An Axiomatic Approach to Mild Distributions

Abstract

The Banach Gelfand Triple $(S_0,L^2,S_0^{\prime })$ consists of the Feichtinger algebra $S_0\left(R^d\right)$ as a space of test functions, the dual space $S_0^{\prime }\left(R^d\right)$, known as the space of mild distributions, and the intermediate Hilbert space $L^2\left(R^d\right)$. This Gelfand Triple is very useful for the description of mathematical problems in the area of time-frequency analysis, but also for classical Fourier analysis and engineering applications. Because the involved spaces are Banach spaces, we speak of a Banach Gelfand Triple, in contrast to the widespread concept of rigged Hilbert spaces, which usually involve nuclear Frechet spaces. Still, both concepts serve very similar purposes. Based on the manifold properties of $S_0\left(R^d\right)$, it has found applications in the derivation of mathematical statements related to Gabor Analysis but also in providing an alternative and more lucid description of classical results, such as the Shannon sampling theory, with a potential to renew the way how Fourier and time-frequency analysis, but also signal processing courses for engineers (or physicists and mathematicians) could be taught in the future. In the present study, we will demonstrate that one could choose a relatively large variety of similar Banach Gelfand Triples, even if one wants to include key properties such as Fourier invariance (an extended version of Plancherel’s Theorem). Some of them appeared naturally in the literature. It turns out, that $S_0\left(R^d\right)$ is the smallest member of this family. Consequently $S_0^{\prime }\left(R^d\right)$ is the largest dual space among all these spaces, which may be one of the reasons for its universal usefulness. This article provides a study of the basic properties following from a short list of relatively simple assumptions and gives a list of non-trivial examples satisfying these basic axioms.

1. Introduction

It is the aim of this article to demonstrate that there exists a variety of Banach algebras which may serve as the basis for a theory of (mild) distributions. We are using this vocabulary in order to describe objects (signals, generalized functions, etc.) which can be defined over general locally compact Abelian (LCA) groups. Given this goal, with the idea of providing a natural setting for the definition of a (generalized) Fourier transform, well suited for the description of engineering problems or providing a solid basis for heuristic derivations in physics (using Dirac measures, for example), we have to provide a setting where “ordinary functions” (say integrable or square integrable) have a Fourier transform as described in standard books on Fourier transforms. However, in some cases, unbounded measures (such as Dirac combs) are also supposed to have a Fourier transform, and such cases should be included in the approach as well.

We will present a list of basic properties which suffice to establish the basic properties of Fourier invariant spaces of “mild distributions”. In order to avoid unnecessary technicalities, we present our results in the context of ${\mathbb{R}}^d$. However, we avoid the use of specific properties that are only available in the Euclidean context. We will not make use of the dilations, the differentiability of functions, or of the existence of a (Fourier invariant) Gauss function.

The prototype of the construction will be Feichtinger’s Algebra $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$, which is the smallest in the proposed family of Fourier invariant Banach algebras of test functions. Consequently, the dual space $({\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0^{\prime }})$, the space of the so-called mild distributions is the largest within the family under consideration. Both space can be introduced for general LCA groups see [1] or [2]. While listing a few others, we will also point out that some of them appeared already earlier in the literature, such as the space ${\mathcal{M}}_{dT}\left({\mathbb{R}}^d\right)$ of twice transformable (potentially unbounded Radon) measures following L. Argabright and J. de Lamadrid (see [3,4] or [5] for an elaboration on this topic, suitable for the theory of quasi-crystals).

The abstract approach to Banach Gelfand Triples starts with the assumption that one has a continuous embedding of a given Banach space $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ of test functions in its dual space $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$. We will present an axiomatic approach in the context of ${\mathbb{R}}^d$, although most of the results can be formulated and proved in the much more general context of general locally compact Abelian (LCA) groups G, not just for the Euclidean case $G={\mathbb{R}}^d$. In order to start, we recall an alternative approach to the concept of convolution and the Fourier (Stieltjes) transform for bounded measures as developed by the author in [6].

However, this is usually far from enough to call the elements of ${\mathit{A}}^{\prime }$ generalized functions (or distributions). One wants to apply the operators which are possible for “ordinary functions” (such as translation, pointwise multiplication, convolution, dilations, rotations, Fourier transform, etc.). Usually, the existence of the Haar measure (in particular, its invariance with respect to translations) helps with most of these properties.

In the sequel, let us describe in an axiomatic way what is needed in order to establish a Banach Gelfand Triple, with $({\mathit{S}}_0,{\mathit{L}}^2,{\mathit{S}}_0^{\prime })\left({\mathbb{R}}^d\right)$ as the most important special case.

1.1. Foundations: Convolution, Bounded Measures, and Segal Algebras

Among others, we want to have a double algebra structure on our space of test functions, i.e., both with respect to pointwise multiplication as well as convolution. Hence, it is natural to assume that $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})=(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach algebra with respect to both multiplicative structures to be continuously embedded in both $\left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$ (Banach algebra with respect to pointwise multiplication) and $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$ (Banach algebra with respect to convolution). Observe that ${\mathit{C}}_0\left({\mathbb{R}}^d\right)$ is the closure of ${\mathit{C}}_c\left({\mathbb{R}}^d\right):=\{k\in {\mathit{C}}_b\left({\mathbb{R}}^d\right)|k\left(x\right)=0 \mathrm{for} \left|x\right|\ge R>0\}$, the compactly supported functions on ${\mathbb{R}}^d$, inside of the (pointwise) Banach algebra $\left({\mathit{C}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$ being bounded, continuous, complex-valued, endowed with the sup-norm ${\parallel f\parallel }_{\infty }:={sup}_{x\in {\mathbb{R}}^d}\left|f\left(x\right)\right|.$

Let us recall a few basic facts regarding $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$ and the Fourier transform on ${\mathbb{R}}^d$. Recall that the space of bounded measures $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$ can be introduced simply as the dual space of $\left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$, the space of continuous, complex-valued functions vanishing at infinity with the sup-norm. We refer to [6] for a bottom up description of the situation, providing an alternative approach to the important concept of the convolution of measures as follows:

Proposition 1.

1. 

$({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$ is a Banach convolution algebra. The mapping $\mu \mapsto C_{\mu }$, assigning to each $\mu \in {\mathit{M}}_b\left({\mathbb{R}}^d\right)$ the convolution operator given by $C_{\mu }\left(f\right)\left(x\right)=\mu \left(T_x{f}^{✓}\right)$, realizes a representation of this Banach algebra on the Banach space $\left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$, with ${\parallel \mu \parallel }_{{\mathit{M}}_b}=|\parallel C_{\mu }{|\parallel }_{{\mathit{C}}_0}$ (functional resp. operator norm on ${\mathit{C}}_0\left({\mathbb{R}}^d\right)$). We write $\mu \ast f$ for $C_{\mu }\left(f\right)$.

2. 

Under this correspondence, a Dirac measure ${\delta }_x:f\mapsto f\left(x\right)$ for $f\in {\mathit{C}}_0\left({\mathbb{R}}^d\right)$ corresponds to a translation operator. In other words,

$${\delta }_x\ast f=T_{x}f,f\in {\mathit{C}}_0\left({\mathbb{R}}^d\right).$$

3. 

${\mathit{M}}_d\left({\mathbb{R}}^d\right)$, the subspace of discrete, bounded measures of the form $\mu ={\sum }_{k=1}^{\infty }{c}_k{\delta }_{x_k}$ with ${\sum }_{k=1}^{\infty }\left|c_k\right|<\infty$ form a closed, proper subalgebra of ${\mathit{M}}_b\left({\mathbb{R}}^d\right)$ under convolution.

4. 

$\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$ can be described as the closure of measures of the form

$${\mu }_k\left(f\right)={\int }_{{\mathbb{R}}^d}f\left(x\right)k\left(x\right)dx,f\in {\mathit{C}}_0\left({\mathbb{R}}^d\right),k\in {\mathit{C}}_c\left({\mathbb{R}}^d\right),$$

or equivalently the usual space of Lebesgue integrable functions, with

$$\parallel {\mu }_g{\parallel }_{{\mathit{M}}_b}={\parallel g\parallel }_1={\int }_{{\mathbb{R}}^d}\left|g\left(x\right)\right|dx,g\in {\mathit{L}}^1\left({\mathbb{R}}^d\right).$$

(1)

5. 

$\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$ is a closed ideal of $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$ and coincides with the subspace of ${\mathit{M}}_b\left({\mathbb{R}}^d\right)$ with the property ${lim}_{x\to 0}\parallel T_x\mu -\mu {\parallel }_{{\mathit{M}}_b}=0.$

Due to the separability of $\left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$, the $w^{\ast }-$density of a subspace in ${\mathit{M}}_b\left({\mathbb{R}}^d\right)$ can be characterized via $w^{\ast }-$convergent sequences as follows: measure ${\mu }_0\in {\mathit{M}}_b\left({\mathbb{R}}^d\right)$ is the $w^{\ast }-$limit of a bounded sequence of measures ${\left({\mu }_n\right)}_{n\ge 1}$ if one has the following:

$$\underset{n\to \infty }{lim}{\mu }_n\left(f\right)={\mu }_0\left(f\right),f\in {\mathit{C}}_0\left({\mathbb{R}}^d\right).$$

(2)

Given the boundedness assumption, i.e., the assumption that

$$\underset{n}{sup}\parallel {\mu }_n{\parallel }_{{\mathit{M}}_b}<\infty ,\mathrm{with}{\parallel \mu \parallel }_{{\mathit{M}}_b}=\underset{{\parallel f\parallel }_{\infty }\le 1}{sup}\left|\mu \left(f\right)\right|,$$

it is enough to have the convergence of the left hand side of (2) for any $k\in {\mathit{C}}_c\left({\mathbb{R}}^d\right)$, only in order to find that the limit defines a bounded measure ${\mu }_0$.

Recall that ${St}_{\rho }g\left(x\right)={\rho }^{-d}g(x/\rho ),x\in {\mathbb{R}}^d,\rho >0$. Note that we have ${\parallel {St}_{\rho }g\parallel }_1={\parallel g\parallel }_1$ for $\rho >0$, $g\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$. In fact, it provides an automorphism of the Banach convolution algebra. Obviously, $D_{\rho }$ is isometric on $\left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$ and compatible with pointwise multiplication. The Fourier transform intertwines ${St}_{\rho }$ with another dilation operator, namely $D_{\rho }$, given by $D_{\rho }f\left(x\right)=f\left(\rho x\right)$. In fact, we have $\mathcal{F}\circ {St}_{\rho }=D_{\rho }\circ \mathcal{F}{\mathit{L}}^1$ for $\rho >0$.

Proposition 2.

• $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_1\right)$ has bounded approximate units, the so-called Dirac sequences. They can be obtained by the ${\mathit{L}}^1$-norm preserving compression of functions $g\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$ with $\widehat{g}\left(0\right)={\int }_{{\mathbb{R}}^d}g\left(x\right)dx=1$, and they satisfy

  $$\underset{\rho \to 0}{lim}{\parallel {St}_{\rho }g\ast f-f\parallel }_1=0,f\in {\mathit{L}}^1\left({\mathbb{R}}^d\right).$$
• For any $\mu \in {\mathit{M}}_b\left({\mathbb{R}}^d\right)$, we have ${St}_{\rho }g\ast \mu \in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$ and $\mu ={w^{\ast }\text{-}lim}_{\rho \to 0}{St}_{\rho }g\ast \mu .$

Summarizing the information obtained so far we can say that we have a dual pair, consisting of the space ${\mathit{C}}_0\left({\mathbb{R}}^d\right)$ and the dual space ${\mathit{M}}_b\left({\mathbb{R}}^d\right)$, and the concept of convolution, both inside of ${\mathit{M}}_b\left({\mathbb{R}}^d\right)$ and as an action of measures on ${\mathit{C}}_0\left({\mathbb{R}}^d\right)$ via the convolution operators. However, obviously there are many functions in ${\mathit{C}}_0\left({\mathbb{R}}^d\right)$ that are not integrable or bounded measures (e.g., Dirac measures) that are not representable via continuous functions. Moreover, this pair behaves badly with respect to the Fourier transform, which at first sight is only defined on say ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$ and gives functions in ${\mathit{C}}_0\left({\mathbb{R}}^d\right)$ (by the Riemann–Lebesgue Lemma), but not necessarily in ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$, which makes Fourier inversion a delicate matter. Thus, in order to take the next step, we need some dense subalgebras of ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$, known as Segal algebras (in H. Reiter’s sense, see [2,7] or [8]).

Definition 1.

A Segal algebra on ${\mathbb{R}}^d$ is a Banach space $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$, continuously embedded as a dense subspace into $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$, with the following two extra properties:

1. 

Translations are isometric on $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$, i.e.,

$$\parallel T_x{f\parallel }_{\mathit{B}}={\parallel f\parallel }_{\mathit{B}},f\in \mathit{B},x\in {\mathbb{R}}^d.$$

2. 

Translation acts as a strongly continuous group of operators (continuous shift property), as follows:

$$\underset{x\to 0}{lim}\parallel T_x{f-f\parallel }_{\mathit{B}}=0,f\in \mathit{B}.$$

One could characterize Segal algebras as homogeneous Banach spaces that are dense subspaces inside of $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$. As pointed out in [8], one can use vector-valued integration to derive that they are (Banach) ideals in $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$, but we prefer to point to the more elementary approach developed in [9], also justifying the term “algebra”:

Lemma 1.

Every Segal algebra $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is an essential Banach ideal in $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$, i.e., ${\mathit{L}}^1\left({\mathbb{R}}^d\right)\ast \mathit{B}\subseteq \mathit{B}$ with the norm estimate

$${\parallel g\ast f\parallel }_{\mathit{B}}\le {\parallel g\parallel }_1{\parallel f\parallel }_{\mathit{B}},g\in {\mathit{L}}^1\left({\mathbb{R}}^d\right),f\in \mathit{B}.$$

(3)

Moreover, ${lim}_{\rho \to 0}\parallel {St}_{\rho }{g\ast f-f\parallel }_{\mathit{B}}=0$ for any $f\in \mathit{B}$, which implies ${\mathit{L}}^1\ast \mathit{B}=\mathit{B}$ due to the Cohen–Hewitt Factorization Theorem ([10], 11.8). As a consequence, the band-limited functions from ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$ form a dense ideal in any Segal algebra.

Remark 1.

Segal algebras have been quite popular in the 70th, see, e.g., the papers [11,12,13,14] or [15] (showing that $\mathcal{S}\left({\mathbb{R}}^d\right)\hookrightarrow \left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$).

Another useful observation is formulated in the following lemma.

Lemma 2.

Assume that two Segal algebras $({\mathit{B}}^1,\parallel ·{\parallel }^{\left(1\right)})$ and $({\mathit{B}}^2,\parallel ·{\parallel }^{\left(2\right)})$ are given. If ${\mathit{B}}^1\subseteq {\mathit{B}}^2$, then we have automatically $({\mathit{B}}^1,\parallel ·{\parallel }^{\left(1\right)})\hookrightarrow ({\mathit{B}}^2,\parallel ·{\parallel }^{\left(2\right)})$, continuously embedded as a dense subspace.

Proof. 

By the Closed Graph Theorem, the embedding is automatically continuous. Since the band-limited functions from ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$ are a dense subspace of any Segal algebra, the density follows. □

Obviously the above results apply to the Segal algebra $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$.

1.2. Interlude: Wiener Amalgam Spaces

Since Wiener Amalgams are an important source for the examples to be given, let us mention that particular examples have been used in many classical and recent papers, such as [16,17], or the survey article [18], for the classical setting. In the modern literature on time-frequency analysis, they play a significant role in the work of, e.g., E. Cordero (such as [19,20] or [21]) and J. Toft [22,23,24] or [25].

Since we need some basic results concerning the so-called Wiener amalgam spaces, as introduced in [26], we need the concept of a regular BUPU in $\mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$, i.e., bounded partitions of unity in the Fourier algebra.

Definition 2.

A family $\mathrm{\Psi }={\left(T_k\psi \right)}_{k\in {\mathbb{Z}}^d}={\left({\psi }_k\right)}_{k\in {\mathbb{Z}}^d}$ generated from a compactly supported function $\psi \in {\mathit{C}}_c\left({\mathbb{R}}^d\right)$ is called a regular BUPU (Bounded Uniform Partition of Unity for ${\mathit{C}}_0\left({\mathbb{R}}^d\right)$) if one has

$$\sum _{k\in {\mathbb{Z}}^d}\psi (x-k)=\sum _{k\in {\mathbb{Z}}^d}{\psi }_k\left(x\right)\equiv 1.$$

For the case $\psi \in \mathcal{F}{\mathit{L}}^1$, we will talk about a $\mathcal{F}{\mathit{L}}^1$-BUPU.

For the case $supp\left(\psi \right)\subseteq B_R\left(0\right)$, we will say that the BUPU is of size $\le R$ and write $\left|\mathrm{\Psi }\right|\le R$. The existence of “arbitrary fine” BUPUs follows easily by a dilation argument. We write $D_{\tau }\Psi$ for the family ${\left(D_{\tau }{\psi }_k\right)}_{k\in {\mathbb{Z}}^d}$, and we let $\tau$ tend to infinity. One thus obtains $supp\left(D_{\tau }\psi \right)=(1/\tau )supp\left(\psi \right)\subseteq \rho B_R\left(0\right)=B_{R/\tau }\left(0\right)$ or $|D_{\tau }\mathrm{\Psi }|\le R/\tau$. Such BUPUs play an important role in many branches of Fourier analysis.

We refer to [18,27] for survey papers concerning basic facts about Wiener amalgam spaces. Their usefulness for the description of convolutions and the Fourier transform of bounded measures is explored in [6].

Proposition 3.

• For any $\mu \in {\mathit{M}}_b\left({\mathbb{R}}^d\right)$, the discrete measures $D_{\rho \mathrm{\Psi }}:={\sum }_{k\in {\mathbb{Z}}^d}\mu \left(D_{\rho }{\psi }_k\right){\delta }_{\rho k}$, forming a bounded and $w^{\ast }-$convergent net (for $\rho \to 0$) of bounded discrete measures with limit μ in $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$, with $\parallel D_{\rho \mathrm{\Psi }}{\left(\mu \right)\parallel }_{{\mathit{M}}_b}\le {\parallel \mu \parallel }_{{\mathit{M}}_b},$, i.e.,

  $$\underset{\rho \to \infty }{lim}{D}_{\rho \mathrm{\Psi }}\left(\mu \right)\left(f\right)=\mu \left(f\right),f\in {\mathit{C}}_0\left({\mathbb{R}}^d\right).$$
• For any $f\in {\mathit{C}}_0\left({\mathbb{R}}^d\right)$, we have ${lim}_{\left|\mathrm{\Psi }\right|\to 0}{\parallel D_{\mathrm{\Psi }}\mu \ast f-f\parallel }_{\infty }=0.$

Obviously, one has $f\left(x\right)={\sum }_{k\in {\mathbb{Z}}^d}f\left(x\right){\psi }_k\left(x\right)$ in the pointwise sense, in fact, even norm convergence of the finite partial sums to f in $\left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$ (uniform convergence). Setting $\mu ·h$ by the convention $[\mu ·h]\left(f\right)=\mu \left(hf\right),f\in {\mathit{C}}_0\left({\mathbb{R}}^d\right).$ As has been observed in [6], one has ${\parallel \mu \parallel }_{{\mathit{M}}_b}={\sum }_{k\in {\mathbb{Z}}^d}\parallel \mu {\psi }_k{\parallel }_{{\mathit{M}}_b}$, and hence, $\mu ={\sum }_{k\in {\mathbb{Z}}^d}\mu {\psi }_k$ as absolutely convergent series in $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$. This fact could be described more precisely by the following equality of spaces: ${\mathit{M}}_b\left({\mathbb{R}}^d\right)=\left(\mathit{W}({\mathit{M}}_b,{\mathcal{l}}^1),\parallel ·{\parallel }_{\mathit{W}({\mathit{M}}_b,{\mathcal{l}}^1)}\right)\left({\mathbb{R}}^d\right)$.

On the other hand, the Wiener algebra $\mathit{W}({\mathit{C}}_0,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)$ or Segal algebra of Wiener’s type (as described in Wiener’s work on Tauberian Theorems, see [7,17], Chap. I and 5) is defined by requiring the absolute convergence of this series representation. In fact, ${\parallel f\parallel }_{\mathit{W}({\mathit{C}}_0,{\mathcal{l}}^1)}:={\sum }_{k\in {\mathbb{Z}}^d}{\parallel f{\psi }_k\parallel }_{\infty }<\infty$ defines a norm on this subspace of $\left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$. We have the following continuous and dense embeddings

$${\mathit{C}}_c\left({\mathbb{R}}^d\right)\hookrightarrow \left(\mathit{W}({\mathit{C}}_0,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\mathit{W}}\right)\hookrightarrow \left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right).$$

Different BUPUs define the same space and equivalent norms.

Feichtinger’s algebra $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ was originally introduced in [1] as the Wiener amalgam space $(\mathit{W}(\mathcal{F}{\mathit{L}}^1,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\mathit{W}(\mathcal{F}{\mathit{L}}^1,{\mathcal{l}}^1)})$. Many basic properties have been described there, see also [28]. This particular example also explains why BUPUs in the Fourier algebra $\left(\mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\mathcal{F}{\mathit{L}}^1}\right)$ are relevant (such as the sequence of cubic splines over $\mathbb{R}$). The reader is referred to [28] for a summary of the equivalent characterizations. As a member of the wider family of modulation spaces, ${\mathit{S}}_0\left({\mathbb{R}}^d\right)={\mathit{M}}_0^{1,1}\left({\mathbb{R}}^d\right)$ also appears in the book [29] and in several chapters of [30]. It will provide the prototype of a Banach algebra of test functions as discussed in the rest of this paper, but we want to show that there are a couple of other possible choices. Modulation spaces appear in a large number of recent books and papers, such as [31,32] or [33].

The dual space for $\left(\mathit{W}({\mathit{C}}_0,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\mathit{W}}\right)$ is the space of translation-bounded (Radon) measures, i.e., the space of all Radon measures $\nu$ with either ${sup}_{k\in {\mathbb{Z}}^d}\left|\nu \right|\left({\psi }_k\right)<\infty ,$ respectively, with the (equivalent) property that for any non-zero function $\phi \in {\mathit{C}}_c\left({\mathbb{R}}^d\right)$, one has

$$\underset{x\in {\mathbb{R}}^d}{sup}\parallel \nu ·T_x{\phi \parallel }_{{\mathit{M}}_b}<\infty .$$

This space also plays a role in connection with the theory of transformable measures ${\mathcal{M}}_T\left({\mathbb{R}}^d\right)$, following L. Argabright and J. Gil de Lamadrid (see [3,4]), see Proposition 9 below, and [5].

The dual space of Feichtinger’s algebra is $({\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0^{\prime }})$, the space of mild distributions. The Fourier transform can be extended from ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$—or if one wants to think of Plancherel’s Theorem from ${\mathit{L}}^2\left({\mathbb{R}}^d\right)$—to all of ${\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right)$, by the usual rule, setting for $\sigma \in {\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right)$$\widehat{\sigma }\left(f\right)=\sigma \left(\widehat{f}\right)$ for all $f\in {\mathit{S}}_0\left({\mathbb{R}}^d\right)$.

The well-known principle that the forward, respectively, inverse Fourier transform exchanges the roles of convolution and pointwise multiplication can be roughly summarized in the following lemma.

Lemma 3.

Given $g\in {\mathit{S}}_0\left(\mathbb{R}\right)$ and $\sigma \in {\mathit{S}}_0^{\prime }\left(\mathbb{R}\right)$, we have

$$\mathcal{F}(g\ast \sigma )=\widehat{g}·\widehat{\sigma }and\mathcal{F}(g·\sigma )=\widehat{g}\ast \widehat{\sigma }.$$

1.3. Fourier Transform and Convolution

As a third ingredient, we recall the notion of pure frequencies, which are the complex exponential functions of the form ${\chi }_s\left(t\right)=exp\left(2\pi ist\right)$ (here, the product is to be understood as the inner product $st={\sum }_{k=1}^d{s}_k{t}_k$). Due to the obvious fact that ${\parallel {\chi }_s\parallel }_{\infty }=1$, they form a uniformly bounded family of linear functionals of the form $\widehat{f}(-s)={\int }_{{\mathbb{R}}^d}f\left(t\right){\chi }_s\left(t\right)dt$. Moreover, one has $w^{\ast }-$convergence to zero for any sequence $s_n\to \infty$. In fact, these two statements together are known as the Riemann–Lebesgue Lemma, stating that ${\parallel \widehat{f}\parallel }_{\infty }\le {\parallel f\parallel }_1$ and ${lim}_{s\to \infty }\widehat{f}\left(s\right)=0$ for any $f\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$.

By showing that the action of $\mu \in {\mathit{M}}_b\left({\mathbb{R}}^d\right):={\mathit{C}}_0^{\prime }\left({\mathbb{R}}^d\right)$ extends to $\left({\mathit{C}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$ (the space of bounded, complex-valued functions, still with the sup-norm), we find that the following definition of the Fourier(–Stieltjes) transform makes sense (see [6]):

Definition 3.

$$\widehat{\mu }\left(s\right)=\mu \left(\overline{{\chi }_s}\right)=\mu \left({\chi }_{-s}\right),s\in {\mathbb{R}}^d.$$

Using the exponential law ${\chi }_s\left(x\right){\chi }_s\left(y\right)={\chi }_s(x+y),x,y\in {\mathbb{R}}^d$ and through the approximation by discrete measures, one can verify the well-known Convolution Theorem, as follows:

Proposition 4.

The Fourier transform maps the Banach convolution algebra $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$ into the pointwise Banach algebra $\left({\mathit{C}}_b\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_{\infty }\right)$ of bounded, continuous functions, with the sup-norm, and satisfies

$$\mathcal{F}(\mu \ast \nu )=\widehat{\mu \ast \nu }=\widehat{\mu }·\widehat{\nu },\mu ,\nu \in {\mathit{M}}_b\left({\mathbb{R}}^d\right).$$

(4)

Let us recall that a measure $\mu \in {\mathit{M}}_b\left({\mathbb{R}}^d\right)$ is called band-limited if its Fourier transform $\widehat{\mu }$ has compact support $supp\left(\widehat{\mu }\right)$, i.e., $\widehat{\mu }\left(y\right)=0$ for $\left|y\right|\ge R$ (for some $R>0$). Since there exist functions $g\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$ with $\widehat{g}\left(u\right)\equiv 1$ on $B_R\left(0\right)$ (ball of radius R around 0), we can write $\mu =g\ast \mu \in {\mathit{L}}^1\ast {\mathit{M}}_b\left({\mathbb{R}}^d\right)\subseteq {\mathit{L}}^1\left({\mathbb{R}}^d\right)$, and we observe that any band-limited measure belongs in fact to ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$.

Next, let us recall that a Segal algebra is called strongly character invariant (SCI) if one has

$$\parallel {\chi }_s{·f\parallel }_{\mathit{B}}={\parallel f\parallel }_{\mathit{B}},f\in \mathit{B},s\in {\mathbb{R}}^d.$$

In order to make this article self-contained, we verify the strong continuity of the representation of $\widehat{{\mathbb{R}}^d}$ on SCI Segal algebras via the modulation operators $M_s\left(f\right)={\chi }_s·f$. This will pave the way for the integrated action of $\mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$ on $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ by pointwise multiplication, as discussed in [9]. The key argument used in the lemma below is quite similar to Step III of the proof of Theorem 1 in [1]. It is also a consequence of the considerations in Section 2.2 of [34]. However, in both cases, the results are derived in the general context of LCA groups, and Reiter is even starting from an alternative definition of ${\mathit{S}}_0\left(G\right)$, denoting it by ${\mathfrak{S}}^1\left(G\right)$.

Lemma 4.

Let $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ be a SCI Segal algebra on ${\mathbb{R}}^d$. Then, for any $f\in \mathit{B}$, the mapping $s\mapsto {\chi }_{s}f$ is continuous from $\widehat{{\mathbb{R}}^d}={\mathbb{R}}^d$ into $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$.

Proof. 

First, we observe that the modulation operators $M_s,s\in {\mathbb{R}}^d$ form a commutative group of isometric operators on $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$. Hence, we can reduce the consideration to continuity at $s=0$ using the chain of equalities

$$\parallel M_{s+t}g-M_s{g\parallel }_{\mathit{B}}=\parallel M_s(M_{t}g-g){\parallel }_{\mathit{B}}=\parallel M_t{g-g\parallel }_{\mathit{B}},s,t\in {\mathbb{R}}^d.$$

For the rest of the proof, we will assume that $\left|s\right|\le 1$.

Secondly, we recall that the band-limited elements from $\mathit{B}$ form a dense subspace. Thus, given $g\in \mathit{B}$ and $\epsilon >0$, we can find some $h\in \mathit{B}$ which is band-limited, say $supp\left(\widehat{h}\right)\subset B_R\left(0\right)$ for some $R>0$. Using the well-known fact that $\mathcal{F}\left(M_{s}g\right)=T_s\widehat{g}$, we observe that $supp\left(\mathcal{F}\left(M_{s}g\right)\right)\subset B_R\left(0\right)+B_1\left(0\right)\subseteq B_{R+1}\left(0\right):=Q$, some compact subset of $\widehat{{\mathbb{R}}^d}$, whenever $\left|s\right|\le 1$.

As a third fact about Segal algebras, we recall that for any fixed compact set $Q\subseteq \widehat{{\mathbb{R}}^d}$, the norms of $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$ and $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ are equivalent on the subspace ${\mathit{B}}_Q:=\{h\in \mathit{B}|supp\widehat{h}\subseteq Q\}.$ This claim is verified as follows: by the continuous embedding of $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ into $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$, we anyway have for any $f\in \mathit{B}$: ${\parallel f\parallel }_{{\mathit{L}}^1}\le C_{\mathit{B}}{\parallel f\parallel }_{\mathit{B}}$. the band-limited function $h\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$ with $\widehat{h}\left(u\right)\equiv 1$ on Q. However, as a consequence of Wiener’s Inversion Theorem ([7], Chap. 1), any such function belongs to any dense ideal of ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$, and thus, we have ${\parallel h\parallel }_{\mathit{B}}<\infty$. Since $\widehat{f}=\widehat{f}·\widehat{h}$ implies $f=h\ast f$ for $f\in {\mathit{B}}_Q$, we obtain (via the estimate (3) for Segal algebras)

$${\parallel f\parallel }_{\mathit{B}}\le {\parallel h\parallel }_{\mathit{B}}{\parallel f\parallel }_{{\mathit{L}}^1},f\in {\mathit{B}}_Q.$$

This allows us to come back to our estimate for f, as given above. Recall that $M_{s}f-f\in {\mathit{B}}_Q$ since we assume $\left|s\right|\le 1$. Using the fact that ${lim}_{s\to 0}\parallel M_{s}f-f{\parallel }_{{\mathit{L}}^1}=0$, which is easy to derive for $f\in {\mathit{C}}_c\left({\mathbb{R}}^d\right)$ and by approximation otherwise, we can choose $\delta >0$ such that

$$\parallel M_{s}f-f{\parallel }_{{\mathit{L}}^1}<\epsilon /{\parallel h\parallel }_{\mathit{B}},$$

and finally,

$$\parallel M_s{f-f\parallel }_{\mathit{B}}\le \epsilon .$$

□

As a consequence of Lemma 4, we can apply the general principle of an integrated group action via [9], which results, in this concrete case, in the following proposition:

Proposition 5.

Let $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ be a SCI Segal algebra on ${\mathbb{R}}^d$. Then, $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is a Banach module over $\left(\mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\mathcal{F}{\mathit{L}}^1}\right)$ with respect to pointwise multiplication, i.e.,

$${\parallel h·f\parallel }_{\mathit{B}}\le {\parallel h\parallel }_{\mathcal{F}{\mathit{L}}^1}{\parallel f\parallel }_{\mathit{B}},h\in \mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right),f\in \mathit{B}.$$

(5)

Proof. 

Starting from the isometric and strongly continuous representation of $\widehat{{\mathbb{R}}^d}$ on $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ via the modulation operators, it is clear that any finite, discrete measure of the form $\nu ={\sum }_{k=1}^K{c}_k{\delta }_{s_k}$ defines a pointwise multiplication operator by the corresponding trigonometric polynomial $\tau \left(t\right)={\sum }_{k=1}^K{c}_k{\chi }_{s_k}\left(t\right)$. By a simple approximation argument, this representation is also valid for general bounded, discrete measures of the form $\nu ={\sum }_{k=1}^{\infty }{c}_k{\delta }_{s_k}$ with ${\sum }_{k=1}^{\infty }\left|c_k\right|<\infty$, providing a multiplication operator with the almost period function ${\sum }_{k=1}^{\infty }{c}_k{\chi }_{s_k}$.

Thus, we only have to note that, for the Fourier–Stieltjes transforms of a sequence of discretized versions of the form $D_{\mathrm{\Psi }}\left(\nu \right)$, the corresponding Fourier–Stieltjes transforms converge uniformly over compact subsets of ${\mathbb{R}}^d$ to $\widehat{\nu }$. This can be verified in different ways. The most convenient reference seems to be Corollary 4 in [6].

Thus, overall, we have shown that the integrated group action of $\nu \in {\mathit{M}}_b\left({\mathbb{R}}^d\right)$ on $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is just the pointwise multiplication by the bounded and continuous function $\widehat{\nu }$. Since $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$ is a closed ideal in $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$, the claim of the proposition is verified. □

Remark 2.

Combined with the characteristic property of any Segal algebra as a Banach ideal in $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$, one thus finds that any SCI Segal algebra $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is a double module in the spirit of [35]. In this sense, it is a minimal one, i.e., $\mathit{B}={\mathit{B}}_0$ in the notation used there. It is also a minimal translation and modulation invariant Banach space (MTMIB), following [36], and thus, the corresponding statements about Tauberian results apply. We do not repeat the details here. Note that this minimality property (equivalent to the density of the Schwartz space $\mathcal{S}\left({\mathbb{R}}^d\right)$) should not be confused with the fact that ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ is the smallest SCI Segal algebra according to [1].

Remark 3.

By combining the action of translation and modulation, one has the representation of phase space ${\mathbb{R}}^d\times {\widehat{\mathbb{R}}}^d$ on such Segal algebras, $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$, via a projective representation, as follows: $\lambda =(t,s)\to \pi \left(\lambda \right)=M_s{T}_t$. Equivalently, one can view them as (essential) Heisenberg modules. This perspective is described in [37]. The integrated group representation of ${\mathit{L}}^1\left(H^d\right)$ arising from the corresponding group representation, known as the Schrödinger representation of the reduced Heisenberg group $H^d={\mathbb{R}}^d\times {\widehat{\mathbb{R}}}^d\times \mathbb{T}$ generated compact and, in fact, trace class operators in ${\mathit{L}}^2\left({\mathbb{R}}^d\right)$. A detailed discussion of this aspect is outside the scope of this article.

In preparation for the proof of Theorem 2, we formulate the following simple results:

Lemma 5.

Given the two Segal algebras $({\mathit{B}}^1,\parallel ·{\parallel }^{\left(1\right)})$ and $({\mathit{B}}^2,\parallel ·{\parallel }^{\left(2\right)})$, their intersection ${\mathit{B}}^1\cap {\mathit{B}}^2$ is a Segal algebra as well, with the natural norm ${\parallel f\parallel }_{{\mathit{B}}^1\cap {\mathit{B}}^2}:={\parallel f\parallel }_{{\mathit{B}}^1}+{\parallel f\parallel }_{{\mathit{B}}^2}.$ In particular, it is dense in both of the original Segal algebras.

If both are strongly character invariant, the same is true for ${\mathit{B}}^1\cap {\mathit{B}}^2$, and thus, $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ is continuously and densely embedded as well.

Proof. 

It is an easy exercise to verify that the norm for ${\mathit{B}}^1\cap {\mathit{B}}^2$ is complete, since convergence for both of these norms implies ${\mathit{L}}^1$-convergence.

Since both Segal algebras contain the band-limited functions in ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$, this is also true for their intersection, which is thus also dense in $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$. The isometric and continuous translation properties are obvious. The fact that the band-limited functions are contained in the intersection combined with the existence of band-limited approximate units (Dirac sequences) of band-limited functions (such as Fejer or De la Valle Poussin kernels) implies that the intersection is also dense in the given Segal algebras.

Assuming that both of them are strongly character invariant, it is well known (due to the main result of [1]) that $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ is continuously embedded into both of them, and hence also into the intersection. In fact, it is embedded as a dense subspace. Since TF-shifts act continuously, this property extends to all of ${\mathit{B}}^1\cap {\mathit{B}}^2$ by the continuity of both the time and the frequency shifts. □

Remark 4.

Note that the proof of Lemma 5 is valid if ${\mathit{B}}^2$ is some other Banach space of (mild or tempered) distributions containing ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$. In particular, it can be used for spaces of the form ${\mathit{B}}^2=\mathcal{F}\mathit{B}$, where $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is some Segal algebra or a homogeneous Banach space, with the norm $\parallel \widehat{f}{\parallel }_{\mathcal{F}\mathit{B}}:={\parallel f\parallel }_{\mathit{B}}$, for $f\in \mathit{B}$.

There is a similar statement for the sum of two Segal algebras or, in a similar way, for two Banach spaces of mild distributions.

Lemma 6.

For two Segal algebras $({\mathit{B}}^1,\parallel ·{\parallel }^{\left(1\right)})$ and $({\mathit{B}}^2,\parallel ·{\parallel }^{\left(2\right)})$, the sum of the two spaces

$${\mathit{B}}^1+{\mathit{B}}^2:=\{f=f_1+f_2|f_1\in {\mathit{B}}^1,f_2\in {\mathit{B}}^2\}$$

is a Segal algebra with the (quotient) norm given by

$${\parallel f\parallel }_{{\mathit{B}}^1+{\mathit{B}}^2}:=\underset{f=f_1+f_2}{inf}(\parallel f_1{\parallel }_{{\mathit{B}}^1}+\parallel f_2{\parallel }_{{\mathit{B}}^2}).$$

This Segal algebra is strongly character invariant if both $({\mathit{B}}^1,\parallel ·{\parallel }^{\left(1\right)})$ and $({\mathit{B}}^2,\parallel ·{\parallel }^{\left(2\right)})$ have this property. Furthermore, one can allow ${\mathit{B}}^2=\mathcal{F}\mathit{B}$ for any strongly character invariant Segal algebra and still have the same conclusion.

Proof. 

It is a matter of routine to demonstrate that absolutely convergent series in ${\mathit{B}}^1+{\mathit{B}}^2$ with the norm provided above are norm convergent. Hence, ${\mathit{B}}^1+{\mathit{B}}^2$ is a Banach space with respect to this natural norm. Obviously, it is still continuously and densely embedded into $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$, and it satisfies all the properties required for a Segal algebra. Moreover, ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ is continuously embedded into this space in the strongly character invariant case.

For the case ${\mathit{B}}^2=\mathcal{F}\mathit{B}$, one only has to observe that the continuous shift property for $\mathcal{F}\mathit{B}$ results from the continuous modulation property for $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$, as shown in [1], as follows:

$$\underset{s\to 0}{lim}\parallel {\chi }_s·f-f{\parallel }_{\mathit{B}}=0,f\in \mathit{B}$$

implies

$$\underset{s\to 0}{lim}\parallel T_s\widehat{f}-\widehat{f}{\parallel }_{{\mathit{B}}^2}=0,\widehat{f}\in {\mathit{B}}^2.$$

□

2. Setting the Stage: Banach Algebras of Test Functions

Using these ingredients, we now come to the crucial definition. It is set up in a way which is maximally symmetric with respect to the Fourier transform, on the one hand. In this sense, condition (A1) could (or almost should) be replaced by the condition $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})\hookrightarrow {\mathit{L}}^1\cap \mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$, which is a consequence of (A1) combined with (A5). Nevertheless, the combined conditions (A1), (A2), and (A4) are closer to the spirit of Segal algebras following H. Reiter and will allow the construction of appropriate Banach algebras. We also avoid beginning with $\left({\mathit{L}}^2\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_2\right)$, because this would still require the use of Lebesgue integration theory in order to start. In contrast, it is our aim to explore the possibility of establishing most of the crucial facts about the Fourier transform (as relevant for engineers, for example) without diving too much into Lebesgue integration theory or making use of deep facts about (nuclear) topological function spaces.

Let us not forget that the material of this section will be the axiomatic basis for the development of a theory of generalized Fourier transforms. As the examples later on will show, there is still a significant freedom at this point. Only further additional requirements will exclude most of the ones fitting to the present axiomatic regime, finally narrowing the setting down to $(\mathit{A},{\parallel ·\parallel }_{\mathit{A}})=\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$, if one wants to have the largest domain and the maximal number of additional properties.

Definition 4.

A Banach space $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach space of test functions if

(A1) 

$(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is continuously embedded into $({\mathit{L}}^1\cap {\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1+\parallel ·{\parallel }_{\infty })$;

(A2) 

$(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach convolution module over $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$;

(A3) 

$(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach module with respect to pointwise multiplication

over the Fourier–Stieltjes algebra $\left(\mathcal{F}{\mathit{M}}_b\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_{\mathcal{F}{\mathit{M}}_b}\right)$;

(A4) 

${\mathit{A}}_c={\mathit{C}}_c\left({\mathbb{R}}^d\right)\cap \mathit{A}$, the compactly supported functions are dense in $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$;

(A4b) 

The band-limited functions in $\mathit{A}$ are dense in $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$;

(A5) 

$(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is invariant under the Fourier transform.

(A6) 

$(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is invariant under the involutions $f\mapsto \overline{f}$ (complex conjugation)

as well as $f\mapsto f^{✓},hencealsof\mapsto f^{*},with{f}^{*}\left(x\right)=\overline{f^{✓}}\left(x\right).$

Remark 5.

Let us recall the notion of a Banach module by illustrating the concrete case of condition (A2), as follows: we assume that there is an associative action of the bounded measures (which form a Banach algebra with respect to convolution, by the arguments given in [38]) on the Banach space $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$, meaning that we have

$$\mu \ast f\in \mathit{A}and{\parallel \mu \ast f\parallel }_{\mathit{A}}\le {\parallel \mu \parallel }_{{\mathit{M}}_b}{\parallel f\parallel }_{\mathit{A}},\mu \in {\mathit{M}}_b\left({\mathbb{R}}^d\right),f\in \mathit{A}$$

(6)

and furthermore the associativity law (between internal and external convolution)

$$({\mu }_1\ast {\mu }_2)\ast f={\mu }_1\ast ({\mu }_2\ast f),f\in {\mathit{C}}_0\left({\mathbb{R}}^d\right).$$

Moreover, the convolution between $\mu \in {\mathit{M}}_b\left({\mathbb{R}}^d\right)={\mathit{C}}_0^{\prime }\left({\mathbb{R}}^d\right)$ and $f\in \mathit{A}\subseteq {\mathit{C}}_0\left({\mathbb{R}}^d\right)$ is equivalently described by a pointwise definition, as follows:

$$\mu \ast f\left(x\right)=\mu \left(T_x{f}^{✓}\right),x\in {\mathbb{R}}^d.$$

Remark 6.

Any Banach space satisfying A5) is also invariant under the flip operator $f\mapsto f^{✓}={\mathcal{F}}^2\left(f\right).$ However, for $h=\widehat{f}\in \mathcal{F}\left(\mathit{A}\right)=\mathit{A}$, we also have $\overline{h}=\mathcal{F}\left(f^{\ast }\right)=\mathcal{F}\left(\overline{f^{✓}}\right)$, which implies that, given (A5), (A6) is, in fact, equivalent to the condition that $\mathit{A}$ is closed under $f\mapsto \overline{f}$ (complex conjugation).

Remark 7.

It is obvious that conditions (A4) and (A4b) coincide if (A5) applies. Hence, (A4b) will be automatically included if we assume that (A1) to (A5) are valid throughout the main part of this article.

We also observe that the embedding $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})\hookrightarrow \left({\mathit{L}}^1\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_1\right)$, combined with (A5), implies $\mathit{A}=\mathcal{F}\left(\mathit{A}\right)\hookrightarrow \mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)\hookrightarrow {\mathit{C}}_0\left({\mathbb{R}}^d\right)$. Consequently, the continuous embedding into ${\mathit{L}}^1\cap {\mathit{C}}_0\left({\mathbb{R}}^d\right)$ (as required in A1) is automatically satisfied in this case.

Proposition 6.

1. 

Any Segal algebra in $\left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$ satisfies conditions (A1) and (A2).

2. 

Any strongly character invariant Segal algebra satisfies (A1) to (A3).

3. 

$\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ is the smallest Banach space inside of ${\mathit{L}}^2\left({\mathbb{R}}^d\right)$ (or ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$) which satisfies (A1)–(A4). It also satisfies (A5) and (A6).

Proof. 

First, we observe that the set-theoretic inclusion $\mathit{B}\subset {\mathit{C}}_0$ and thus $\mathit{B}\subset {\mathit{L}}^1\cap {\mathit{C}}_0$ implies that the embedding has to be continuous by the closed graph theorem. In fact, we only have to verify that the inclusion mapping is continuous by showing that $f_n\to 0$ in $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ and $f_n\to f_0$ in $\left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$ implies $f_0=0$. This is obvious because any convergent sequence in $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is also convergent in $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$ and thus has a subsequence which is convergent almost everywhere. Thus, we have ${lim}_{k\to \infty }{f}_{n_k}\left(x\right)=0$ a.e., which obviously implies that $f_0\left(x\right)=0$ a.e. or $f_0=0$ in ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$.

Property (A3) follows from the fact that the Fourier image $\mathcal{F}\left(\mathit{B}\right)$ of the Segal algebra is, in fact, a homogeneous Banach space and thus ${\mathit{M}}_b\left({\mathbb{R}}^d\right)\ast \mathcal{F}\left(\mathit{B}\right)\subseteq \mathcal{F}\left(\mathit{B}\right)$ (by [9] again). This rests on the fact that in the case of Segal algebras, the strong character invariance implies

$$\underset{s\to 0}{lim}\parallel {\chi }_s{·f-f\parallel }_{\mathit{B}}=0,f\in \mathit{B}.$$

The details are given in [1], p. 274. □

Lemma 7.

Given the properties (A1) to (A3), condition (A4) is equivalent to the assumption that ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ continuously embedded into $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ as a dense subspace.

Proof. 

Assume that ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ is contained in $\mathit{A}$, then—again by the Closed Graph Theorem—it is continuously embedded. Since ${\mathit{S}}_0\left({\mathbb{R}}^d\right)\cap {\mathit{C}}_c\left({\mathbb{R}}^d\right)=\mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)\cap {\mathit{C}}_c\left({\mathbb{R}}^d\right)$ forms a dense subspace of $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$, it follows that it is also contained in $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$, and it is dense there (in the $\mathit{A}$-norm), i.e., A4) is valid.

Conversely, assume that ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ is a dense subspace of $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$. Since conditions (A1)–(A3) imply that $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)\hookrightarrow (\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ by the main result of [1], we have that ${\mathit{S}}_0\left({\mathbb{R}}^d\right)\cap {\mathit{C}}_c\left({\mathbb{R}}^d\right)\subset \mathit{A}\cap {\mathit{C}}_c\left({\mathbb{R}}^d\right)$ is dense in $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$. For similar arguments, compare Remark 2 in [1], p. 275. □

Lemma 8.

Assume that $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ satisfies (A1)–(A4). Then, $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a minimal translation and modulation invariant (MINTSTA) Banach space following [36]. In particular, ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ is dense in $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ and $\mathit{A}$ satisfies the metric approximation property by [39].

Just as an illustration of the consequences of these observations, let us recall a couple of special features valid for general Banach algebras of the test functions (cf. Section 3.2.4 of [40] for details, explained in detail for $\mathit{A}={\mathit{S}}_0$).

Lemma 9.

For $g\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$ with $\widehat{g}\left(0\right)=1$ or $h=\widehat{g}\in \mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$ with h(0) = 1, one has the following:

$$\underset{\rho \to 0}{lim}\parallel {St}_{\rho }g\ast f-f{\parallel }_{\mathit{A}}=0,f\in \mathit{A}.$$

$$\underset{\rho \to 0}{lim}\parallel D_{\rho }h·f-f{\parallel }_{\mathit{A}}=0,f\in \mathit{A}.$$

$$li{m}_{\rho \to 0}\parallel {St}_{\rho }g\ast (D_{\rho }h·f)-f{\parallel }_{\mathit{A}}=0,f\in \mathit{A}.$$

The axioms imply a couple of additional properties.

Theorem 1.

Any Banach space $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ of test functions satisfies

• $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach algebra with respect to pointwise multiplication;
• $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach algebra with respect to convolution;
• $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is isometrically translation invariant;
• $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is isometrically invariant under modulation operators
  (multiplication with pure frequencies).

Proof. 

First, we observe that according to assumption (A1), we have, for some constant $C>0$, the following estimate for the corresponding natural norms:

$${\parallel f\parallel }_1+{\parallel f\parallel }_{\infty }\le C{\parallel f\parallel }_{\mathit{A}},f\in \mathit{A}.$$

In particular, we have a continuous embedding of $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ into $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$, which is a closed ideal in ${\mathit{M}}_b\left({\mathbb{R}}^d\right)$. Consequently, we have by A2) for $g\in \mathit{A}\subset {\mathit{L}}^1\left({\mathbb{R}}^d\right)$

$${\parallel g\ast f\parallel }_{\mathit{A}}\le {\parallel g\parallel }_{{\mathit{L}}^1}{\parallel f\parallel }_{\mathit{A}}\le C{\parallel g\parallel }_{\mathit{A}}{\parallel f\parallel }_{\mathit{A}},g,f\in \mathit{A}.$$

Thus, $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach algebra with respect to convolution. If necessary, the $\mathit{A}$-norm has to be replaced by the equivalent norm $C{\parallel f\parallel }_{\mathit{A}}$ in order to avoid the constant C in the estimate above.

The isometric invariance under translation is obtained by choosing the bounded measure acting on $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ to be a Dirac measure ${\delta }_x$, recalling that ${\delta }_x\ast f=T_{x}f$. Using again (A2), we obtain

$$\parallel T_{x}f{\parallel }_{\mathit{A}}\le \parallel {\delta }_x{\parallel }_{{\mathit{M}}_b\left({\mathbb{R}}^d\right)}{\parallel f\parallel }_{\mathit{A}}={\parallel f\parallel }_{\mathit{A}},x\in {\mathbb{R}}^d,f\in \mathit{A}.$$

But since the same argument applies to $T_{-x}$, we have

$${\parallel f\parallel }_{\mathit{A}}=\parallel T_{-x}\left(T_{x}f\right){\parallel }_{\mathit{A}}\le \parallel T_{x}f{\parallel }_{\mathit{A}}$$

and thus finally

$${\parallel f\parallel }_{\mathit{A}}=\parallel T_{x}f{\parallel }_{\mathit{A}},x\in {\mathbb{R}}^d,f\in \mathit{A}.$$

In a similar way, we derive the isometric character invariance of the $\mathit{A}-$norm, i.e., the isometry $\parallel M_{s}f{\parallel }_{\mathit{A}}={\parallel f\parallel }_{\mathit{A}}$ for $s\in {\mathbb{R}}^d$. The fact that translation on the Fourier transform side is just modulation on the time side, due to the intertwining property $\mathcal{F}\left(M_{s}f\right)=T_x\widehat{f}$, implies that $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a pointwise Banach module over $\left(\mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\mathcal{F}{\mathit{L}}^1}\right)$. By an argument similar to the one given above, we finally conclude that $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is also a pointwise Banach algebra. □

Observing that any Segal algebra is a homogeneous Banach space following Katznelson (cf. [41]), we can invoke the main result of [9] and claim the following:

Theorem 2.

The collection of Fourier invariant Banach spaces $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ for the test functions can be characterized as the family of Banach spaces of the form $\mathit{B}\cap \mathcal{F}\mathit{B}$, where $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is a strongly character invariant Segal algebra, which is closed under complex conjugation.

Proof of Theorem 2.

We start by verifying that the intersection of the form $\mathit{B}\cap \mathcal{F}\mathit{B}$ satisfies (with their natural norm) the conditions (A1)–(A5), whenever $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is a strongly character invariant Segal algebra.

We start by observing that $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is not only ideal in $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$, but even in $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$, according to [9]. However, similar arguments apply to the homogeneous Banach space ${\mathcal{F}}^{-1}\left(\mathit{B}\right)$ (with its natural norm), which is a Banach convolution module over $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$ as well. Applying the Fourier transform, one obtains that $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ is a pointwise module over the Fourier–Stieltjes algebra $\mathcal{F}{\mathit{M}}_b\left({\mathbb{R}}^d\right)$ as well, i.e., (A3) is satisfied.

Of course, $\mathit{B}\cap \mathcal{F}\mathit{B}$ is Fourier invariant, since the properties (A4) combined with (A6) imply that $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is also invariant under the inverse Fourier transform, which can be described as ${\mathcal{F}}^{-1}\left(f\right)=\mathcal{F}\left(f^{✓}\right)$, for $f\in \mathit{A}$ (hence, $\widehat{f}\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$).

Since the Fourier transform (following mild distributions) interchanges the properties (A2) and (A3) (since obviously $\mathit{B}\cap \mathcal{F}\mathit{B}\hookrightarrow {\mathit{L}}^1\cap \mathcal{F}{\mathit{L}}^1$), we have also verified that the intersection satisfies (A1)–(A3).

Since (A5) is obvious, we only have to verify (A4), but this results from the fact that, in ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$, the compactly supported elements are dense (with respect to its natural norm). This can be derived in many different ways, e.g., by using the Fourier invariance of $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ combined with the fact that the band-limited functions are dense in any Segal algebra.

Finally, let us verify the converse. We assume that a Banach space $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ satisfies (A1) to (A5). Then, according to (A1), it is continuously embedded into $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$. It follows from (A2) that it is an Banach ideal in $({\mathit{M}}_b\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{M}}_b})$ (with respect to convolution) and thus particularly isometrically translation invariant. In a similar way, we obtain the strong character invariance from (A3).

Since we assume the Fourier invariance property (A5), we derive from (A4) the consequence that the band-limited functions in $\mathit{A}$ form a dense subspace of $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$. However, this in turn implies that translation is continuous, and consequently, $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a strongly character invariant Segal algebra. Thus, we trivially have by assumption (A5): $\mathit{A}=\mathit{A}\cap \mathcal{F}\mathit{A}$. □

Corollary 1.

A Banach space $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach algebra of the test functions if and only if it is a Fourier invariant Segal algebra which is closed under complex conjugation.

Proof. 

We just have to recall that by assumption we have $\mathit{A}=\mathcal{F}\mathit{A}$, which is isometrically translation invariant, hence, $\mathit{A}$ is strongly character invariant. □

Some of the ideas behind the following considerations are already found in [35].

Theorem 3.

Assume that we are given two Banach algebras ${\mathit{A}}^1$ and ${\mathit{A}}^2$ of the test functions. Then, their intersection, their sum, or the complex interpolation spaces ${[{\mathit{A}}^1,{\mathit{A}}^2]}_{\theta }$ for $\theta \in [0,1]$, derived from these spaces, all satisfy the conditions (A1) to (A6).

Proof. 

That the intersection and the sum of two spaces of test functions belongs to the family is a consequence of Lemmas 5 and 6. That the interpolation of two Banach modules (over a fixed Banach algebra) are again Banach modules follows from the general principles of interpolation theory (see [42]). □

Remark 8.

Similar statements apply by the same arguments to other interpolation spaces, for example, to real interpolation spaces obtained with the help of Peetre’s K-functional (see [42]).

Remark 9.

Obviously, the conclusions just obtained would allow us to assume from the very beginning that $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach algebra with respect to pointwise multiplication (or convolution). However, in order to keep the list of axiomatic assumptions as short as possible, we only require that $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach space with a double module structure (following [35]) and deduce the double Banach algebra from the axioms in Theorem 1.

It is an interesting question to ask about the local structure of such Banach algebras of test functions. If fact, it coincides with the local structure of the Fourier algebra.

Lemma 10.

For any Banach algebra of the test functions, we have to following chain of continuous embeddings:

$${\mathit{S}}_0\left({\mathbb{R}}^d\right)=\mathit{W}(\mathcal{F}{\mathit{L}}^1,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)\hookrightarrow \mathit{A}\hookrightarrow {\mathcal{M}}_{\mathit{A}}\hookrightarrow \mathit{W}(\mathcal{F}{\mathit{L}}^1,{\mathcal{l}}^{\infty })\left({\mathbb{R}}^d\right).$$

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Proof. 

Since any Banach space of test functions is known to be sandwiched between ${\mathit{S}}_0\left({\mathbb{R}}^d\right)=\mathit{W}(\mathcal{F}{\mathit{L}}^1,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)$ and ${\mathit{L}}^1\cap \mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$, it is clear that the local structure equals that of the Fourier algebra $\mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$. Moreover, since $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach algebra, as observed in Theorem, it is clear that $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})\hookrightarrow \left({\mathcal{M}}_{\mathit{A}},\parallel ·{\parallel }_{{\mathcal{M}}_{\mathit{A}}}\right)$. □

3. A List of Key Examples

The criterion provided by Theorem 2 gives a good recipe for generating (all possible) Banach algebras of the test functions by starting with any strongly character invariant Segal algebra. By the main result of [1] $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ is a smallest element in the family, continuously (and densely) embedded into any other one. There is also clearly a biggest element, namely ${\mathit{L}}^1\cap \mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$, obtained by starting with the full Banach convolution algebra $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$ and intersecting it with the Fourier algebra $\left(\mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\mathcal{F}{\mathit{L}}^1}\right)\hookrightarrow \left({\mathit{C}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{\infty }\right)$. This is one of the classical Segal algebras as follows: $\{f\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)|\widehat{f}\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)\}$.

As we will see in a moment, there are many more such Banach algebras, which will be studied in more detail in a subsequent article. In fact, the plan is to check the different properties of all these alternative candidates generating Fourier invariant Banach Gelfand Triples. In this follow-up article, we will also compare the properties of these alternative examples, case by case, and derive the validity (or failure) of additional properties. This approach will show that $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ is not only the smallest such Banach algebra, but it is also the one fulfilling a maximum of desirable properties, often turning it into a good substitute for the space of rapidly decreasing Schwartz functions $\mathcal{S}\left({\mathbb{R}}^d\right)$, with the space ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right)$ of tempered distributions as the (Fourier invariant) dual space. In contrast to the Schwartz–Bruhat space, defined over general LCA groups (see [43], used in [44] or [45]), it is much easier to use, as explained, among others, in [34].

Theorem 4.

The following Banach spaces (with their respective natural norms) are Banach algebras of test functions satisfying A1) to A6) (and often more):

1. 

${\mathit{S}}_0\left({\mathbb{R}}^d\right)$, the prototypical example (and the smallest);

2. 

${\mathit{L}}^1\cap \mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$ (and is the biggest);

3. 

$\mathit{W}\cap \mathcal{F}\mathit{W}\left({\mathbb{R}}^d\right)$, with $\mathit{W}=\mathit{W}({\mathit{C}}_0,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)$;

4. 

$\mathit{W}+\mathcal{F}\mathit{W}\left({\mathbb{R}}^d\right)$;

5. 

${\mathit{W}}^p\cap \mathcal{F}{\mathit{W}}^p\left({\mathbb{R}}^d\right)$, for $1\le p<\infty$, with ${\mathit{W}}^p=\mathit{W}({\mathit{L}}^p,{\mathcal{l}}^1)$;

6. 

${\mathit{M}}^{p,1}\left({\mathbb{R}}^d\right)\cap \mathit{W}(\mathcal{F}{\mathit{L}}^p,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)$ for $1\le p\le 2$.

Proof. 

All these examples can be derived with the help of the characterization in Theorem 2, in one way or the other. ${\mathit{S}}_0\left({\mathbb{R}}^d\right)={\mathit{M}}_0^{1,1}\left({\mathbb{R}}^d\right)$ is known to be Fourier invariant from the very beginning, see [1,29] or [28]. For $1<p<2$, we have—based on the classical Hausdorff–Young principle—

$${\mathit{M}}_0^{1,1}\left({\mathbb{R}}^d\right)\hookrightarrow \mathit{W}(\mathcal{F}{\mathit{L}}^p,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)\hookrightarrow \mathit{W}({\mathit{L}}^q,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)\hookrightarrow \mathit{W}({\mathit{L}}^1,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)={\mathit{L}}^1\left({\mathbb{R}}^d\right),$$

thus $\mathit{W}(\mathcal{F}{\mathit{L}}^p,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)$ is (for obvious reasons) an SCSI. Finally, one has to recall that the description of ${\mathit{M}}^{p,1}\left({\mathbb{R}}^d\right)$ as an (inverse) Fourier image of $\mathit{W}(\mathcal{F}{\mathit{L}}^p,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)$ describes an alternative characterization of such modulation spaces. Thus, Ex.6 is covered.

The chain of dense inclusions

$${\mathit{S}}_0\left({\mathbb{R}}^d\right)\hookrightarrow \mathit{W}({\mathit{C}}_0,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)\hookrightarrow \mathit{W}({\mathit{L}}^p,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)\hookrightarrow {\mathit{L}}^1\left({\mathbb{R}}^d\right)$$

immediately implies that $\mathit{W}({\mathit{L}}^p,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)$ is a (proper) Segal algebra for $1<p<\infty$. For $p=\infty$, the space $\mathit{W}({\mathit{C}}_0,{\mathcal{l}}^1)$ should be used. This is the closure of ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ in $\mathit{W}({\mathit{L}}^{\infty },{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)$. It is a Segal algebra as well. Strong character invariance is obvious in all these cases. □

A detailed study of the properties and strict inclusion between the examples given above will be given in a subsequent paper. For now, let us just point out that ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ is a proper subspace of $\mathit{W}\cap \mathcal{F}\mathit{W}$ according to an early result by Losert ([46]). We thus have the following chain of strict inclusions, as a justification of the visualization in Figure 1:

$${\mathit{S}}_0\hookrightarrow \mathit{W}\cap \mathcal{F}\mathit{W}\hookrightarrow \mathit{W}+\mathcal{F}\mathit{W}\hookrightarrow {\mathit{L}}^1\cap \mathcal{F}{\mathit{L}}^1.$$

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Remark 10.

According to [1], the Banach algebra $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ is the smallest strongly character invariant Segal algebra on ${\mathbb{R}}^d$. Thus, it coincides with the intersection over the family of Banach spaces of test functions and is a member of this family, unlike the intersection of all Segal algebras, which can be shown to coincide with the family of all band-limited ${\mathit{L}}^1$-functions. It is easy to show that any norm imposed on this space will have to be incomplete. In fact, if it was a Banach space for some given norm, it would always be possible to write many functions with non-compact support on the Fourier transform side as absolutely convergent infinite series.

Using a different terminology we can describe $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ as a Banach space $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$, continuously embedded into the Hilbert space $\left({\mathit{L}}^2\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_2\right)$ such that TF-shifts $\pi \left(\lambda \right)$ define a strongly continuous, isometric, and projective representation of ${\mathbb{R}}^d\times {\widehat{\mathbb{R}}}^d$ on $(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$, embedded into any other space of this type.

Compactness in the Algebra of Test Functions

This subsection relies on the Kolomogorov-type characterization of compact subsets in Banach spaces of distributions as given in [47].

Lemma 11.

Let $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ be a Banach algebra of test functions. Then, a bounded, closed subset M in $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is compact if and only if it is equicontinuous and tight, i.e., if and only if one has

1. [$\mathit{A}$-equicontinuity] For all $\epsilon >0$ there exists some $\delta >0$ such that

$$\parallel T_{z}f-f{\parallel }_{\mathit{A}}\le \epsilon ,\forall \left|z\right|\le \delta ,\forall f\in M,$$

or equivalently one has the following: for any $\epsilon >0$ there exists $k\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$ (or even in ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$) such that

$${\parallel k\ast f-f\parallel }_{\mathit{A}}\le \epsilon ,\forall f\in M;$$

2. [${\mathit{L}}^1$-tightness] For all $\epsilon >0$ there exists some compactly supported function $\phi \in \mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$ such that

$${\parallel \phi ·f-f\parallel }_{\mathit{A}}\le \epsilon ,\forall f\in M.$$

It is not difficult to derive from this criterion (relating the question to the classical Ascoli-Arcela Theorem) that for any pair of functions $h,k\in \mathit{A}\cap {\mathit{C}}_c\left({\mathbb{R}}^d\right)$ the mapping $f\mapsto k\ast \left(hf\right)$ is a compact operator on $\mathit{A}$.

In fact, using this argument, we can even strengthen the argument. Recall that the tightness of M in the ${\mathit{L}}^1$-sense, can be characterized in the following way: for any $\epsilon >0$ there exists some compact set $Q\subset {\mathbb{R}}^d$ such that

$${\int }_{{\mathbb{R}}^d\setminus Q}\left|f\left(x\right)\right|dx={\parallel f-f·{\mathbf{1}}_Q\parallel }_1\le \epsilon ,\forall f\in M.$$

Preposition 7.

Assume that M is a bounded, closed subset of $\mathit{A}$ which is tight following ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$ and equicontinuous following $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$. Then, the set is compact following $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$.

Proof. 

Given $\epsilon >0$, we first choose some $k\in \mathit{A}\cap {\mathit{C}}_c\left({\mathbb{R}}^d\right)$ (which is a dense subspace of $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$) with

$${\parallel k\ast f-f\parallel }_{\mathit{A}}<\epsilon /2,f\in M.$$

Next, we make use of the ${\mathit{L}}^1$-tightness and choose (for this particular choice of k) some $h\in \mathit{A}\cap {\mathit{C}}_c\left({\mathbb{R}}^d\right)$ such that

$${\parallel hf-f\parallel }_1<\epsilon /\left(2{\parallel k\parallel }_{\mathit{A}}\right),f\in M,$$

which implies due to (A2)

$${\parallel k\ast (hf-f)\parallel }_{\mathit{A}}<\epsilon /2,f\in M.$$

Then, with the help of the splitting

$$k\ast \left(hf\right)-f=[k\ast (hf)-k\ast f]+[k\ast f-f]$$

we obtain altogether

$${\parallel k\ast hf-f\parallel }_{\mathit{A}}<\epsilon ,f\in M.$$

This implies that the set M can be approximated by the relatively compact set $k\ast (h·M)$ to any given precision, and consequently, it has to be compact itself. □

Any such Banach space of test functions also satisfies further relevant properties as follows:

Preposition 8.

Assume that $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ satisfies the conditions (A1) to (A4). Then, it is a separable Banach space and satisfies the metric approximation property, i.e., for any compact set M in $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ and $\epsilon >0$, there exists some finite rank operator T on $\mathit{A}$ such that

$${\parallel Tf-f\parallel }_{\mathit{B}}\le \epsilon ,f\in M.$$

Proof. 

The separability follows easily from the fact that ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ is dense in $\mathit{A}$ and separable. In fact, the finite Gabor sums with rational coefficients form a dense subset of $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$. The verification that the metric approximation property applies in the given situation is discussed in [39], under much more general assumptions. These finite rank operators arise as finite dimensional approximation to discretized regularization operators. □

Remark 11.

The separability of $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is relevant for the description of the corresponding dual spaces as follows. The books [48] (Chap. IV) or [49] (2.3) demonstrate that separability implies that, on bounded subsets of $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$, the $w^{\ast }-$topology is metrizable. Consequently, the continuity of functionals or operators can be characterized using sequences. We will make use of this principle in order to avoid the more involved concept of the convergence of nets.

4. Banach Spaces of (Mild) A-Distributions

In this section, we will describe a couple of basic properties of the dual spaces $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$. Their generic elements $\mathit{A}$-distributions will be denoted by $\sigma \in {\mathit{A}}^{\prime }$. For the case $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})=\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$, we will use the established terminology of mild distributions (instead of calling them ${\mathit{S}}_0$-distributions). Since ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ is a dense subspace of $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$, for any Banach algebra of test functions, any space $\mathit{A}$-distribution discussed here is also a Banach space of mild distributions. Consequently, the generalized Fourier transform (following ${\mathit{S}}_0^{\prime }$) is well defined, even without recurrence to the theory of tempered distributions. On the other hand, one can view it as a restriction of the Fourier transform following tempered distributions, which leaves ${\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right)$ and, in fact, all the other spaces of $\mathit{A}$-distributions invariant.

As we shall see, the triple $(\mathit{A},{\mathit{L}}^2,{\mathit{A}}^{\prime })$ forms a Banach Gelfand Triple (see [50] for a summary of this concept), which shares many properties with the well-known Rigged Hilbert Spaces, with the typical special case being the Schwartz triple $(\mathcal{S}\left({\mathbb{R}}^d\right),{\mathit{L}}^2\left({\mathbb{R}}^d\right),{\mathcal{S}}^{\prime }\left({\mathbb{R}}^d\right))$. However, in contrast to the usual setting there, the inner space is not a nuclear Frechet space, and its topology is created by a countable family of (semi-)norms. We are instead dealing with Banach spaces and their duals, endowed with (sequential) $w^{\ast }-$convergence.

Nevertheless one of the key points we want to make in this paper is to demonstrate that, for quite a few questions in physics and engineering, this not yet popular setting is equally useful and often much easier to use. Instead of deep methods from the theory of topological vector spaces, a few fundamental principles from linear functional analysis suffice.

Lemma 12.

Given $h\in \mathit{A}$ or more generally any locally integrable function h, which can be represented as a finite sum of functions from the spaces ${\mathit{L}}^p\left({\mathbb{R}}^d\right)$, defines a regular distribution given by

$${\sigma }_h\left(f\right)={\int }_{{\mathbb{R}}^d}f\left(x\right)h\left(x\right)dx.$$

The mapping $h\mapsto {\sigma }_h$ is injective and bounded between suitable function spaces.

Proof. 

Let us start with $h\in \mathit{A}$. Using A1), we can identify functions $h\in \mathit{A}$ with linear functions ${\sigma }_h$ of the form

$${\sigma }_h\left(f\right)={\int }_{{\mathbb{R}}^d}f\left(x\right)h\left(x\right)dx,f\in \mathit{A}.$$

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In other words, every $h\in \mathit{A}$ defines a regular mild distribution. In fact, using the assumption $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})\hookrightarrow {\mathit{L}}^1\cap {\mathit{C}}_0\left({\mathbb{R}}^d\right)$, we have, for some $C_{\mathit{A}}>0$, the following:

$$|{\sigma }_h\left(f\right)|\le {\int }_{{\mathbb{R}}^d}\left|f\left(x\right)\right|\left|h\left(x\right)\right|dx\le {\parallel f\parallel }_{\infty }{\parallel h\parallel }_1\le C_{\mathit{A}}{\parallel f\parallel }_{\mathit{A}}{\parallel h\parallel }_{\mathit{A}},$$

and thus

$$\parallel {\sigma }_h{\parallel }_{{\mathit{A}}^{\prime }}=\underset{{\parallel f\parallel }_{\mathit{A}}\le 1}{sup}\left|{\sigma }_h\left(f\right)\right|\le C_{\mathit{A}}{\parallel h\parallel }_{\mathit{A}},$$

thus showing that the mapping $h\to {\sigma }_h$ is a bounded linear operator from $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ into $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$. The reader is encouraged to verify that it is injective, thus providing a continuous embedding.

The proof for a corresponding bounded linear injection of $\left({\mathit{L}}^p\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_p\right)$ follows the same lines. □

Theorem 5.

Given a Banach algebra $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ of test functions we have the following:

1. 

For any $p\in [1,\infty ]$, one has $\left({\mathit{L}}^p\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_p\right)\hookrightarrow ({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$;

2. 

$(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})\hookrightarrow ({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$, as a $w^{\ast }-$dense subspace;

3. 

$({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$ is Fourier invariant, with $\widehat{\sigma }\left(f\right)=\sigma \left(\widehat{f}\right),f\in \mathit{A},\sigma \in {\mathit{A}}^{\prime };$

4. 

By way of the adjointness relation $[g\ast \sigma ]\left(f\right):=\sigma (g^{✓}\ast f)$

$({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$ is a Banach convolution module over $\left({\mathit{L}}^1\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_1\right)$;

5. 

The pointwise action of $\mathcal{F}{\mathit{L}}^1$ on $\mathit{A}$ extends in a unique $w^{\ast }\text{-}{w}^{\ast }\text{-}$continuous fashion

to a pointwise module action on $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$.

Proof. 

Let us collect the key arguments to verify the claims of Theorem 5. Points (3) to (5) will follow from the general extension principle given in the next section.

Let us thus start with the chain of continuous inclusions (each space with its natural norm) obtained by applying (A1) and Lemma 8, as follows:

$${\mathit{S}}_0\hookrightarrow \mathit{A}\hookrightarrow {\mathit{L}}^1\cap {\mathit{C}}_0\hookrightarrow {\mathit{L}}^1\cap {\mathit{L}}^p\hookrightarrow {\mathit{L}}^p\hookrightarrow {\mathit{A}}^{\prime }\hookrightarrow {\mathit{S}}_0^{\prime }.$$

The norm density of ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ in $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ implies the $w^{\ast }-$density if $\mathit{A}$ in ${\mathit{A}}^{\prime }$ (as in the case of $\mathit{A}={\mathit{S}}_0$). In fact, following a standard criterion for $w^{\ast }-$density, it suffices to show that one has the following: given $f\in \mathit{A}$, then, the condition ${\sigma }_g\left(f\right)=0$ for all $g\in \mathit{A}$ implies that $f=0$ (in $\mathit{A}$). This is easily verified because one can choose $g=\overline{f}$ and thus obtain

$$0={\sigma }_g\left(f\right)={\int }_{{\mathbb{R}}^d}{\left|f\left(x\right)\right|}^{2}dx,$$

which in turn implies $f=0$ in ${\mathit{L}}^2\left({\mathbb{R}}^d\right)$, hence $f\left(x\right)=0$ a.e. But since we have $f\in \mathit{A}\subset {\mathit{L}}^1\cap {\mathit{C}}_0\left({\mathbb{R}}^d\right)$, we know that f is represented by a continuous function, thus implying $f\left(x\right)=0$ for all $x\in {\mathbb{R}}^d$ or $f=0$ in $\mathit{A}$.

Whether the Fourier transform is defined as in 3 or via the restriction of the Fourier transform for mild or even tempered distributions does not matter. In each case, the Fourier invariance of $\mathit{A}$ implies that of ${\mathit{A}}^{\prime }$. Since the Fourier transform on ${\mathit{A}}^{\prime }$ is essentially the (real) adjoint mapping to the (ordinary) Fourier transform on $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$, it is also $w^{\ast }\text{-}{w}^{\ast }\text{-}$continuous and thus uniquely determined in this way.

As an alternative argument, one might use the sequential approach to mild distributions (see [51]) and verify the remaining properties 3, 4, and 5 based on such an approach.

We simply point to the discussion of the remaining points (mostly 4 and 5) to the general, more functional analytic principles described in the subsequent section. □

Our next point in this section is to demonstrate that specific $\mathit{A}$-distributions following Example 4 already appeared in the literature a couple of years ago, in a different context. The connection between the space ${\mathcal{M}}_T\left({\mathbb{R}}^d\right)$ of transformable measures following Argabright/Lamadrid (see [3,4]) and mild distributions was already established in an earlier paper [52]. Their method allows to obtain the Fourier transform for certain unbounded Radon measures, such as $\mu =\bigsqcup \bigsqcup ={\sum }_{k\in {\mathbb{Z}}^d}{\delta }_k$, the Dirac comb over the discrete subgroup ${\mathbb{Z}}^d⊲{\mathbb{R}}^d$, as used extensively in engineering applications (see [53]).

In the context of mild distributions, their space ${\mathcal{M}}_T\left({\mathbb{R}}^d\right)$ can be described as the set of all Radon measures μ that define mild distributions (i.e., are continuous linear functionals in $\mathcal{F}{\mathit{L}}^1\cap {\mathit{C}}_c\left({\mathbb{R}}^d\right)$, with respect to both the inductive limit topology and the norm of $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$), with the additional property that the Fourier transform $\mathcal{F}\left(\mu \right)$ (taken following mild distributions) belongs to $\mathit{W}(\mathit{M},{\mathcal{l}}^{\infty })\left({\mathbb{R}}^d\right)$. Correspondingly, one can show (see [5], Theorem 8.2) that the set of transformable measures which have a transformable Fourier transform can be described as a subset of ${\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right)$ as follows:

Preposition 9.

A mild distribution $\mu \in {\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right)$ is a transformable measure with transformable Fourier transform $\widehat{\mu }$ if and only if both μ and $\widehat{\mu }$ belong to $\mathit{W}(\mathit{M},{\mathcal{l}}^{\infty })\left({\mathbb{R}}^d\right)$. In other words, these measures constitute the Fourier invariant space

$$\mathit{B}=\mathit{W}(\mathit{M},{\mathcal{l}}^{\infty })\left({\mathbb{R}}^d\right)\cap \mathcal{F}\mathit{W}(\mathit{M},{\mathcal{l}}^{\infty })\left({\mathbb{R}}^d\right)={\mathcal{M}}_T\left({\mathbb{R}}^d\right)\cap \mathcal{F}{\mathcal{M}}_T\left({\mathbb{R}}^d\right).$$

(10)

This space is the dual of the space $\mathit{W}({\mathit{C}}_0,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)+\mathcal{F}\left(\mathit{W}({\mathit{C}}_0,{\mathcal{l}}^1)\left({\mathbb{R}}^d\right)\right)$, see Ex.4. above.

5. Extended Domains for Operators, Including the Fourier Transform

The situation described so far allows us to enter the discussion of Banach Gelfand Triple, (we could use the term $\mathit{A}$-BGTr) of the form $(\mathit{A},{\mathit{L}}^2,{\mathit{A}}^{\prime })$, following the outline given in [54] (in particular, Theorem 7.3.3 there), or the general considerations given in [50]. Instead, we will focus on the extension of operators well defined on the underlying Banach algebra of test functions and discuss their extension to the corresponding space of $\mathit{A}$-distributions.

The overall setting aims at enlarging the domain of operators such as translation, pointwise-multiplication, convolution or the Fourier transform from a space $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ of test functions to a larger “extended” domain, particularly $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$. Such an idea requires the discussion of at least four issues as follows:

• How can we identify test functions with elements of the space of generalized functions?
• How can we define operators on the extended domain which can serve as an extension of existing operators on the space of test functions?
• We should then be able to justify the particular form of the extension. We will talk of the consistency question and we say that the consistence principle applies to a particular extension if the operator (usually keeping the same name) applied in the new form corresponds to the original operator on the test functions (up to the natural identification). In other words, it should not matter whether one applies the old operator first and then identifies the resulting test function with a generalized function, or applies the identification first and applies the extended operator afterwards.
• Finally, one would like to argue that the extension is unique if one allows various natural forms of (weaker than norm) continuity. Typically, it will be in the form of distributional convergence, often in the form of a $w^{\ast }-$convergence in some dual space.

Remark 12.

The last and the third issue are often closely related due to the $w^{\ast }-$density of $\mathit{A}$ in ${\mathit{A}}^{\prime }$. Once the consistency is established, the uniqueness can be derived using the arguments provided by the alternative sequential approach in [51]. Such a method employs more elementary mathematical arguments, but it requires some lengthy derivations, as compared to the (equivalent) but more elegant functional analytic method which we propose below.

There is a simple recipe which allows to extend an operator T that is the adjoint of another operator S, which leaves $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ invariant to the dual space $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$. We summarize the idea already used in a theorem formulated in the context ${\mathit{S}}_0\left({\mathbb{R}}^d\right)$ for unitary operators ($T=S$) as Theorem 7.3.3 in [54]. Similar ideas are also quite familiar in the context of rigged Hilbert spaces, where, however, more involved topological considerations are required.

Lemma 13.

Given a bounded operator T on $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$, we assume that there exists another bounded operator on S on $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ such that

$${\int }_{\mathbb{R}}Tf\left(x\right)g\left(x\right)={\int }_{\mathbb{R}}f\left(y\right)Sg\left(y\right)dy,f,g\in {\mathit{S}}_0\left({\mathbb{R}}^d\right),$$

(11)

then, it extends to a bounded operator $\tilde{T}$ on $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$. Such an extension is unique by the property that preserves the $w^{\ast }-$convergence of the sequences, and it can be given explicitly by the rule

$$\left[\tilde{T}\left(\sigma \right)\right]\left(f\right)=\sigma \left(Sf\right),f\in \mathit{A}.$$

(12)

Remark 13.

Due to the uniqueness of this (canonical) extension we can use the (same) symbol T for the original or the extended operator $\tilde{T}$ once the lemma is established, due to the $w^{\ast }-$density of $\mathit{A}$ in ${\mathit{A}}^{\prime }$. In fact, the original operator T can be recovered by restricting $\tilde{T}$ to $\mathit{A}\subset {\mathit{A}}^{\prime }$, which coincides with the adjoint operator $S^{\prime }$ to the operator S, which is defined on the dual of ${\mathit{A}}^{\prime }$, endowed with the $w^{\ast }-$topology if (11) is satisfied.

Proof. 

First of all, we observe that $\tilde{T}$, as given by (12), is a well-defined linear mapping. We claim that it defines a bounded linear operator on $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$, which is also a $w^{\ast }\text{-}{w}^{\ast }\text{-}$continuous operator on $({\mathit{A}}^{\prime },\parallel ·{\parallel }_{{\mathit{A}}^{\prime }})$, which due to (11), coincides with T on $\mathit{A}$, viewed as a subspace of ${\mathit{A}}^{\prime }$.

Obviously we have the following estimate, valid for $f\in \mathit{A}$ with ${\parallel f\parallel }_{\mathit{A}}\le 1$:

$$|\tilde{T}\left(\sigma \right)\left(f\right)|=|\sigma \left(Sf\right)|\le {\parallel \sigma \parallel }_{{\mathit{A}}^{\prime }}{\parallel Sf\parallel }_{\mathit{A}}\le {\parallel \sigma \parallel }_{{\mathit{A}}^{\prime }}{|\parallel S|\parallel }_{\mathit{A}}{\parallel f\parallel }_{\mathit{A}}\le {\parallel \sigma \parallel }_{{\mathit{A}}^{\prime }}{|\parallel S|\parallel }_{\mathit{A}}$$

By taking the supremum over all these functions, we come up with the estimate

$$\parallel \tilde{T}\left(\sigma \right){\parallel }_{{\mathit{A}}^{\prime }}\le {|\parallel S|\parallel }_{\mathit{A}}{\parallel \sigma \parallel }_{{\mathit{A}}^{\prime }},\sigma \in \mathit{A},$$

and thus, we come up with the estimate

$$|\parallel \tilde{T}{|\parallel }_{{\mathit{A}}^{\prime }}\le {|\parallel S|\parallel }_{\mathit{A}}.$$

(13)

Next, we have to take care of the $w^{\ast }-$continuity. Due to the separability of $\left({\mathit{S}}_0\left(\mathbb{R}\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$, we can restrict our attention here to the convergence of sequences. Given a $w^{\ast }-$convergent sequence ${\left({\sigma }_n\right)}_{n\ge 1}$ with limit ${\sigma }_0\in {\mathit{S}}_0^{\prime }\left(\mathbb{R}\right)$. Then, clearly,

$$\underset{n\to \infty }{lim}[\tilde{T}\left({\sigma }_n\right)]\left(f\right)=\underset{n\to \infty }{lim}{\sigma }_n\left(Tf\right)={\sigma }_0\left(Tf\right)=[\tilde{T}\left({\sigma }_0\right)]\left(f\right),$$

using only the fact that $Sf$ is just another element from ${\mathit{S}}_0\left(\mathbb{R}\right)$.

Finally, we obtain the converse estimate to (13) from the observation that the dual to $\mathit{A}$, endowed with the $w^{\ast }-$topology, is just the original spaces $\mathit{A}$ (cf. [49]) and that that for the $w^{\ast }\text{-}{w}^{\ast }\text{-}$continuous operator $\tilde{T}$ on ${\mathit{A}}^{\prime }$ are the adjoint operators ${\tilde{T}}^{\ast }$ (taken with respect to $w^{\ast }-$convergence), given for $\sigma \in {\mathit{A}}^{\prime }$ by

$${\tilde{T}}^{\ast }\left(f\right)\left(\sigma \right)=[\tilde{T}\left(\sigma \right)]\left(f\right)=\sigma \left(Tf\right),f\in \mathit{A},$$

coinciding with the original operator T. This implies the opposite estimate

$${|\parallel T|\parallel }_{\mathit{A}}=|\parallel {\tilde{T}}^{\ast }{|\parallel }_{\mathit{A}}\le |\parallel \tilde{T}{|\parallel }_{{\mathit{A}}^{\prime }},$$

(14)

thus implying the overall equality of norms

$$|\parallel \tilde{T}{|\parallel }_{{\mathit{A}}^{\prime }}={|\parallel S|\parallel }_{\mathit{A}}$$

(15)

□

We now apply this principle to the Fourier transform, using the Fundamental Identity (FIFT), which is an easy derivative from Fubini’s Theorem, as follows:

$${\int }_{{\mathbb{R}}^d}\widehat{f}\left(y\right)g\left(y\right)dy={\int }_{{\mathbb{R}}^d}f\left(x\right)\widehat{g}\left(x\right)dx.$$

(16)

In this case (cf. [54]), $T=\mathcal{F}$ is a unitary operator with $S=T$. In a similar way, multiplication operators, and finally, (with a slight modification) convolution operators can be dealt with in this way. Thus, we define the usual operations which are well defined for ordinary functions for ${\mathit{S}}_0\left(\mathbb{R}\right)$ in the usual way, as follows:

• For $\sigma \in {\mathit{S}}_0^{\prime }$, we define the Fourier transform via $\widehat{\sigma }\left(f\right):=\sigma \left(\widehat{f}\right),f\in \mathit{A}$.
• For $h\in \mathcal{F}{\mathit{L}}^1\left(\mathbb{R}\right)$, we set $\sigma ·h\left(f\right):=\sigma (h·f),f\in \mathit{A}$.
• For $g\in {\mathit{L}}^1\left(\mathbb{R}\right)$, we set $\sigma \ast g\left(f\right):=\sigma (g^{✓}\ast f),f\in \mathit{A}$.

The last convention relies on the fact that one has the following for $h\in {\mathit{C}}_b\left(\mathbb{R}\right)$:

$${\sigma }_h(g^{✓}\ast f)={\int }_{\mathbb{R}}h\left(x\right)(g\ast f)\left(x\right)dx={\int }_{\mathbb{R}}{\int }_{\mathbb{R}}h\left(x\right)g^{✓}(x-y)f\left(y\right)dydx$$

which by a change of variable and Fubini’s Theorem gives

$${\int }_{\mathbb{R}}\left({\int }_{\mathbb{R}}h\left(x\right)g(y-x)\right)f\left(y\right)dy={\sigma }_{h\ast g}\left(f\right).$$

Recall that the convolution with bounded measures (in particularm with functions from ${\mathit{L}}^1\left(\mathbb{R}\right)$) defines a bounded operator on $\mathit{A}$ (see, e.g., [9]), because $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Segal algebra. Since $g\mapsto g^{✓}$ is an isometric involution on ${\mathit{L}}^1\left({\mathbb{R}}^d\right)$, the convolution by $g^{✓}$ has the same operator norm. Note that one also has ${(g\ast f)}^{✓}=g^{✓}\ast f^{✓},f,g\in {\mathit{L}}^1\left({\mathbb{R}}^d\right).$

Consequently, the extended version of convolution still allows us to define $g\ast \sigma$ for $g\in {\mathit{L}}^1\left({\mathbb{R}}^d\right)$ or $\sigma ·h$ for any $\sigma \in {\mathit{S}}_0^{\prime }\left(\mathbb{R}\right)$ and $h\in \mathcal{F}{\mathit{L}}^1\left({\mathbb{R}}^d\right)$, as above, and we have

$${\parallel g\ast \sigma \parallel }_{{\mathit{A}}^{\prime }}\le {\parallel g\parallel }_1{\parallel \sigma \parallel }_{{\mathit{A}}^{\prime }},g\in {\mathit{L}}^1\left({\mathbb{R}}^d\right),\sigma \in {\mathit{A}}^{\prime }.$$

In a similar way, we observe the general estimates

$${\parallel \sigma ·h\parallel }_{{\mathit{A}}^{\prime }}\le {\parallel \sigma \parallel }_{\mathcal{F}{\mathit{L}}^1\left(\mathbb{R}\right)}\parallel {\mathit{A}}^{\prime }{\parallel }_{\sigma }h\in \mathcal{F}{\mathit{L}}^1\left(\mathbb{R}\right),\sigma \in {\mathit{S}}_0^{\prime }\left(\mathbb{R}\right).$$

The last property of all the Banach algebras of the test functions listed above is satisfied, and this is also very useful for the invariance under dilations, especially under the general automorphisms of ${\mathbb{R}}^d$. Any such automorphism α can be realized by a (unique) non-singular $d\times d$-matrix $\mathit{B}$ via matrix multiplication $x\mapsto \alpha \left(x\right)=\mathit{B}\ast x$. This mapping can be extended to ${\mathit{M}}_b\left({\mathbb{R}}^d\right)$, satisfying the rule ${\delta }_x\mapsto {\delta }_{\alpha \left(x\right)}$, as the adjoint of the following dilation operator $\left[D_{\alpha }\phi \right]\left(x\right)=\phi \left(\alpha \left(x\right)\right)$, for $x\in {\mathbb{R}}^d$.

For this purpose, we introduce property (A7), as follows:

Definition 5

(A7). $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is invariant under the general automorphisms of the underlying group ${\mathbb{R}}^d$ (specifically dilations, or rotations, if any of the operators $D_{\alpha }$ maps $\mathit{A}$ into itself.

Remark 14.

Since all of these operators are bounded on $\left({\mathit{L}}^2\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_2\right)$, one can apply the Closed Graph Theorem. Since $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ is a Banach space by assumption, this implies that any of these operators is bounded on $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$. In other words, there exists $C_{\alpha }>0$ such that

$$\parallel D_{\alpha }f{\parallel }_{\mathit{A}}\le C_{\alpha }{\parallel f\parallel }_{\mathit{A}},f\in \mathit{A}.$$

(17)

Preposition 10.

If $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ satisfies (A7), then, the operator $D_{\alpha }$ extends to a $w^{\ast }\text{-}{w}^{\ast }\text{-}$continuous operator on $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$. All the examples listed above satisfy this condition.

Proof. 

We can apply Lemma 13 after observing that the validity of Formula (11) follows from the standard transformation formula for integrals, with S being a suitably normalized version of $D_{1/\alpha }$. The verification of the details (concerning the various examples listed above) is left to the interested reader. □

6. Conclusions and Summary

The purpose of this article is to show how one can set up a theory of generalized functions comparable to the theory of mild distributions from an axiomatic basis, requesting only a few properties from a given Banach space of test functions $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$. The invariance properties of such a space under translation and modulation, formulated properly, allow us to introduce the convolution and pointwise multiplication of the test functions by the test functions and even the multipliers of test functions. Overall, the theory of (strongly character invariant) Segal algebra helps us understand the situation.

Starting from these first principles, which among others imply that such Banach spaces, $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$, or Fourier invariant Banach algebras are under both convolution and pointwise multiplication, and a classical version of the convolution theorem easily established in this context, we can move on to the dual space. The assumptions allow us to embed $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$ (the space of test functions) into the dual space $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})$, which is called the space of $\mathit{A}$-distributions. Many (invariance) properties and operators, including the Fourier transform, extend to such a larger space of “signals”.

While this article points out that there is a variety of choices for such Fourier invariant Banach algebras of test-functions (and thus a corresponding variety of dual spaces), it also observes that the Segal algebra $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$ is the smallest one, and consequently, the space $({\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0^{\prime }})$ (nowadays known as the space of mild distributions on ${\mathbb{R}}^d$) is the largest. As will be discussed in a subsequent paper, there are many other good arguments for the choice of $(\mathit{A},\parallel ·{\parallel }_{\mathit{A}})=\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$, among others, the tensor product property, the invariance under fractional Fourier transforms, or the validity of Poisson’s formula (explored in [53] extensively), which turns the Segal algebra $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0}\right)$, together with the space of mild distribution, into the most useful setting following [55].

We hope that our work will help engineers and physicists in their presentation of results by offering a mathematical setting that is both relatively simple to use yet mathematically well based.