Localization of Fr\'echet frames and expansion of generalized functions

TL;DR

This paper extends the theory of Fréchet frames by analyzing matrix operators with decay properties under weaker assumptions, and applies localization techniques for frame expansions of distributions.

Contribution

It introduces weaker assumptions for matrix operator continuity and extends localization results from Banach to Fréchet spaces, enabling frame expansions of distributions.

Findings

Weaker assumptions still ensure operator continuity.

Localization of Fréchet frames facilitates distribution expansions.

Extension from Banach to Fréchet spaces achieved.

Abstract

Matrix type operators with the off-diagonal decay of polynomial or sub-exponential types are revisited with weaker assumptions concerning row or column estimates, still giving the continuity results for the frame type operators. Such results are extended from Banach to Fr\'{e}chet spaces. Moreover, the localization of Fr\'{e}chet frames is used for the frame expansions of tempered distributions and a class of Beurling ultradistributions.

Equations

$$f=\sum _{n=1}^{\infty }⟨f,f_n⟩g_n=\sum _{n=1}^{\infty }⟨f,g_n⟩f_n,f\in H.$$

$$H_{G,G^d}^{\Theta }:=\{f\in H:(⟨f,g_n^d⟩{)}_{n=1}^{\infty }\in \Theta ,\Vert f{\Vert }_{H_{G,G^d}^{\Theta }}:=\Vert |[\Vert |(⟨f,g_n^d⟩{)}_{n=1}^{\infty }{]}_{\Theta }\}.$$

$$\max \{|⟨e_m,g_n⟩|,|⟨e_m,\widetilde{g_n}⟩|\}\le C_{\gamma }(1+|m-n|{)}^{-\gamma },m,n\in \mathbb{N};$$

$$\max \{|⟨e_m,g_n⟩|,|⟨e_m,\widetilde{g_n}⟩|\}\le C_{\gamma }{\mathrm{e}}^{-\gamma |m-n|},m,n\in \mathbb{N}.$$

$$\max \{|⟨e_m,g_n⟩|,|⟨e_m,\widetilde{g_n}⟩|\}\le C_s{\mathrm{e}}^{-\gamma |m-n{|}^{\beta }},m,n\in \mathbb{N}.$$

$$\{0\}\ne {\cap }_{k\in {\mathbb{N}}_0}{[}_{k}Y]\subseteq \dots \subseteq Y_2\subseteq Y_1\subseteq Y_0,|\cdot {|}_0\le |\cdot {|}_1\le |\cdot {|}_2\le \dots$$

$$Y_F:={\cap }_{k\in {\mathbb{N}}_0}{[}_{k}Y]\text{mbox}isdensein{Y}_k,k\in {\mathbb{N}}_0({\mathbb{N}}_0=\mathbb{N}\cup \{0\}).$$

$$({\varphi }_n(f){)}_{n=1}^{\infty }\in {\Theta }_F,f\in X_F,$$

$$A_k\Vert f{\Vert }_{s_k}\le \Vert |[\Vert |\{{\varphi }_n(f){\}}_{n=1}^{\infty }{]}_k\le B_k\Vert f{\Vert }_{{\tilde{s}}_k},f\in X_F,k\in {\mathbb{N}}_0,$$

$${\ell }_{\mu }^p:=\{(a_n{)}_{n=1}^{\infty }:\Vert |[\Vert |(a_n{)}_n{]}_{p,\mu }:=(\sum _{n=1}^{\infty }|a_n{|}^p\mu (n{)}^p{)}^{1/p}<\infty \}$$

$$1=\mu_{0}(x)\leq\mu_{1}(x)\leq\mu_{2}(x)\leq...,\ \mbox{ for every $x\in\mathbb{R}$}.$$

$${\mathcal{S}}_k(\mathbb{R}):=\{f\in L^2(\mathbb{R}):||f|{|}_k=\sum _{m=0}^k||(1+|\cdot {|}^2{)}^{k/2}{f}^{(m)}|{|}_{L^2(\mathbb{R})}\},k\in \mathbb{N}.$$

$$||f|{|}_k^{\alpha }=\underset{n\in {\mathbb{N}}_0}{\sup }||\frac{k^n{e}^{k|x{|}^{1/\alpha }}|f^{(n)}(x)|}{n{!}^{\alpha }}|{|}_{L^2(\mathbb{R})},k\in \mathbb{N}.$$

$$\Vert f{\Vert }_{H^k}:=\Vert |[\Vert |(a_n{n}^k{)}_n{]}_{{\ell }^2}<\infty \},k\in {\mathbb{N}}_0,$$

$$\Vert f{\Vert }_{H^k}:=\Vert |[\Vert |(a_n{e}^{k{n}^{1/(2\alpha )}}{)}_n{]}_{{\ell }^2}<\infty \},k\in {\mathbb{N}}_0.$$

$$|A_{m,n}|\le \frac{C_{\gamma }}{(1+|m-n|{)}^{\gamma }}(\text{mbox}resp.|A_{m,n}|\le C_{\gamma }{\mathrm{e}}^{-\gamma |m-n|}),\forall n,m\in \mathbb{N}.$$

$$(*)\ \ \ |A_{m,n}|\leq C\left\{\begin{array}[]{ll}\frac{(1+m)^{\gamma}}{(1+n)^{2\gamma}},&n\geq m,\\ \frac{(1+n)^{\gamma}}{(1+m)^{2\gamma}},&n\leq m,\mbox{ for some }C>0.\end{array}\right.$$

$$(\ast \ast )|A_{m,n}|\le C(1+|n-m|{)}^{-\gamma },\text{mbox}forsomeC>0.$$

$$(***)\ \ \ |A_{m,n}|\leq C\left\{\begin{array}[]{ll}\frac{m^{\gamma}}{n^{\gamma}},&n\geq m,\\ \frac{n^{\gamma}}{m^{\gamma}},&n\leq m,\mbox{ for some }C>0.\end{array}\right.$$

$$|A_{m,n}|\leq\left\{\begin{array}[]{ll}C_{0}\,n^{\gamma_{0}},&n>m,\\ C_{1}\,n^{\gamma_{1}}m^{-\gamma_{1}},&n\leq m,\end{array}\right.$$

$$|A_{m,n}{c}_n|$$

$$|\sum _{n=1}^m{A}_{m,n}{c}_n|$$

$$|a_m|$$

$$|A_{m,n}|\leq\left\{\begin{array}[]{ll}C_{0}n^{\gamma_{0}},&n>m\\ C_{\gamma}n^{\gamma}m^{-\gamma},&n\leq m.\end{array}\right.$$

$$(\exists \epsilon >0,{\gamma }_1\in \mathbb{N})(\exists C_0>0,C_1>0)\text{mbox}suchthat$$

$$|A_{m,n}|\leq\left\{\begin{array}[]{ll}C_{0}n^{-1-\varepsilon},&n>m,\\ C_{1}n^{\gamma_{1}}m^{-\gamma_{1}-1-\varepsilon},&n\leq m.\end{array}\right.$$

$$|A_{m,n}|\le C(1+|n-m|{)}^{-{\gamma }_1-1-\epsilon }$$

$$|A_{m,n}|\leq\left\{\begin{array}[]{ll}C_{0}\mathrm{e}^{\gamma_{0}n^{\beta}},&n>m,\\ C_{1}\mathrm{e}^{-\gamma_{1}(m^{\beta}-n^{\beta})},&n\leq m.\end{array}\right.$$

$$|A_{m,n}{c}_n|$$

$$|\sum _{n=1}^m{A}_{m,n}{c}_n|$$

Full text

1 Introduction, Motivation and Main Aims

Localized frames were introduced independently by Gröchenig [21] and Balan, Casazza, Heil, and Landau [2, 3]. The localization conditions in [21] are related to off-diagonal decay (of polynomial or exponential type) of the matrix determined by the inner products of the frame elements and the elements of a given Riesz basis. A localized frame in this sense leads to the same type of localization of the canonical dual frame as well as to the convergence of the frame expansions in all associated Banach spaces. We refer to [12, 13, 19, 18], where various interesting properties and applications of localized frames were considered. The localization and self-localization, considered independently in [2, 3, 1], are directed to the over-completeness of frames and the relations between frame bounds and density with applications to Gabor frames. For the present paper we have chosen to stick to the localization concept from [21], because the results obtained for a family of Banach spaces there can naturally be related to Fréchet frames (cf. [28, 29, 30, 31]).

The aims of this paper are:

First, to extend the continuity results on matrix-type operators acting on elements of a Banach or Fréchet spaces expanded by frames. Such results are related to the off-diagonal decay conditions considered in the literature and the aim now is to use relaxed version of off-diagonal decay which requires column-decay but allows row-increase of a matrix.

Second, to present the frame expansions of tempered distributions $\mathcal{S}(\mathbb{R})$ and tempered ultradistributions of Beurling type by the use of localization.

Beside of the main aims, the important novelty is the analysis related to sub-exponential off-diagonal decay without assumption of the exponential off-diagonal decay as it was considered in [21]. More precisely, in [21] the presumed exponential off-diagonal decay of matrices implies the analysis of sub-exponentially weighted spaces. Probably the most important impact in applications is related to Hermite basis which is almost always used for the global expansion of $L^{2}$-functions or tempered generalized functions over ${\mathbb{R}}^{n}$. Our results by the use of localization, show that the same is true if one uses a kind of perturbation of Hermite functions through localization.

The paper is organized as follows. We recal in Section 2 the notation, basic definitions and the needed known results. In Section 3 we consider matrices with column decay and possible row increase. For such type of matrices, we obtain in Section 4 continuity results for the frame related operators using less restrictive conditions in comparison to the localization conditions known in the literature. Sub-exponential localization is introduced and analyzed in Section 5. The use of Jaffard’s Theorem and [21, Theorems 11 and 13] is intrinsically connected with the sub-exponential localization. Section 6 is devoted to Fréchet frames and series expansions in certain classes of Fréchet spaces based on polynomial, exponential, and sub-exponential localization. In particular, we obtain frame expansions in the Schwartz space $\mathcal{S}$ of rapidly decreasing functions and its dual, the space of tempered distributions, as well as in the spaces $\Sigma^{\alpha}$, $\alpha>1/2$, and their duals, spaces of tempered ultradistributions. In order to illustrate some results, we provide examples with the Hermite orthonormal bases $h_{n},n\in\mathbb{N}$, and construct a Riesz basis which is polynomially and exponentially localized to $h_{n},n\in\mathbb{N}$. Finally, in the Appendix, we add some details in the proof of the Jaffard’s theorem.

2 Notation, Definitions and Preliminaries

Throughout the paper, $({H},\langle\cdot,\cdot\rangle)$ denotes a separable Hilbert space and $G$ (resp. $E$) denotes the sequence $(g_{n})_{n=1}^{\infty}$ (resp. $(e_{n})_{n=1}^{\infty}$) with elements from ${H}$. Recall that $G$ is called:

• frame for ${H}$ [14] if there exist positive constants $A$ and $B$ (called frame bounds) so that $A\|f\|^{2}\leq\sum_{n=1}^{\infty}|\langle f,g_{n}\rangle|^{2}\leq B\|f\|^{2}$ for every $f\in{H}$;

• Riesz basis for ${H}$ [4] if its elements are the images of the elements of an orthonormal basis under a bounded bijective operator on ${H}$.

Recall (see e.g. [11]), if $G$ is a frame for ${H}$, then there exists a frame $([_{n})_{n=1}^{\infty}f]$ for ${H}$ so that

[TABLE]

Such $([_{n})_{n=1}^{\infty}f]$ is called a dual frame of $([_{n})_{n=1}^{\infty}g]$. Furthermore, the analysis operator $U_{G}$, given by $U_{G}f=(\langle f,g_{n}\rangle)_{n=1}^{\infty}$, is bounded from ${H}$ into $\ell^{2}$; the synthesis operator $T_{G}$, given by $T_{G}f=\sum_{n=1}^{\infty}c_{n}g_{n}$, is bounded from $\ell^{2}$ into ${H}$; the frame operator $S_{G}:=T_{G}U_{G}$ is a bounded bijection of ${H}$ onto ${H}$ with unconditional convergence of the series $S_{G}f=\sum_{n=1}^{\infty}\langle f,g_{n}\rangle g_{n}$. The sequence $(S_{G}^{-1}g_{n})_{n=1}^{\infty}$ is a dual frame of $([_{n})_{n=1}^{\infty}g]$, called the canonical dual of $([_{n})_{n=1}^{\infty}g]$, and it will be denoted by $(\widetilde{g_{n}})_{n=1}$ or $\widetilde{G}$. When $G$ is a Riesz basis of ${H}$ (and thus a frame for ${H}$), then only $\widetilde{G}$ is a dual frame of ${H}$, it is the unique biorthogonal sequence to $G$ and it is also a Riesz basis for ${H}$. A frame $G$ which is not a Riesz basis has other dual frames in addition to the canonical dual and in that case we use notation $G^{d}$ or $(g_{n}^{d})_{n=1}^{\infty}$ for a dual frame of $G$.

Next, $(X,\|\cdot\|)$ denotes a Banach space and $(\Theta,{\||{[}\||}\cdot])$ denotes a Banach sequence space; $\Theta$ is called a $BK$-space if the coordinate functionals are continuous. If the canonical vectors form a Schauder basis for $\Theta$, then $\Theta$ is called a $CB$-space. A $CB$-space is clearly a $BK$-space.

Given a $BK$-space $\Theta$ and a frame $G$ for ${H}$ with a dual frame $G^{d}=(g_{n}^{d})_{n=1}^{\infty}$, one associates to $\Theta$ the Banach space

[TABLE]

When $G$ is a Riesz basis for ${H}$, then we use notation $H^{\Theta}_{G}$ for $H^{\Theta}_{G,\widetilde{G}}$.

2.1 Localization of frames

In this paper we consider polynomially and exponentially localized frames in the way defined in [21], and furthermore, sub-exponential localization. Let $G$ be a Riesz basis for the Hilbert space ${H}$. A frame $E$ for ${H}$ is called:

polynomially localized with respect to $G$ with decay $\gamma>0$ (in short, $\gamma$-localized wrt $([_{n})_{n=1}^{\infty}g]$) if there is a constant $C_{\gamma}>0$ so that

[TABLE]

• -

exponentially localized with respect to $G$ if for some $\gamma>0$ there is a constant $C_{\gamma}>0$ so that

[TABLE]

• -

$\beta$-sub-exponentially localized with respect to $G$ (for $\beta\in(0,1)$) if for some $\gamma>0$ there is $C_{\gamma}>0$ so that

[TABLE]

2.2 Fréchet frames

We consider Fréchet spaces which are projective limits of Banach spaces as follows. Let $\{Y_{k},|\cdot|_{k}\}_{k\in\mathbb{N}_{0}}$ be a sequence of separable Banach spaces such that

[TABLE]

[TABLE]

Under the conditions (1)-(2), $Y_{F}$ is a Fréchet space. We will use such type of sequences in two cases:

1. $Y_{k}=X_{k}$ with norm $\|\cdot\|_{k},k\in\mathbb{N}_{0};$
2. $Y_{k}=\Theta_{k}$ with norm ${\||{[}\||}\cdot]_{k},k\in\mathbb{N}_{0}$.

Let $\{\Theta_{k},{\||{[}\||}\cdot]_{k}\}_{k\in\mathbb{N}_{0}}$ be a sequence of $CB$-spaces satisfying (1). Then (2) holds, because every sequence $(c_{n})_{n=1}^{\infty}\in\Theta_{F}$ can be written as $\sum_{n=1}^{\infty}c_{n}\delta_{n}$ with the convergence in $\Theta_{F}$, where $\delta_{n}$ denotes the $n$-th canonical vector, $n\in\mathbb{N}$. Furthermore, $\Theta_{F}^{*}$ can be identified with the sequence space $\Theta_{F}^{\circledast}:=\{(U\delta_{n})_{n=1}^{\infty}\ :\ U\in\Theta_{F}^{*}\}$ with convergence naturally defined in correspondence with the convergence in $\Theta_{F}^{*}$.

We use the therm operator for a linear mapping, and by invertible operator on $X$ we mean a bounded bijective operator on $X$. Given sequences of Banach spaces, $\{X_{k}\}_{k\in\mathbb{N}_{0}}$ and $\{\Theta_{k}\}_{k\in\mathbb{N}_{0}}$, which satisfy (1)-(2), an operator $T:\Theta_{F}\to X_{F}$ is called $F$-bounded if for every $k\kern-2.20001pt\in\kern-1.49994pt\mathbb{N}_{0}$, there exists a constant $C_{k}>0$ such that $\|T([_{n})_{n=1}^{\infty}c]\|_{k}\leq C_{k}{\||{[}\||}\{c_{n}\}_{n=1}^{\infty}]_{k}$ for all $([_{n})_{n=1}^{\infty}c]\kern-2.20001pt\in\kern-1.49994pt\Theta_{F}$.

Definition 2.1**.**

[31]* Let $\{X_{k},\|\cdot\|_{k}\}_{k\in\mathbb{N}_{0}}$ be a sequence of Banach spaces satisfying (1)-(2) and let $\{\Theta_{k},{\||{[}\||}\cdot]_{k}\}_{k\in\mathbb{N}_{0}}$ be a sequence of $BK$-spaces satisfying (1)-(2). A sequence $([_{n})_{n=1}^{\infty}\phi]$ with elements from ${X_{F}^{*}}$ is called a General Fréchet frame (in short, General $F$-frame) for $X_{F}$ with respect to $\Theta_{F}$ if there exist sequences $\{\widetilde{s}_{k}\}_{k\in\mathbb{N}_{0}}\subseteq\mathbb{N}_{0}$, $\{s_{k}\}_{k\in\mathbb{N}_{0}}\subseteq\mathbb{N}_{0}$, which increase to $\infty$ with the property $s_{k}\leq\widetilde{s}_{k}$, $k\in\mathbb{N}_{0}$, and there exist constants $0<A_{k}\leq B_{k}<\infty$, $k\in\mathbb{N}_{0}$, satisfying*

[TABLE]

[TABLE]

and there exists a continuous operator $V:\Theta_{F}\rightarrow X_{F}$ so that $V(\phi_{n}(f))_{n=1}^{\infty}=f$ for every $f\kern-2.20001pt\in\kern-1.49994ptX_{F}$.

When $s_{k}=\widetilde{s}_{k}=k$, $k\in\mathbb{N}_{0}$, and the continuity of $V$ is replaced by the stronger condition of $F$-boundedness of $V$, then the above definition reduces to the definition of a Fréchet frame (in short, $F$-frame) for $X_{F}$ with respect to $\Theta_{F}$ introduced in [29]. We will consider such frames in the sequel.

In the particular case when $X_{k}=X$, and $\Theta_{k}=\Theta$, $k\in\mathbb{N}_{0}$, a Fréchet frame for $X_{F}$ with respect to $\Theta_{F}$ becomes a Banach frame for $X$ with respect to $\Theta$ as introduced in [20].

For another approach to frames in Fréchet spaces we refer to [5]. For more on frames for Banach spaces, see e.g. [8, 7, 34] and the references therein.

2.3 Sequence and function spaces

Recall that a positive continuous function $\mu$ on $\mathbb{R}$ is called: a $k$-moderate weight if $k\geq 0$ and there exists a constant $C>0$ so that $\mu(t+x)\leq C(1+|t|)^{k}\mu(x),\ t,x\in\mathbb{R};$ a $\beta$-sub-exponential (resp. exponential) weight, if $\beta\in(0,1)$ (resp. $\beta=1$) and there exist constants $C>0,\gamma>0$, so that $\mu(t+x)\leq Ce^{\gamma|t|^{\beta}}\mu(x),\ t,x\in\mathbb{R}.$ If $\beta$ is clear from the context, we will write just sub-exponential weight. Let $\mu$ be a $k$-moderate, sub-exponential, or exponential weight so that $\mu(n)\geq 1$ for every $n\in\mathbb{N}$, and $p\in[1,\infty)$. Then the Banach space

[TABLE]

is a $CB$-space. We refer, for example, to [25, Ch. 27] for the so called Köthe sequence spaces. We will need the following, easy to prove, statements.

Lemma 2.2**.**

Let $G$ be a frame for ${H}$ and let $G^{d}=([_{n})_{n=1}^{\infty}g^{d}]$ be a dual frame of $G$. Let $\mu_{k}$ be $k$-moderate (resp. sub-exponential or exponential) weights, $k\in\mathbb{N}_{0},$ so that

[TABLE]

Then the spaces $\Theta_{k}:=\ell^{2}_{\mu_{k}}$, $k\in\mathbb{N}_{0}$, satisfy (1)-(2). Denote $M:=\{(\langle f,g^{d}_{n}\rangle)_{n=1}^{\infty}\ :\ f\in{H}\}$. The assumption that $M\cap\Theta_{F}$ is dense in $M\cap\Theta_{k}\neq\{0\}$ with respect to the ${\||{[}\||}\cdot]_{k}$-norm for every $k\in\mathbb{N}$, leads to the conclusion that the spaces $X_{k}:=H_{G,G^{d}}^{\Theta_{k}}$, $k\in\mathbb{N}$, satisfy (1)-(2).

If $G$ is a Riesz basis for ${H}$, then the density assumption of $M\cap\Theta_{F}$ in $M\cap\Theta_{k}\neq\{0\}$, $k\in\mathbb{N}$, is fulfilled and in addition one has that $g_{n}\in X_{F}$ for every $n\in\mathbb{N}$.

Throughout the paper we also consider specific weights, relevant to the function spaces of interest and the corresponding sequence spaces. Let $k\in\mathbb{N}_{0}$ and $\mu_{k}(x)=(1+|x|)^{k}$ (resp. $\mu_{k}(x)=e^{k|x|^{\beta}},\beta\in(0,1]$), $x\in\mathbb{R}$. Then, with $\Theta_{k}:=\ell^{p}_{\mu_{k}}$, $k\in\mathbb{N}_{0}$, the projective limit $\cap_{k}\Theta_{k}$ is the space ${\bf s}$ of rapidly (resp. ${\mathfrak{s}}^{\beta}$ of sub-exponentially when $\beta<1$ and exponentially when $\beta=1$) decreasing sequences determined by $\{([_{n})_{n=1}^{\infty}a]\in\mathbb{C}^{\mathbb{N}}\ :\ (\sum_{n=1}^{\infty}|a_{n}\mu_{k}(n))|^{p})^{1/p}<\infty,\ \forall k\in\mathbb{N}_{0}\}$, which is the same set for any $p\in[1,\infty)$. The space ${\bf s}$ (resp. ${\mathfrak{s}}^{\beta}$) can also be derived as the projective limit of the Banach spaces ${\bf s}_{k}$ (resp. ${\mathfrak{s}}^{\beta}_{k}$) defined as $\{(a_{n})_{n=1}^{\infty}\in\mathbb{C}^{\mathbb{N}}\ :\ {\||{[}\||}(a_{n})_{n=1}^{\infty}]_{\sup,k}:=\sup_{n\in\mathbb{N}}|a_{n}|n^{k}<\infty\}$, (resp. ${\||{[}\||}\cdot]_{\sup,k}^{\beta}:=\sup_{n\in\mathbb{N}}|a_{n}|e^{kn^{\beta}}<\infty\}$), $k\in\mathbb{N}_{0}$; note that here instead of $k\in\{0,1,2,3,\ldots\}$ one can also use any strictly increasing sequence of non-negative numbers $k\in\{0,q_{1},q_{2},q_{3},\ldots\}$.

Recall that the well known Schwartz space $\mathcal{S}$ is the intersection of Banach spaces

[TABLE]

The dual ${\mathcal{S}}^{\prime}(\mathbb{R})$ is the space of tempered distributions.

The space of sub-exponentially decreasing functions of order $1/\alpha$, $\alpha>1/2,$ is $\Sigma^{\alpha}:=X_{F}=\cap_{k\in\mathbb{N}_{0}}\Sigma^{k,\alpha}$ where $\Sigma^{k,\alpha}$ are Banach spaces of $L^{2}-$ functions with finite norms

[TABLE]

Its dual $({\Sigma^{\alpha}}(\mathbb{R}))^{\prime}$ is the space of Beurling tempered ultradistributions, cf. [27, 17].

Remark 2.3**.**

The case $\alpha=1/2$ leads to the trivial space $\Sigma^{1/2}=\{0\}.$ There is another way in considering the test space which corresponds to that limiting Beurling case $\alpha=1/2$ and can be considered also for $\alpha<1/2$ (cf. [26, 17, 10, 9]). We will not treat these cases in the current paper.

In the sequel, $(h_{n})_{n=1}^{\infty}$ is the Hermite orthonormal basis of $L^{2}(\mathbb{R})$ re-indexed from $1$ to $\infty$ instead of from [math] to $\infty$. Recall that $h_{n}\in\Sigma^{\alpha}$, $\alpha>1/2$, $n\in\mathbb{N}$. Moreover, we know [33]:

• If $f\in\mathcal{S}$, then $(\langle f,h_{n}\rangle)_{n=1}^{\infty}\in{\bf s}$; conversely, if $(a_{n})_{n=1}^{\infty}\in{\bf s}$, then $\sum_{n=1}^{\infty}a_{n}h_{n}$ converges in $\mathcal{S}$ to some $f$ with $(\langle f,h_{n}\rangle)_{n=1}^{\infty}=(a_{n})_{n=1}^{\infty}$.

• If $F\in\mathcal{S}^{\prime}$, then $(b_{n})_{n=1}^{\infty}:=(F(h_{n}))_{n=1}^{\infty}\in{\bf s}^{\prime}$ and $F(f)=\sum_{n=1}^{\infty}\langle f,h_{n}\rangle b_{n}$, $f\in\mathcal{S}$; conversely, if $(b_{n})_{n=1}^{\infty}\in{\bf s}^{\prime}$, then the mapping $F:f\to\sum_{n=1}^{\infty}\langle f,h_{n}\rangle b_{n}$ is well defined on $\mathcal{S}$, it determines $F$ as an element of $\mathcal{S}^{\prime}$ and $(F(h_{n}))_{n=1}^{\infty}=(b_{n})_{n=1}^{\infty}$.

The above two statements also hold when $\mathcal{S}$, $\mathcal{S}^{\prime}$, ${\bf s}$, and ${\bf s}^{\prime}$ are replaced by $\Sigma^{\alpha}$, $(\Sigma^{\alpha})^{\prime}$, ${\mathfrak{s}}^{1/(2\alpha)}$, and $({\mathfrak{s}}^{1/(2\alpha)})^{\prime}$ with $\alpha>1/2$, respectively ([26], [27], [17]).

We can consider $\mathcal{S}$ and $\Sigma^{\alpha}$ as the projective limit of Hilbert spaces $H^{k},k\in\mathbb{N}_{0},$ with elements $f=\sum_{n}a_{n}h_{n},$ in the first case with norms

[TABLE]

and in the second case with norms

[TABLE]

Thus, $(h_{n})_{n}$ is an $F$-frame for $\mathcal{S}(\mathbb{R})$ with respect to ${\bf s}$ as well as an $F$-frame for $\Sigma^{\alpha}$ with respect to ${\mathfrak{s}}^{1/2\alpha}$, $\alpha>1/2$, ($F$- boundedness is trivial).

3 Matrix type operators

Papers [21, 13, 19] concern matrices with off-diagonal decay of the form: for some $\gamma>0$ there is $C_{\gamma}>0$ such that

[TABLE]

In this paper we consider matrices with more general off-diagonal type of decay (see $(***)$ below which is weaker condition compare to the polynomial type condition in (6)). Moreover, we consider matrices which have column decrease but allow row increase (see Propositions 3.2 and 3.6) allowing sub-exponential type conditions as well. For such more general matrices, we generalize some results from [21] with respect to certain Banach spaces and, furthermore, proceed to the Fréchet case.

In the sequel, for a given matrix $(A_{mn})_{m,n\in\mathbb{N}}$, the letter $\mathcal{A}$ will denote the mapping $(c_{n})_{n=1}^{\infty}\to(a_{m})_{m=1}^{\infty}$ determined by $a_{m}=\sum_{n=1}^{\infty}A_{m,n}c_{n}$ (assuming convergence), $m\in\mathbb{N}$; conversely, for a given mapping $\mathcal{A}$ determined on a sequence space containing the canonical vectors $\delta_{n}$, $n\in\mathbb{N}$, the correspondent matrix $(A_{mn})_{m,n\in\mathbb{N}}$ is given by $A_{m,n}=\langle\mathcal{A}\delta_{n},\delta_{m}\rangle$. We will sometimes use $\mathcal{A}$ with the meaning of $(A_{mn})_{m,n\in\mathbb{N}}$ and vice-verse.

3.1 Polynomial type conditions

Let us begin with some comparison of polynomial type of off-diagonal decay:

Lemma 3.1**.**

Let $\gamma>0$. Consider the following conditions:

[TABLE]

[TABLE]

[TABLE]

Then, the implications $(*)\Rightarrow(**)\Rightarrow(***)$ hold. The converse implications are not valid.

**Proof.  **Implications $(*)\Rightarrow(**)\Rightarrow(***)$ follow from the inequalities $\frac{(1+min(m,n))^{\gamma}}{(1+max(m,n))^{2\gamma}}\leq(1+|n-m|)^{-\gamma}\leq\frac{(min(m,n))^{\gamma}}{(max(m,n))^{\gamma}},n,m\in\mathbb{N},$ which are easy to be verified. To show that $(***)$ does not imply $(**)$ even up to a multiplication with a constant, take a matrix $A_{m,n}$ which satisfies $|A_{m,n}|=\frac{Cn^{\gamma}}{m^{\gamma}},\ n\leq m$, for some $\gamma>0$ and some positive constant $C,$ and assume that there exist $\gamma_{1}(\gamma)\in\mathbb{N}$ and a positive constant $K$ so that for $m\geq n$ one has $\frac{Cn^{\gamma}}{m^{\gamma}}\leq\frac{K}{(1+m-n)^{\gamma_{1}}}$; then taking $m=2n$, one obtains 0<C\cdot 2^{-\gamma}\leq\frac{K}{(1+n)^{\gamma_{1}}}\to 0\mbox{ as n\to\infty,} which leads to a contradiction. In a similar spirit, one can show that $(**)$ does not imply $(*)$. $\Box$

Below we show that the relaxed polynomial type conditions, as well as conditions allowing row-increase, still lead to continuous operators.

Proposition 3.2**.**

Assume that the matrix $(A_{mn})_{m,n\in\mathbb{N}}$ satisfies the condition

[TABLE]

for some $\gamma_{0}\geq 0,\gamma_{1}>0,C_{0}>0,C_{1}>0$. Then $\mathcal{A}$ is a continuous operator from ${\bf s}_{\gamma_{1}+\gamma_{0}+1+\varepsilon}$ into ${\bf s}_{\gamma_{1}}$ for any $\varepsilon\in(0,1)$.

**Proof.  **Let $\varepsilon\in(0,1)$ and let $(c_{n})_{n=1}^{\infty}\in{\bf s}_{\gamma_{0}+\gamma_{1}+1+\varepsilon}$. For every $n>m$,

[TABLE]

Next,

[TABLE]

Therefore,

[TABLE]

Since $\sum_{n=m+1}^{\infty}\frac{1}{n^{\gamma_{1}+1+\varepsilon}}\leq m^{-\gamma_{1}}\sum_{n=m+1}^{\infty}\frac{1}{n^{1+\varepsilon}},$ the assertion follows. $\Box$

A direct consequence of Proposition 3.2 is:

Corollary 3.3**.**

Assume that the matrix $(A_{mn})_{m,n\in\mathbb{N}}$ satisfies: there exist $\gamma_{0}\geq 0$ and $C_{0}>0$, and for every $\gamma>0$ there is $C_{\gamma}>0$ so that

[TABLE]

Then, $\mathcal{A}$ is a continues operator from ${\bf s}$ into ${\bf s}$.

In order to determine $\mathcal{A}$ as a mapping from a space ${\bf s}_{\gamma_{1}}$ into the same space, we have to change the decay condition.

Proposition 3.4**.**

Let $(A_{mn})_{m,n\in\mathbb{N}}$ satisfy:

[TABLE]

[TABLE]

Then $\mathcal{A}$ is a continuous operator from ${\bf s}_{\gamma_{1}}$ into ${\bf s}_{\gamma_{1}}$.

Remark 3.5**.**

For the same conclusion as above, one has in [21] another condition non-comparible to (7):

[TABLE]

3.2 Sub-exponential and exponential type conditions

Up to the end of the paper $\beta$ will be a fixed number of the interval $(0,1];$ $\beta=1$ is related to the exponential growth order while $\beta\in(0,1)$ corresponds to the pure sub-exponential growth order.

Proposition 3.6**.**

Assume that the matrix $(A_{mn})_{m,n\in\mathbb{N}}$ satisfies the condition: there exist positive constants $C_{0},C_{1}$ and $\gamma_{0}\geq 0$, $\gamma_{1}>0$, so that

[TABLE]

Then $\mathcal{A}$ is a continues operator from ${\mathfrak{s}}^{\beta}_{\gamma_{1}+\gamma_{0}+\varepsilon}$ into ${\mathfrak{s}}^{\beta}_{\gamma_{1}}$ for any $\varepsilon\in(0,1)$.

**Proof.  **Let $\varepsilon\in(0,1)$ and let $(c_{n})_{n=1}^{\infty}\in{\mathfrak{s}}^{\beta}_{\gamma_{1}+\gamma_{0}+\varepsilon}$. Then for $n>m$,

[TABLE]

Further on,

[TABLE]

Therefore,

[TABLE]

This completes the proof. $\Box$

Remark 3.7**.**

Since $e^{-\gamma(m-n)^{\beta}}\leq e^{-\gamma(m^{\beta}-n^{\beta})}$ for $n\leq m$ ($\beta\in(0,1],\gamma\in(0,\infty)$), in (9) we consider $e^{-\gamma(m^{\beta}-n^{\beta})}$ instead of $e^{-\gamma{(m-n)^{\beta}}}.$

As a consequence of Proposition 3.6, we have:

Corollary 3.8**.**

Assume that the matrix $(A_{mn})_{m,n\in\mathbb{N}}$ satisfies the condition: there exist constants $C_{0}>0$ and $\gamma_{0}\geq 0$, and for every $\gamma>0$, there is a positive constant $C_{\gamma}$ so that

[TABLE]

Then $\mathcal{A}$ is a continuous operator from ${\mathfrak{s}}^{\beta}$ into ${\mathfrak{s}}^{\beta}$.

Proposition 3.9**.**

Let $(A_{mn})_{m,n\in\mathbb{N}}$ satisfy the condition: there exist positive constants $\varepsilon,\gamma_{1},C_{0},C_{1}$, so that

[TABLE]

Then $\mathcal{A}$ is a continues operator from ${\mathfrak{s}}^{\beta}_{\gamma_{1}}$ into ${\mathfrak{s}}^{\beta}_{\gamma_{1}}$.

Remark 3.10**.**

One can simply show that the assumption $|A_{m,n}|\leq Ce^{-\gamma|m-n|^{\beta}},m,n\in\mathbb{N}$, leads to similar continuity results. We will consider this condition later in relation to the the invertiblity of such matrices and the Jaffard theorem.

4 Continuity of the frame-related operators under relaxed “decay” conditions

We now determine weaker localization-conditions which are still sufficient to imply continuity of the frame-related operators.

Proposition 4.1**.**

Let $G$ be a frame for ${H}$, $G^{d}$ be a dual frame of $G$, and $\mu_{k}(x)=(1+|x|)^{k}$, $k\in\mathbb{N}_{0}$. Under the notations in Lemma 2.2, assume that $M\cap\Theta_{F}$ is dense in $M\cap\Theta_{k}\neq\{0\}$ with respect to the ${\||{[}\||}\cdot]_{k}$-norm for every $k\in\mathbb{N}$ and let $E=(e_{n})_{n=1}^{\infty}$ be a sequence with elements from $X_{F}$ which is a frame for ${H}$. Then the following statements hold.

• (i)

Assume that there exist $s_{0}\in\mathbb{N}$, $C>0$ and for every $k\in\mathbb{N}$ there exists $C_{k}>0$ such that

[TABLE]

Then the analysis operator $f\to U_{E}f=(\langle f,e_{m}\rangle)_{m=1}^{\infty}$ is continuous one from $X_{F}$ into ${\bf s}$.

• (ii)

Assume that there exist $\widetilde{s}_{0}\in\mathbb{N}_{0}$, $\widetilde{C}>0$ and for every $k\in\mathbb{N}$ there exists $\widetilde{C}_{k}>0$ such that

[TABLE]

Then the synthesis operator $(c_{n})_{n}\to T_{E}(c_{n})=\sum c_{n}e_{n}$ is a continuous one from ${\bf s}$ into $X_{F}$.

• (iii)

Under the assumptions of (i) and (ii), the frame operator $T_{E}U_{E}$ is continuous one from $X_{F}$ into $X_{F}$.

**Proof.  **Note that under the given assumptions, $\Theta_{F}$ is the space ${\bf s}$.

(i) Let $A_{m,n}=\langle g_{n},e_{m}\rangle$, $m,n\in\mathbb{N}$, and $\mathcal{A}$ be the corresponding operator for the matrix $A$. Let $f\in X_{F}$. Then $(\langle f,g^{d}_{n}\rangle)_{n=1}^{\infty}\in{\bf s}$ and

[TABLE]

By Corollary 3.3 it follows that $(\langle f,e_{m}\rangle)_{n=1}^{\infty}\in{\bf s}$. Furthermore, by Proposition 3.2, for every $k\in\mathbb{N}$ there is a constant $K_{s_{0},k,C,C_{k}}$ so that

[TABLE]

Therefore, the analysis operator $U_{E}$ is continuous from $X_{F}$ into ${\bf s}$.

(ii) Let $(c_{n})\in{\bf s}$. First we show that $\sum_{n=1}^{\infty}c_{n}e_{n}$ converges in $X_{F}$ and then the continuity of $T_{E}$. Since $(c_{n})_{n=1}^{\infty}\in\ell^{2}$, $x=\sum_{n}c_{n}e_{n}\in{H}$. Denote $A_{m,n}=\langle e_{n},g^{d}_{m}\rangle$ and consider the corresponding operator $\mathcal{A}$. Then, $(\langle x,g^{d}_{m}\rangle)_{m}=(\sum_{n}A_{m,n}c_{n})_{m}=\mathcal{A}(c_{n})\in{\bf s}$, which implies that $x\in X_{F}$, and furthermore, for every $k\in\mathbb{N}$, one has $\|T_{E}(c_{n})_{n}\|_{k}=\|x\|_{k}={\||{[}\||}(\langle x,g^{d}_{m}\rangle)_{m}]_{\Theta_{k}}$. For every $k\in\mathbb{N}$ there is a constant $R_{k}$ such that ${\||{[}\||}(d_{n})]_{\Theta_{k}}\leq R_{k}{\||{[}\||}(d_{n})]_{\sup,k+2}$ for every $(d_{n})\in{\bf s}_{k+2}$. By Proposition 3.2, we conclude that

[TABLE]

Thus, the synthesis operator $T_{E}$ is well defined and continues from ${\bf s}$ into $X_{F}$.

(iii) follows from (i) and (ii). $\Box$

It is of interest to consider case when $X_{F}$ is $\mathcal{S}$.

Corollary 4.2**.**

Let $(e_{n})_{n=1}^{\infty}$ be a frame of $L2(\mathbb{R})$ with elements in $\mathcal{S}(\mathbb{R})$. Assume that for every $k\in\mathbb{N}$ there are constants $C_{k},\widetilde{C}_{k}$ such that

[TABLE]

Then the analysis operator $U_{E}$ is continuous from $\mathcal{S}$ into ${\bf s}$, the synthesis operator $T_{E}$ is continuous from ${\bf s}$ into $\mathcal{S}$, and the frame operator $T_{E}U_{E}$ is continuous from $\mathcal{S}$ into $\mathcal{S}$.

Now, we consider sub-exponential weights.

Proposition 4.3**.**

Let $\beta\in(0,1)$ and let the assumptions of the first part of Lemma 2.2 hold with the weights $\mu_{k}(x)=e^{k|x|^{\beta}}$, $k\in\mathbb{N}_{0}$. Let $E=(e_{n})_{n=1}^{\infty}$ be a sequence with elements from $X_{F}$ which is a frame for ${H}$. Then the following statements hold.

• (i)

Assume that there exist constants $\gamma_{0}\in\mathbb{N}$, $C>0$ such that for every $k\in\mathbb{N}$ there exists $C_{k}>0$ such that

[TABLE]

Then the analysis operator $f\to U_{E}f=(\langle f,e_{m}\rangle)_{m=1}^{\infty}$ is a continuous one from $X_{F}$ into ${\mathfrak{s}}^{\beta}$.

• (ii)

Assume that there exist constants $\tilde{\gamma}_{0}\in\mathbb{N}$, $\tilde{C}>0$ such that for every $k\in\mathbb{N}$ there exists $\tilde{C}_{k}>0$ such that

[TABLE]

Then the synthesis operator $(c_{n})_{n}\to T_{E}(c_{n})=\sum c_{n}e_{n}$ is a continuous one from ${\mathfrak{s}}^{\beta}$ into $X_{F}$.

• (iii)

If (11) and (12) hold, then the frame operator $T_{E}U_{E}$ is continuous from $X_{F}$ into $X_{F}$.

**Proof.  **Under the given assumptions, $\Theta_{F}$ is the space ${\mathfrak{s}}^{\beta}$. The rest of the proof can be done in a similar way as the proof of Proposition 4.1, using Corollary 3.8 instead of Corollary 3.3. $\Box$

If in the above proposition one chooses $G$ to be the Hermite basis $(h_{n})_{n=1}^{\infty}$ and $\beta=1/(2\alpha)$, $\alpha>1/2$, then $X_{F}=\Sigma^{\alpha}$.

Corollary 4.4**.**

Let $\alpha>1/2$. Let $(e_{n})_{n=1}^{\infty}$ be a sequence with elements from $\Sigma^{\alpha}$ which is a frame for $L^{2}(\mathbb{R})$ and such that for every $k\in\mathbb{N}$ there are constants $C_{k},\widetilde{C}_{k}$ such that

[TABLE]

Then the analysis operator $U_{E}$ is continuous from $\Sigma^{\alpha}$ into ${\mathfrak{s}}^{1/(2\alpha)}$, the synthesis operator $T_{E}$ is continuous from ${\mathfrak{s}}^{1/(2\alpha)}$ into $\Sigma^{\alpha}$, and the frame operator $T_{E}U_{E}$ is continuous from $\Sigma^{\alpha}$ into $\Sigma^{\alpha}$.

5 Boundedness and Banach frames derived from sub-exponential localization of frames

In this section we extend statements from [21] for polynomially and exponentially localized frames to the case of sub-exponentially localized frames (Theorem 5.4 below). We will use the Jaffard’s theorem [24] given there for the sub-exponential and exponential case (see Theorem 5.2 below).

First recall the Schur’s test: *If $(A_{m,n})_{m,n\in\mathbb{N}}$ is an infinite matrix satisfying $\sup_{s\in\mathbb{N}}\sum_{t\in\mathbb{N}}|A_{m,n}|\leq K_{1}$ and $\sup_{t\in\mathbb{N}}\sum_{s\in\mathbb{N}}|A_{m,n}|\leq K_{2}$, then the correspondent matrix frame type operator $\mathcal{A}$ is well defined and bounded from $\ell^{p}$ into $\ell^{p}$ for $1\leq p\leq\infty$ and the operator norm $\|A\|_{\ell^{p}\to\ell^{p}}\leq K_{1}^{1/p^{\prime}}K_{2}^{1/p}$. *

Let $\beta\in(0,1]$ and $\gamma\in(0,\infty)$. Define $\mathcal{E}_{\gamma,\beta}$

to be the space of matrices $(A_{m,n})_{m,n\in\mathbb{N}}$ satisfying the following condition:

[TABLE]

By the Schur’s test, when $(A_{m,n})_{m,n\in\mathbb{N}}\in\mathcal{E}_{\gamma,\beta}$, then the correspondent matrix type operator $\mathcal{A}$ is well defined and bounded from $\ell^{2}$ into $\ell^{2}$, and for the operator norm one has that $\|\mathcal{A}\|_{\ell^{2}\to\ell^{2}}\leq 2C_{\gamma}P_{\gamma,\beta}$, where $C_{\gamma}$ is the constant from (13) and $P_{\gamma,\beta}$ denotes the sum of the convergent series $\sum_{j=0}^{\infty}e^{-\gamma j^{\beta}}$.

We will also need the following statements, which extend [21, Lemmas 2 and 3] to the case of sub-exponential localization.

Lemma 5.1**.**

For every $\gamma\in(0,\infty)$ and $\beta\in(0,1]$, the following holds.

(a)* There exists a positive number $C$ so that $\sum_{k\in\mathbb{N}}e^{-\gamma|m-k|^{\beta}}e^{-\gamma|k-n|^{\beta}}\leq Ce^{-(\gamma/2)|m-n|^{\beta}}$ for every $s,t\in\mathbb{N}$.*

(b)* If the matrix $(A_{m,n})_{m,n\in\mathbb{N}}$ belongs to $\mathcal{E}_{\gamma,\beta}$ and $\mu$ is a $\beta_{\mu}$-sub-exponential weight with $\beta_{\mu}<\beta$, then $\mathcal{A}$ maps boundedly $\ell^{p}_{\mu}(\mathbb{N}_{0})$ into $\ell^{p}_{\mu}(\mathbb{N}_{0})$ for every $p\in[1,\infty]$.*

**Proof.  **(a) can be proved following the idea of [21, Lemma 2].

(b) Let $\gamma_{\mu}$ comes from the $\beta_{\mu}$-sub-exponential weight $\mu$, i.e., $\mu(t+x)\leq C_{\mu}e^{\gamma_{\mu}|t|^{\beta_{\mu}}}\mu(x),\ t,x\in\mathbb{R}.$ Take $k>\max(1,\gamma\beta/(\gamma_{\mu}\beta_{\mu}))$ and use the assumption $\beta_{\mu}<\beta$ to observe that there is a constant $C_{1}\in(0,\infty)$ so that $e^{-(\gamma/k)|x-n|^{\beta}}\mu(n)^{-1}\mu(x)\leq C_{1}$. The rest of the proof can be done using a similar approach as in [21, Lemma 3]. $\Box$

Theorem 5.2**.**

(Jaffard)* Given $\beta\in(0,1]$ and $\gamma\in(0,\infty)$, let $(A_{m,n})_{m,n\in\mathbb{N}}\in\mathcal{E}_{\gamma,\beta}$ and let the corresponding matrix type operator $\mathcal{A}$ be invertible on $\ell^{2}$. Then $\mathcal{A}^{-1}\in\mathcal{E}_{\gamma_{1},\beta}$ for some $\gamma_{1}\in(0,\gamma)$.*

In the appendix we will give a sketch of the Jaffard’s proof.

Remark 5.3**.**

The exponential localization type condition $|A_{m,n}|\leq C_{\gamma}e^{-\gamma|m-n|}$, $m,n\in\mathbb{N}$, considered in [21], implies that

[TABLE]

Here we consider the more general case intrinsically related to $\beta\in(0,1)$.

Theorem 5.4**.**

Let $p\in[1,\infty)$ and $G$ be a Riesz basis for ${H}$, and let $E$ be a frame for ${H}$ which is $\beta$-sub-exponentially or exponentially localized (respectively, $(k+1+\varepsilon)$-localized for some $\varepsilon>0$) with respect to $G$. Let $\mu$ be $\beta_{\mu}$-sub-exponential weight and let $\beta_{\mu}<\beta$ in the case of $\beta$-sub-exponentially localized frame $E$ (respectively, let $\mu$ be a $k$-moderate weight) with $\mu(n)\geq 1$ for every $n\in\mathbb{N}$.

Then for every $p\in[1,\infty)$ the following statements hold.

• (i)

The analysis operator $U_{E}$ maps boundedly ${H}_{G}^{\ell^{p}_{\mu}}$ into $\ell^{p}_{\mu}$.

• (ii)

The synthesis operator $T_{E}$ maps boundedly $\ell^{p}_{\mu}$ into ${H}_{G}^{\ell^{p}_{\mu}}$.

• (iii)

The frame operator $S_{E}=T_{E}U_{E}$ is invertible on ${H}_{G}^{\ell^{p}_{\mu}}$ and the series in $S_{E}f=\sum_{n=1}^{\infty}\langle f,e_{n}\rangle e_{n}$ converges unconditionally.

• (iv)

The canonical dual frame $(\widetilde{e_{n}})_{n=1}^{\infty}$ of $(e_{n})_{n=1}^{\infty}$ has the same type of localization as $(e_{n})_{n=1}^{\infty}$, i.e., it is $\beta$-sub-exponentially or exponentially localized (resp. $(k+1+\varepsilon)$-localized) ** with respect to $G$.**

• (v)

The frame expansions $f=\sum_{n=1}^{\infty}\langle f,e_{n}\rangle\widetilde{e_{n}}=\sum_{n=1}^{\infty}\langle f,\widetilde{e_{n}}\rangle e_{n}$ hold with unconditional convergence in ${H}_{G}^{\ell^{p}_{\mu}}$.

• (vi)

There is norm equivalence between $\|f\|_{{H}_{G}^{\ell^{p}_{\mu}}}$, ${\||{[}\||}(\langle f,e_{n}\rangle)_{n=1}^{\infty}]_{\ell^{p}_{\mu}}$, and ${\||{[}\||}(\langle f,\widetilde{e_{n}}\rangle)_{n=1}^{\infty}]_{\ell^{p}_{\mu}}$.

**Proof.  **In the cases of polynomial and exponential localization, the assertions are given in [21, Prop. 8 and Prop. 10]. For the sub-exponential case, one can proceed in a similar way, but to use the Jaffard’s theorem Theorem 5.2 and Lemma 5.1. For the sake of completeness, we sketch a proof.

Consider the matrix $(A_{m,n})_{m,n\in\mathbb{N}}$ with the property $|A_{m,n}|\leq Ce^{-\gamma|m-n|^{\beta}}$, for some $C>0$ and $\gamma>0$.

(i) Let $f\in{H}_{G}^{\ell^{p}_{\mu}}$ and thus $(|\langle f,\widetilde{g_{n}}\rangle|)_{n=1}^{\infty}\in{\ell^{p}_{\mu}}$. By Lemma 5.1(b), we have that $\mathcal{A}(|\langle f,\widetilde{g_{n}}\rangle|)_{n=1}^{\infty}$ belongs to ${\ell^{p}_{\mu}}$. Furthermore, for $m\in\mathbb{N}$ we have

[TABLE]

Therefore, $(\langle f,e_{m}\rangle)_{m=1}^{\infty}$ also belongs to ${\ell^{p}_{\mu}}$ and

[TABLE]

(ii) Let $c=(c_{n})_{n=1}^{\infty}\in\ell^{p}_{\mu}(\subseteq\ell^{2})$. Then the series $\sum_{n=1}^{\infty}c_{n}e_{n}$ converges in ${H}$ and let us denote its sum by $y$. Since $\mathcal{A}(|c_{n}|)_{n=1}^{\infty}\in\ell^{p}_{\mu}$ by Lemma 5.1(b), and since $|\langle y,\widetilde{g}_{m}\rangle|\leq C\sum_{n=1}^{\infty}|c_{n}||\langle e_{n},\widetilde{g}_{m}\rangle|\leq C\sum_{n=1}^{\infty}A_{m,n}|c_{n}|$ for every $m\in\mathbb{N}$, it follows that $U_{\widetilde{G}}y\in\ell^{p}_{\mu}$ and therefore the element $T_{G}U_{\widetilde{G}}y=y$ belongs to ${H}_{G}^{\ell^{p}_{\mu}}$. Hence, $T_{E}$ maps $\ell^{p}_{\mu}$ into ${H}_{G}^{\ell^{p}_{\mu}}$ and furthermore, $\|T_{E}c\|_{{H}_{G}^{\ell^{p}_{\mu}}}=\|y\|_{{H}_{G}^{\ell^{p}_{\mu}}}=\|U_{\widetilde{G}}y\|_{\ell^{p}_{\mu}}\leq C\|\mathcal{A}(|c_{n}|)\|_{\ell^{p}_{\mu}}$, which by Lemma 5.1 implies that $\|T_{E}c\|_{{H}_{G}^{\ell^{p}_{\mu}}}\leq C\|\mathcal{A}\|\|c\|_{\ell^{p}_{\mu}}$.

(iii) By (i) and (ii), $S_{E}$ maps boundedly ${H}_{G}^{\ell^{p}_{\mu}}$ into ${H}_{G}^{\ell^{p}_{\mu}}$. For the unconditional convergence, take any re-ordering $N_{1}$ of $\mathbb{N}$. Let $f\in{H}_{G}^{\ell^{p}_{\mu}}$. Consider $\sum_{n\in N_{1}}\langle f,e_{n}\rangle e_{n}$ and take $\varepsilon>0$. Since $(\langle f,e_{n}\rangle)_{n\in N_{1}}\in\ell^{p}_{\mu}$, there is a finite set $N_{2}\subset N_{1}$ so that $\|(\langle f,e_{n}\rangle)_{n\in N_{1}\setminus N_{2}}\|_{\ell^{p}_{\mu}}<\varepsilon$. Then for every finite $N_{3}$ such that $N_{3}\supset N_{2}$, $N_{3}\subset N_{1}$, one has that $\|Sf-\sum_{n\in N_{3}}\langle f,e_{n}\rangle e_{n}\|\leq\|T_{E}\|\|(\langle f,e_{n}\rangle)_{n\in N_{1}\setminus N_{2}}\|_{\ell^{p}_{\mu}}<\|T_{E}\|\varepsilon.$ Therefore, $\sum_{n\in N_{1}}\langle f,e_{n}\rangle e_{n}$ converges to $Sf$.

Finally, let us show the bijectivity of $S_{E}$ on ${H}_{G}^{\ell^{p}_{\mu}}$. Consider the operator $\mathcal{V}:=U_{\widetilde{G}}S_{E}T_{G}$ and observe that it is invertible on $\ell^{2}$ and it maps boundedly $\ell^{p}_{\mu}$ into $\ell^{p}_{\mu}$. Let $(V_{m,n})_{m,n\in\mathbb{N}}$ be the corresponding matrix of $\mathcal{V}$. Since $|V_{m,n}|=|\langle S_{E}g_{n},\widetilde{g_{m}}\rangle|\leq\sum_{i}|\langle g_{n},e_{i}\rangle|\cdot|\langle e_{i},\widetilde{g_{m}}\rangle|$, by Lemma 5.1(a), there is a positive constant $C$ so that $|V_{m,n}|\leq Ce^{-(\gamma/2)|m-n|^{\beta}}$. Now by Theorem 5.2 it follows that $\mathcal{V}^{-1}\in\mathcal{E}_{\gamma_{1},\beta}$ for some $\gamma_{1}\in(0,\gamma)$. By Lemma 5.1, it follows that $V^{-1}$ maps boundedly $\ell^{p}_{\mu}$ into $\ell^{p}_{\mu}$. Therefore, $V$ is a bounded bijection of $\ell^{p}_{\mu}$ onto $\ell^{p}_{\mu}$. Now the representation $S_{E}=T_{G}VU_{\widetilde{G}}$ implies that $S_{E}$ is a bounded bijection of ${H}_{G}^{\ell^{p}_{\mu}}$ onto ${H}_{G}^{\ell^{p}_{\mu}}$.

(iv) For $m,n\in\mathbb{N}$, $\langle\widetilde{e_{m}},g_{n}\rangle=\sum_{j=1}^{\infty}\langle e_{m},g_{j}\rangle\overline{(V^{-1})_{nj}}$ and $\langle\widetilde{e_{m}},\widetilde{g_{n}}\rangle=\sum_{j=1}^{\infty}\langle e_{m},\widetilde{g_{j}}\rangle\overline{(V^{-1})_{nj}}$ and one can apply Theorem 5.2 and Lemma 5.1(a) to conclude.

(v) follows from (iii) and for (vi) one can use the representations $f=S_{E}^{-1}S_{E}=S_{\widetilde{E}^{-1}}S_{\widetilde{E}}f$ and the already proved (i)-(iv). $\Box$

6

Expansions in Fréchet spaces via localized frames

Our goal is expansion of elements of a Fréchet space and its dual via localized frames and coefficients in a corresponding Fréchet sequence space. First we present in the next theorem general results related to frames localized with respect to a Riesz basis. In the next secton we will apply this theorem using frames localized with respect to the Hermite orthonormal basis in order to obtain frame expansions in the spaces $\mathcal{S}$ and $\Sigma^{\alpha}$, $\alpha>1/2$, and their duals.

Theorem 6.1**.**

Let $G$ be a Riesz basis for ${H}$, $k\in\mathbb{N}_{0}$, and $\mu_{k}$ be a $\beta_{k}$-sub-exponential (resp. $k$-moderate) weight so that (5) holds Let the spaces $\Theta_{k}$ and $X_{k}$ be as in Lemma 2.2. Assume that $E=(e_{n})_{n=1}^{\infty}$ is a sequence with elements in $X_{F}$ forming a frame for ${H}$ which is $\beta$-sub-exponentially localized with $\beta>\beta_{k}$ for all $k\in\mathbb{N}_{0}$ or exponentially localized (respectively, $s$-localized $s$ for every $s\in\mathbb{N}$) with respect to $G$. Then, $\widetilde{e_{n}}\in X_{F}$, $n\in\mathbb{N}$, and the following statements hold:

• (i)

The analysis operator $U_{E}$ is $F$-bounded from $X_{F}$ into $\Theta_{F}$, the synthesis operator $T_{E}$ is $F$-bounded from $\Theta_{F}$ into $X_{F}$, and the frame operator $S_{E}$ is $F$-bounded and bijective from $X_{F}$ onto $X_{F}$ with unconditional convergence of the series in $S_{E}f=\sum_{n=1}^{\infty}\langle f,e_{n}\rangle e_{n}$.

• (ii)

For every $f\in X_{F}$,

[TABLE]

with $(\langle f,\widetilde{e_{n}}\rangle)_{n=1}^{\infty}\in\Theta_{F}$ and $(\langle f,e_{n}\rangle)_{n=1}^{\infty}\in\Theta_{F}$.

• (iii)

If $X_{F}$ and $\Theta_{F}$ have the following property with respect to $(g_{n})_{n=1}^{\infty}$:

[TABLE]

then $X_{F}$ and $\Theta_{F}$ also have the properties $\mathcal{P}_{(e_{n})}$ and $\mathcal{P}_{(\widetilde{e_{n}})}$.

• (iv)

Both sequences $({e_{n}})_{n=1}^{\infty}$ and $(\widetilde{e_{n}})_{n=1}^{\infty}$ form Fréchet frames for $X_{F}$ with respect to $\Theta_{F}$.

• (v)

For every $g\in X_{F}^{*}$,

[TABLE]

with $(g(e_{n}))_{n=1}^{\infty}\in\Theta_{F}^{*}$ and $(g(\widetilde{e_{n}}))_{n=1}^{\infty}\in\Theta_{F}^{*}$.

• (vi)

If $(a_{n})_{n=1}^{\infty}\in\Theta_{F}^{*}$, then $\sum_{n=1}^{\infty}a_{n}{e}_{n}$ (resp. $\sum_{n=1}^{\infty}a_{n}\widetilde{e_{n}}$) converges in $X_{F}^{*}$, i.e., the mapping $f\mapsto\sum_{n=1}^{\infty}\langle f,e_{n}\rangle a_{n}$ (resp. $f\mapsto\sum_{n=1}^{\infty}\langle f,\widetilde{e_{n}}\rangle a_{n}$) determines a continuous linear functional on $X_{F}$.

**Proof.  **(i) The properties for $U_{E}$, $T_{E},$ and $S_{E}$ follow easily using Theorem 5.4(i)-(iii).

Further, the bijectivity of $S_{E}$ on $X_{F}$ implies that $\widetilde{e_{n}}\in X_{F}$ for every $n\in\mathbb{N}$.

(ii) By Theorem 5.4(v), for every $k\in\mathbb{N}$ and every $f\in X_{k}$ we have that $f=\sum_{n=1}^{\infty}\langle f,\widetilde{e_{n}}\rangle e_{n}=\sum_{n=1}^{\infty}\langle f,e_{n}\rangle\widetilde{e_{n}}$ with convergence in $X_{k}$. This implies that for every $f\in X_{F}$, one has that $f=\sum_{n=1}^{\infty}\langle f,\widetilde{e_{n}}\rangle e_{n}=\sum_{n=1}^{\infty}\langle f,e_{n}\rangle\widetilde{e_{n}}$ with convergence in $X_{F}$.

For every $k\in\mathbb{N}$ and every $f\in X_{k}$, by Theorem 5.4(i), we have that $(\langle f,e_{n}\rangle)_{n=1}^{\infty}\in\Theta_{k}$. Therefore, $(\langle f,e_{n}\rangle)_{n=1}^{\infty}\in\Theta_{F}$ for every $f\in X_{F}$. Furthermore, by Theorem 5.4(iv), $(\widetilde{e_{n}})_{n=1}^{\infty}$ has the same type of localization with respect to $G$ as $(e_{n})_{n=1}^{\infty}$. Thus, applying Theorem 5.4(i) with $(\widetilde{e_{n}})_{n=1}^{\infty}$ as a starting frame, we get that $(\langle f,\widetilde{e_{n}}\rangle)_{n=1}^{\infty}\in\Theta_{F}$ for $f\in X_{F}$.

(iii) If $f\in X_{F}$, it is already proved in (i) that $(\langle f,e_{n}\rangle)_{n=1}^{\infty}\in\Theta_{F}$ and $(\langle f,\widetilde{e_{n}}\rangle)_{n=1}^{\infty}\in\Theta_{F}$. To complete the proof of $\mathcal{P}_{(e_{n})}$, assume that $f\in{H}$ is such that $(\langle f,e_{n}\rangle)_{n=1}^{\infty}\in\Theta_{F}$. Consider

[TABLE]

Let $k\in\mathbb{N}$. Since $(\langle f,e_{j}\rangle)_{j=1}^{\infty}\in\ell^{2}_{\mu_{k}}$ and by Theorem 5.4(iv), $(\widetilde{e_{n}})_{n=1}^{\infty}$ has the same type of localization with respect to $G$ as $(e_{n})_{n=1}^{\infty}$, it follows from Lemma 5.1(b) (for the case of sub-exponential localization) and from the way of the proof of [21, Lemma 3] (for the case of polynomial and exponential localization) that $(\sum_{j=1}^{\infty}\langle\widetilde{e_{j}},g_{n}\rangle\langle f,e_{j}\rangle)_{n=1}^{\infty}\in\ell^{2}_{\mu_{k}}$. Therefore, $(\langle f,g_{n}\rangle)_{n=1}^{\infty}\in\Theta_{F}$ and thus, by $\mathcal{P}_{(g_{n})}$, it follows that $f\in X_{F}$. For completing the proof of $\mathcal{P}_{(\widetilde{e_{n}})}$, if $f\in{H}$ is such that $(\langle f,\widetilde{e_{n}}\rangle)_{n=1}^{\infty}\in\Theta_{F}$, it follows in a similar way as above that $f\in X_{F}$.

(iv) By (i), $(\mathbb{e}_{n}(f))_{n=1}^{\infty}\in\Theta_{F}$ for $f\in X_{F}$, and by Theorem 5.4(vi), for $k\in\mathbb{N}$ and $f\in X_{k}$, the norms ${\||{[}\||}(\langle f,e_{n}\rangle)_{n=1}^{\infty}]_{\Theta_{k}}$ and $\|f\|_{X_{k}}$ are equivalent. Furthermore, it follows from Theorem 5.4 that the operator $V:=S_{E}^{-1}T_{E}|_{{}_{\Theta_{F}}}$ maps $\Theta_{F}$ into $X_{F}$ and it is $F$-bounded. Clearly, $V(\mathbb{e}_{n}(f))_{n=1}^{\infty}=f$, $f\in X_{F}$. Therefore, $(\mathbb{e}_{n})_{n=1}^{\infty}$ is an $F$-frame for $X_{F}$ with respect to $\Theta_{F}$. In an analogue way, $(\widetilde{\mathbb{e}_{n}})_{n=1}^{\infty}$ is also an $F$-frame for $X_{F}$ with respect to $\Theta_{F}$.

(v) The representations in (i) can be re-written as $f=\sum_{n=1}^{\infty}\widetilde{{\bf e}_{n}}(f)e_{n}=\sum_{n=1}^{\infty}{\bf e}_{n}(f)\widetilde{e_{n}}$, $f\in X_{F}$, which implies validity of (15) for $g\in X_{F}^{*}$.

For the rest of the proof, consider the $F$-bounded (and hence continuous) operator $V$ from the proof of (iv) and observe that $\widetilde{e_{n}}=V\delta_{n}$, $n\in\mathbb{N}$. This implies that for $g\in X_{F}^{*}$ we have $(g(\widetilde{e_{n}}))_{n=1}^{\infty}=(gV(\delta_{n}))_{n=1}^{\infty}\in\Theta_{F}^{\circledast}$. With similar arguments, considering the operator $\widetilde{V}=S_{\widetilde{E}}^{-1}T_{\widetilde{E}}|_{{}_{\Theta_{F}}}$, it follows that $(g(e_{n}))_{n=1}^{\infty}\in\Theta_{F}^{\circledast}$.

(vi) Let $(a_{n})_{n=1}^{\infty}\in\Theta_{F}^{*}$ and thus there is $k_{0}\in\mathbb{N}$ so that $(a_{n})_{n=1}^{\infty}\in\Theta_{k_{0}}^{*}$, i.e., $C:=\sum_{n=1}^{\infty}|a_{n}|^{2}|\mu(n)|^{-2k_{0}}<\infty$. By Theorem 5.4(vi), there is a positive constant $B_{k_{0}}$ so that ${\||{[}\||}(\langle f,e_{n}\rangle)_{n=1}^{\infty}]_{\Theta_{k_{0}}}\leq B_{k_{0}}\|f\|_{X_{k_{0}}}$ for every $f\in X_{F}$. Let $f\in X_{F}$. By (i), $(\langle f,e_{n}\rangle)_{n=1}^{\infty}\in\Theta_{F}$. Therefore, $\sum_{n=1}^{\infty}\langle f,e_{n}\rangle a_{n}$ converges and furthermore,

[TABLE]

which implies continuity of the linear mapping $f\mapsto\sum_{n=1}^{\infty}\langle f,e_{n}\rangle a_{n}$. In a similar way, it follows that $f\mapsto\sum_{n=1}^{\infty}\langle f,\widetilde{e_{n}}\rangle a_{n}$ determines a continuous linear functional on $X_{F}$. $\Box$

Remark 6.2**.**

Note that in the setting of the above theorem, when $G$ is an orthonormal basis of ${H}$ or more generally, when $G$ is a Riesz basis for ${H}$ satisfying any of the following two conditions:

$(\mathcal{P}_{1})$:  $\forall s\in\mathbb{N}\ \,\exists C_{s}>0\ :\ |\langle g_{m},g_{n}\rangle|\leq C_{s}(1+|m-n|)^{-s},\ m,n\in\mathbb{N}$,

$(\mathcal{P}_{2})$:  $\exists s>0\ \,\exists C_{s}>0\ :\ |\langle g_{m},g_{n}\rangle|\leq C_{s}\mathrm{e}^{-s|m-n|},\ m,n\in\mathbb{N},$

then the property $\mathcal{P}_{(g_{n})}$ is satisfied.

6.1 Frame expansions of tempered distributions and ultradistributions

Here we apply Theorem 6.1 to obtain series expansions in the spaces $\mathcal{S}$ and $\Sigma^{\alpha}$ ($\alpha>1/2$), and their dual spaces, via frames which are localized with respect to the Hermite basis.

Theorem 6.3**.**

Assume that the sequence $(e_{n})_{n=1}^{\infty}$ with elements from $\mathcal{S}(\mathbb{R})$ (resp. in $\Sigma^{\alpha}$) is a frame for $L^{2}(\mathbb{R})$ which is polynomially (resp. sub-exponentially or exponentially) localized with respect to the Hermite basis $(h_{n})_{n=1}^{\infty}$ with decay $\gamma$ for every $\gamma\in\mathbb{N}$. Let $(g_{n})_{n=1}^{\infty}=(h_{n})_{n=1}^{\infty}$. Then $\mathcal{P}_{(g_{n})}$ and the conclusions in Theorem 6.1 hold with $X_{F}$ replaced by $\mathcal{S}$ (resp. $\Sigma^{\alpha}$) and $\Theta_{F}$ replaced by ${\bf s}$ (resp. ${\mathfrak{s}}^{1/(2\alpha)}$).

**Proof.  **For $k\in\mathbb{N}_{0}$, consider the $k$-moderate weight $\mu_{k}(x)=(1+|x|)^{k}$ the spaces $\Theta_{k}:=\ell^{2}_{\mu_{k}}$, $k\in\mathbb{N}_{0}$, satisfy (1)-(2) and their projective limit $\Theta_{F}$ is the space ${\bf s}$. Consider the spaces $X_{k}:=H^{\Theta_{k}}_{(h_{n})}$, $k\in\mathbb{N}_{0}$, which satisfy (1)-(2). As observed after Theorem 6.1, the property $\mathcal{P}_{(h_{n})}$ is satisfied. Since for $f\in L^{2}(\mathbb{R})$ one has that $f\in\mathcal{S}$ if and only if $(\langle f,h_{n}\rangle_{n=0}^{\infty})\in{\bf s}$, it now follows that $X_{F}=\mathcal{S}$. Then the conclusions of Theorem 6.3 follow from Theorem 6.1.

The respective part of the theorem follows in a similar way. $\Box$

As noticed in [32], having in mind the known expansions of tempered distributions $(\mathcal{S}(\mathbb{R}_{+}))^{\prime}$ [35, 15] and Beurling ultradistributions $(G^{\alpha}_{\alpha}(\mathbb{R}_{+}))^{\prime}$ [16, 23, 22], and their test spaces, by the use of the Laguerre orthonormal basis $l_{n},n\in\mathbb{N},$ and validity of the corresponding properties $\mathcal{P}_{(l_{n})}$, we can transfer the above results to the mentioned classes of distributions and ultradistributions over $\mathbb{R}_{+}.$

Remark 6.4**.**

For Proposition 4.1 (resp. 4.3), it is of interest to consider cases when $X_{F}$ is the space $\mathcal{S}$ (resp. $\Sigma^{\alpha}$). Based on Theorem 6.3, we can clarify such cases. If $G$ is a frame for $L^{2}(\mathbb{R})$ with elements from $\mathcal{S}$ and polynomially localized with respect to $(h_{n})_{n=1}^{\infty}$, then one also has $X_{F}=\mathcal{S}$ since in this case Theorem 6.3 implies that $f\in\mathcal{S}$ if and only if $(\langle f,\widetilde{g_{n}}\rangle)\in{\bf s}$, and besides that one also has that $f\in X_{F}$ if and only if $(\langle f,\widetilde{g_{n}}\rangle)\in{\bf s}$.

Concerning Proposition 4.3, with $\alpha>1/2$ and $\beta=1/(2\alpha)$, we have the similar conclusion for $\Sigma^{\alpha}$.

Example 6.5**.**

As an illustration of Theorems 6.1 and 6.3, we give the next example. Let $r\in\mathbb{N}$ and for $i=1,2,\ldots,r$, take $\varepsilon_{i}\geq 0$ and a sequence $(a_{n}^{i})_{n=1}^{\infty}$ of complex numbers satisfying $|a_{n}^{i}|\leq\varepsilon_{i}$ for $n\geq 2$, $\sum_{i=1}^{r}|a^{i}_{1}|\leq 1$, and $\sum_{i=1}^{r}\varepsilon_{i}<1$. For $n\in\mathbb{N}$, consider $e_{n}:=h_{n}+\sum_{i=1}^{r}a_{n}^{i}h_{n+i}$. The sequence $(e_{n})_{n=1}^{\infty}$ is a Riesz basis for $L^{2}(\mathbb{R})$ and it is $s$-localized with respect to the Hermite orthonormal basis $(h_{n})_{n=1}^{\infty}$ for every $s>0$, as well as exponentially localized with respect to $(h_{n})_{n=1}^{\infty}$. In order to show that $(e_{n})_{n=1}^{\infty}$ is a Riesz basis for $L^{2}(\mathbb{R})$, we will represent $(e_{n})_{n=1}^{\infty}$ as a sequence $(Uh_{n})_{n=1}^{\infty}$ for some bounded bijective operator from $L^{2}(\mathbb{R})$ onto $L^{2}(\mathbb{R})$ using similar techniques as in [6, Example 1]. Define $U$ by $Uh_{n}:=h_{n}+\sum_{i=1}^{r}a_{n}^{i}h_{n+i}$, $n\in\mathbb{N}$, and by linearirty on the linear span of $h_{n},n\in\mathbb{N}$. The obtained operator is bounded on the linear span, so extend it by continuity on $L^{2}(\mathbb{R})$, leading to a bounded operator on $L^{2}(\mathbb{R})$. It remains to show the bijectivity of $U$. Let $f\in L^{2}(\mathbb{R})$ Then

[TABLE]

and thus, $\|Uf\|\geq|\langle f,h_{1}\rangle|$. Furthermore, for $f\in L^{2}(\mathbb{R})$,

[TABLE]

leading to

[TABLE]

[TABLE]

Since $\frac{3+\sum_{i=1}^{r}\varepsilon_{i}}{4}<1$, it follows from [6, Lemma 1] that the bounded operator $U$ is bijective on $L^{2}(\mathbb{R})$ and thus $(e_{n})_{n=1}^{\infty}$ is a Riesz basis for $L^{2}(\mathbb{R})$.

Note that under the assumptions of the example, the classical way to obtain invertibility of $U$ does not apply, because $\|U-Id\|$ is not necessarily smaller then $1$.

Remark 6.6**.**

As explained in [21], when dealing with Gabor frames and localization, the natural bases to be considered in this respect are the Wilson bases, but then Gabor frames are not localized with respect to a Wilson basis in the strict sense of the definition of polynomial and exponential localization. However, under appropriate conditions, the authors of [21] still obtain statements in the spirit of Theorem 5.4 which now leads to conclusions as in Theorem 6.1.

7 Appendix

We add in Jaffard’s proof some comments in the end. We also give a simple known assertions for the class $\mathcal{E}_{\gamma,\beta}$ defined on page 5.

Lemma 7.1**.**

Let $\mathcal{A}\in\mathcal{E}_{\gamma,\beta}$.

• (i)

If $B$ belongs to $\mathcal{E}_{\gamma^{\prime},\beta}$ for some $\gamma^{\prime}\in(0,\gamma)$, then $AB$ belongs to $\mathcal{E}_{\gamma^{\prime},\beta}$.

• (ii)

If $B$ belongs to $\mathcal{E}_{\gamma,\beta}$, then $AB$ belongs to $\mathcal{E}_{\gamma^{\prime},\beta}$ for every $\gamma^{\prime}\in(0,\gamma)$.

**Proof.  **(i) Under the assumptions, there exist positive constants $C_{A}$ and $C_{B}$ so that $\ |A_{m,n}|\leq C_{A}e^{-\gamma|m-n|^{\beta}}$ and $\ |B_{m,n}|\leq C_{B}e^{-\gamma^{\prime}|m-n|^{\beta}}$ for $m,n\in\mathbb{N}$. Further, using $|m-k|^{\beta}+|k-n|^{\beta}\geq|m-n|^{\beta},$ for every $m,n\in\mathbb{N}$, one has

[TABLE]

(ii) follows from (i). $\Box$

Now we give some details for the Jaffard’s proof of Theorem 5.2, providing explicit estimates for the bounds.

As in [24], $AA^{*}$ belongs to $\mathcal{E}_{\gamma^{\prime},\beta}$ for $\gamma^{\prime}\in(0,\gamma)$ by Lemma 7.1, $AA^{*}=\|AA^{*}\|(Id-R)$ for some operator $R$ with $||R||=r<1$, and $(AA^{*})^{-1}=\|AA^{*}\|^{-1}\sum_{n=0}^{\infty}R^{n}$. With the method from [24], one obtains

[TABLE]

where $\gamma^{\prime\prime}\in(0,\gamma^{\prime})$, $C_{1}=1+\frac{C_{AA^{*}}}{\|AA^{*}\|}$ (with $C_{AA^{*}}$ satisfying $\ |(AA^{*})_{m,n}|\leq C_{AA^{*}}e^{-\gamma^{\prime}|m-n|^{\beta}}$ for every $s,t\in\mathbb{N}$), and $P=P_{\gamma^{\prime}-\gamma^{\prime\prime},\beta}=\sum_{j=0}^{\infty}e^{-(\gamma^{\prime}-\gamma^{\prime\prime})j^{\beta}}.$

For k=0, $(R^{k})_{m,n}$ equals $1$ when $m=n$ and [math] otherwise, so we have $|(R^{0})_{m,n}|\leq e^{-p|m-n|^{\beta}}$ for every $p$.

Fix $|m-n|$ and let $\varepsilon\in(0,1)$. With $K:=C_{1}2P(>1)$, let

[TABLE]

Denote by $n_{0}$ the highest natural number such that $n_{0}\in\Lambda_{|m-n|}.$ Then

[TABLE]

For $n>n_{0}$ we have $r^{n}<r^{n_{0}}=e^{n_{0}\ln r}$ and hence,

[TABLE]

Let

[TABLE]

Therefore,

[TABLE]

Now using the representation $A^{-1}=A^{*}(AA^{*})^{-1}$ and Lemma 7.1, we can conclude that

[TABLE]

Acknowledgements The authors acknowledge support from OeAD GmbH through the Scientific and Technological Cooperation projects MULT_DR 01/2017 and SRB 01/2018, from the Vienna Science and Technology Fund (WWTF) through project VRG12-009, and from Project 174024 of the Serbian Ministry of Science. The second author is grateful for the hospitality of the University of Novi Sad, where most of the research on the presented topic was done.