^2({\mathbb{R}}^d),$ the Moyal identity holds:

  $$\langle R_{g_1}{f}_1,R_{g_2}{f}_2\rangle =\langle f_1,f_2\rangle \overline{\langle g_1,g_2\rangle }.$$

The Moyal identity implies the inversion formula:

$$f={\displaystyle \frac{1}{\langle g_2,g_1\rangle }}\int \int R_{g_1}f(x,\omega )R{g}_2(x,\omega )dxd\omega .$$

(28)

We refer to [32] for details (explained in terms of the short-time Fourier transform).

Clearly, a smooth function F belongs to ${\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^{2n})$ if

$$(\forall \tau >0)(\exists C>0)\underset{x,\omega }{sup}|x^{{\alpha }_1}{\omega }^{{\alpha }_2}{\partial }_x^{{\beta }_1}{\partial }_{\omega }^{{\beta }_2}F(x,\omega )|\le C{M}_{|{\alpha }_1+{\alpha }_2|}^{\tau ,\sigma }{M}_{|{\beta }_1+{\beta }_2|}^{\tau ,\sigma },$$

for all ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\in {\mathbb{N}}_0^n$, and similarly, $F\in {\mathit{S}}_{\{{\mathcal{M}}_{\sigma }\}}^{\{{\mathcal{M}}_{\sigma }\}}({\mathbb{R}}^{2n})$ if

$$(\exists \tau >0)(\exists C>0)\underset{x,\omega }{sup}|x^{{\alpha }_1}{\omega }^{{\alpha }_2}{\partial }_x^{{\beta }_1}{\partial }_{\omega }^{{\beta }_2}F(x,\omega )|\le C{M}_{|{\alpha }_1+{\alpha }_2|}^{\tau ,\sigma }{M}_{|{\beta }_1+{\beta }_2|}^{\tau ,\sigma }.$$

If $F\in {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^{2n})$, then by performing calculations similar to those presented in the proof of Theorem 4, it follows that ${\mathcal{F}}_{1}F,{\mathcal{F}}_{2}F\in {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^{2n})$, where ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ denote partial Fourier transforms with respect to x and $\omega ,$ respectively.

The following theorem, in the context of classical Gelfand–Shilov spaces, is considered folklore (see, for example, [19,20,21,33]).

Theorem 7. 

Let $\sigma >1$, and let the weight matrix ${\mathcal{M}}_{\sigma }$ be given by (3), $M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}$, $p\in \mathbb{N}$, $M_0^{\tau ,\sigma }=1$. If $f,g\in {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$, and $T\in TFR(f,g)$, then $T\in {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^{2n})$.

Conversely, if $T\in TFR(f,g)$ and $T\in {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^{2n})$, then $f,g\in {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n).$

Proof. 

Since ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^n)$ are closed under reflections, dilations, and modulations, then by Lemma 3, since all elements in the set $TFR(f,g)$ are related and linked by these operations, it suffices to give the proof for some arbitrary $T\in TFR(f,g)$. We will focus on the Grossmann–Royer transform $R_{g}f$ and the same conclusion holds then for all other time–frequency representations in question.

Put $\mathrm{\Phi }(x,t):=f(2x-t)\overline{g(t)}$, $x,t\in {\mathbb{R}}^n$. Then,

$$R_{g}f(x,\omega )=e^{-4\pi i\omega x}\int e^{2\pi i\omega (2t)}\mathrm{\Phi }(x,t)dt.$$

Since ${\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^{2n})$ is closed under dilations and modulations, the first part of the claim will be proved if we show that $\mathrm{\Phi }\in {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^{2n})$, since then ${\mathcal{F}}_2\mathrm{\Phi }\in {\mathit{S}}_{[{\mathcal{M}}_{\sigma }]}^{[{\mathcal{M}}_{\sigma }]}({\mathbb{R}}^{2n})$.

As before, we give the proof for the Beurling case, and the Roumieu case follows by similar arguments.

By Theorem 5, $\widetilde{(M.2)}$, and (4) (see Remark 2), it is enough to show

$$\underset{x,t\in {\mathbb{R}}^n}{sup}|x^{\alpha }{t}^{\beta }\mathrm{\Phi }(x,t)|\le C{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\alpha ,\beta \in {\mathbb{N}}_0^n,$$

and

$$\underset{x,t\in {\mathbb{R}}^n}{sup}|{\partial }_x^{\alpha }{\partial }_t^{\beta }\mathrm{\Phi }(x,t)|\le C{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\alpha ,\beta \in {\mathbb{N}}_0^n,$$

(29)

for any given $\tau >0$, and some constant $C=C_{\tau }>0$.

The decay of $\mathrm{\Phi }(x,t):=f(2x-t)\overline{g(t)}$ can be estimated as follows.

$$\begin{array}{cc}\hfill \underset{x,t\in {\mathbb{R}}^d}{sup}|x^{\alpha }{t}^{\beta }f(2x-t)\overline{g(t)}|& \le 2^{-|\alpha |}\underset{y,t\in {\mathbb{R}}^n}{sup}{|y+t|}^{|\alpha |}{|t|}^{|\beta |}|f(y)||\overline{g(t)}|\hfill \\ \hfill & \le 2^{-|\alpha |}\underset{y,t\in {\mathbb{R}}^n}{sup}\sum _{\gamma \le \alpha }\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\gamma }}\right){|y|}^{|\gamma |}{|t|}^{|\alpha -\gamma |+|\beta |}|f(y)||\overline{g(t)}|\hfill \\ \hfill & \le 2^{-|\alpha |}\sum _{\gamma \le \alpha }\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\gamma }}\right)\underset{y\in {\mathbb{R}}^n}{sup}{|y|}^{|\gamma |}|f(y)|·\underset{t\in {\mathbb{R}}^n}{sup}{|t|}^{|\alpha -\gamma |+|\beta |}|\overline{g(t)}|\hfill \\ \hfill & \le C^2{2}^{-|\alpha |}\sum _{\gamma \le \alpha }\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\gamma }}\right)M_{|\gamma |}^{\tau 2^{-\sigma },\sigma }{M}_{|\alpha -\gamma |+|\beta |}^{\tau 2^{-\sigma },\sigma }\hfill \\ \hfill & \le C^2{M}_{|\alpha |+|\beta |}^{\tau 2^{-\sigma },\sigma }\le C^2{C}_1^{{|\alpha |}^{\sigma }+{|\beta |}^{\sigma }}{M}_{|\alpha |}^{\tau /2,\sigma }{M}_{|\beta |}^{\tau /2,\sigma }\hfill \\ \hfill & \le C_2{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\hfill \end{array}$$

for some $C,C_1,C_2>0$, where we use (4), $\widetilde{(M.2)}$, and (10).

To prove (29), we use the Leibniz formula and the previously outlined properties of the sequence ${(M_p^{\tau ,\sigma })}_{p\in {\mathbb{N}}_0}$:

$$\begin{array}{ccc}\hfill \underset{x,t\in {\mathbb{R}}^n}{sup}|{\partial }_x^{\alpha }{\partial }_t^{\beta }f(2x-t)\overline{g(t)}|& \le & 2^{|\alpha |}\sum _{\gamma \le \beta }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)\underset{x,t\in {\mathbb{R}}^n}{sup}|{\partial }_x^{\alpha }{\partial }_t^{\gamma }f(2x-t){\partial }_t^{\beta -\gamma }\overline{g(t)}|\hfill \\ & \le & 2^{|\alpha |}\sum _{\gamma \le \beta }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)\underset{x,t\in {\mathbb{R}}^n}{sup}|{\partial }_x^{\alpha }{\partial }_t^{\gamma }f(2x-t)|\underset{t\in {\mathbb{R}}^n}{sup}|{\partial }_t^{\beta -\gamma }\overline{g(t)}|\hfill \\ & \le & 2^{|\alpha |}{C}^2\sum _{\gamma \le \beta }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)M_{|\alpha +\gamma |}^{\tau 2^{-\sigma },\sigma }{M}_{|\beta -\gamma |}^{\tau 2^{-\sigma },\sigma }\hfill \\ & \le & 2^{|\alpha |+|\beta |}{C}^2{M}_{|\alpha |+|\beta |}^{\tau 2^{-\sigma },\sigma }\le C_1{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },\hfill \end{array}$$

for some constants $C,C_1>0$.

Thus, $R_{g}f\in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^{2n})$ if $f,g\in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n).$

To prove the second part of theorem, assume that $f\in L^2({\mathbb{R}}^n)$, $g\in L^2({\mathbb{R}}^n)\setminus \{0\}$, and

$$R_{g}f\in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^{2n}).$$

(30)

By the inversion formula (28), we have

$$t^{\alpha }{\partial }_t^{\beta }f(t)={\displaystyle \frac{1}{\langle h,g\rangle }}\int \int R_{g}f(x,\omega )t^{\alpha }{\partial }_t^{\beta }(R(h(t))(x,\omega )dxd\omega ,$$

where we may choose $h\in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ such that $\langle h,g\rangle =1$.

Let there be given $\tau >0$. Then, we have

$$\begin{array}{ccc}\hfill |t^{\alpha }{\partial }_t^{\beta }(R(h(t))(x,\omega )|& =& |t^{\alpha }{\partial }_t^{\beta }(e^{4\pi i\omega (t-x)}h(2x-t))|\hfill \\ & \le & \sum _{\gamma \le \beta }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right){|(4\pi i\omega )}^{\beta -\gamma }||t^{\alpha }{\partial }_t^{\gamma }h(2x-t))|\hfill \\ & \le & {(4\pi )}^{|\beta |}{M}_{|\alpha |}^{\tau /2,\sigma }\sum _{\gamma \le \beta }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)|{\omega }^{\beta -\gamma }|M_{|\gamma |}^{\tau /2,\sigma },\hfill \end{array}$$

wherefrom

$$|t^{\alpha }{\partial }_t^{\beta }f(t)|\le {(4\pi )}^{|\beta |}{M}_{|\alpha |}^{\tau /2,\sigma }\sum _{\gamma \le \beta }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)M_{|\gamma |}^{\tau /2,\sigma }\int \int |{\omega }^{\beta -\gamma }{R}_{g}f(x,\omega )|dxd\omega .$$

(31)

Next, we estimate the double integral in (31) as follows:

$$\begin{array}{cc}\hfill \int \int |{\omega }^{\beta -\gamma }{R}_{g}f(x,\omega )|dxd\omega & \le C_1\underset{x,\omega \in {\mathbb{R}}^n}{sup}{(1+|x|)}^2{(1+|\omega |)}^{2+|\beta -\gamma |}|R_{g}f(x,\omega )|\hfill \\ \hfill & \le C_2{M}_{4+|\beta -\gamma |}^{\tau /2,\sigma },\hfill \end{array}$$

for some constants $C_1,C_2>0,$ wherefrom by (4)

$$\begin{array}{cc}\hfill \sum _{\gamma \le \beta }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)M_{|\gamma |}^{\tau /2,\sigma }\int \int |{\omega }^{\beta -\gamma }{R}_{g}f(x,\omega )|dxd\omega & \le C_2\sum _{\gamma \le \beta }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{\gamma }}\right)M_{|\gamma |}^{\tau /2,\sigma }{M}_{4+|\beta -\gamma |}^{\tau /2,\sigma }\hfill \\ \hfill & \le C_2{2}^{|\beta |}{M}_{4+|\beta |}^{\tau /2,\sigma }.\hfill \end{array}$$

(32)

By (31) and (32), using $\widetilde{{(M.2)}^{\prime }}$ and (10), we obtain

$$\underset{t\in {\mathbb{R}}^n}{sup}|t^{\alpha }{\partial }_t^{\beta }f(t)|\le C_2{(8\pi )}^{|\beta |}{M}_{|\alpha |}^{\tau /2,\sigma }{M}_{4+|\beta |}^{\tau /2,\sigma }\le C_3{C}_4^{{|\beta |}^{\sigma }}{M}_{|\alpha |}^{\tau /2,\sigma }{M}_{|\beta |}^{\tau /2,\sigma }\le C_5{M}_{|\alpha |}^{\tau ,\sigma }{M}_{|\beta |}^{\tau ,\sigma },$$

i.e., $f\in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$.

Finally, $g\in {\mathit{S}}_{({\mathcal{M}}_{\sigma })}^{({\mathcal{M}}_{\sigma })}({\mathbb{R}}^n)$ follows from $R_{g}f=\overline{R_{f}g}$. □

Remark 7. 

An alternative proof of Theorem 7 can be given using arguments based on the kernel theorem for Gelfand–Shilov spaces with extended Gevrey regularity. This is beyond the scope of the current contribution and will be studied elsewhere. For details on kernel theorems and their connection to nuclearity, we refer to [15].

6. Discussion

The family of extended Gevrey classes introduced in [9] provides a convenient framework for studying regularity properties beyond the usual Gevrey-type regularity. In this contribution, we consider both regularity and decay, introducing Gelfand-Shilov-type spaces in the context of extended Gevrey regularity. Our approach is related to, and inspired by, the recently explored weighted matrices/functions approach to Gelfand–Shilov spaces [15,16,17,26]. The results exhibited in this paper can be useful in situations where, on one hand, Gevrey-type regularity is too restrictive, and on the other hand, Schwartz-type regularity is too general.