Horváth Spaces and a Representations of the Fourier Transform and Convolution

Abstract

This paper explores the structural representation and Fourier analysis of elements in Horváth distribution spaces ${\mathcal{S}}_k^{\prime }$, for $k<-n$. We prove that any element in ${\mathcal{S}}_k^{\prime }$ can be expressed as a finite sum of derivatives of continuous $L^1\left({\mathbb{R}}^n\right)$-functions acting on Schwartz test functions. This representation leads to an explicit expression for their distributional Fourier transform in terms of classical Fourier transforms. Additionally, we present a distributional representation for the convolution of two such elements, showing that the convolution is well-defined over $\mathcal{S}$. These results deepen our understanding of non-tempered distributions and extend Fourier methods to a broader functional framework.

1. Introduction

The theory of distributions is a powerful analytical framework whose effectiveness is significantly enhanced when combined with Fourier analysis. One of its principal applications is in the study of partial differential equations, and more specifically, in the search for fundamental solutions.

The structure theorem for tempered distributions states that any $T\in {\mathcal{S}}^{\prime }$ can be expressed as a finite linear combination of derivatives of smooth functions with, at most, polynomial growth:

$$T=\sum _{\left|\alpha \right|\le m}{\partial }^{\alpha }{f}_{\alpha },f_{\alpha }\in C^{\infty }\left({\mathbb{R}}^n\right),|f_{\alpha }\left(x\right){|\le C(1+\left|x\right|)}^k$$

This representation ensures that the Fourier transform of T is well-defined in ${\mathcal{S}}^{\prime }$, and enables the use of Fourier analytic techniques for solving partial differential equations.

In this work, we derive an explicit expression for the distributional Fourier transform of elements in the Horváth space ${\mathcal{S}}_k^{\prime }$, with $k\in \mathbb{Z}$ and $k<-n$, as a finite sum of classical Fourier transforms of continuous functions in $L^1\left({\mathbb{R}}^n\right)$, each multiplied by a monomial. To achieve this, we first establish a structural representation for elements of ${\mathcal{S}}_k^{\prime }$ as finite sums of derivatives of continuous $L^1$-functions, acting on test functions in the Schwartz space $\mathcal{S}$. This provides a more refined understanding of the internal structure of the space ${\mathcal{S}}_k^{\prime }$.

We also establish a representation, on the Schwartz space $\mathcal{S}$, for the convolution of two elements in the Horváth space ${\mathcal{S}}_k^{\prime }$, with $k\in \mathbb{Z}$ and $k<-n$.

Structure theorems for distributions in general, and for tempered distributions in particular, have been the subject of extensive research due to their usefulness in solving partial differential equations. In this context, it is worth highlighting the following results [1,2,3,4,5,6,7], among others.

2. Notation and Preliminary Results

Throughout this paper, we adopt the following standard notations.

Let $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ denote a point in Euclidean space, and let $p=(p_1,\dots ,p_n)\in {\mathbb{N}}^n$ be a multi-index, that is, an n-tuple whose components are natural numbers. The order of p is defined by

$$\left|p\right|=p_1+\cdots +p_n.$$

For any such multi-index, we define the monomial and factorial as:

$$x^p=x_1^{p_1}{x}_2^{p_2}\cdots x_n^{p_n},p!=p_1!·p_2!\cdots p_n!.$$

We denote by ${\partial }_j$ the partial derivative with respect to $x_j$, i.e., ${\partial }_j={\displaystyle \frac{\partial }{\partial x_j}}$, for $j=1,\dots ,n$. Given $p\in {\mathbb{N}}^n$, we write:

$${\partial }^p={\partial }_1^{p_1}\cdots {\partial }_n^{p_n}={\displaystyle \frac{{\partial }^{\left|p\right|}}{\partial x_1^{p_1}\cdots \partial x_n^{p_n}}},$$

where $\left|p\right|$ denotes both the total order of the multi-index and of the corresponding derivative operator.

Let $k\in \mathbb{Z}$ be fixed. We denote by ${\mathcal{S}}_k$ the space of all complex-valued functions $\varphi :{\mathbb{R}}^n\to \mathbb{C}$ that are infinitely differentiable and satisfy the following decay condition: for every multi-index $p\in {\mathbb{N}}^n$ and for every $\epsilon >0$, there exists a constant $A=A(\varphi ,p,\epsilon )>0$ such that

$$\left|{(1+|x|}^2{)}^k{\partial }^p\varphi \left(x\right)\right|\le \epsilon \mathrm{for}\mathrm{all}\left|x\right|>A.$$

(1)

The topology on ${\mathcal{S}}_k$ is generated by the countable family of seminorms ${\left\{q_{k,p}\right\}}_{p\in {\mathbb{N}}^n}$, where

$$q_{k,p}\left(\varphi \right)=\underset{x\in {\mathbb{R}}^n}{sup}\left|{(1+|x|}^2{)}^k{\partial }^p\varphi \left(x\right)\right|.$$

(2)

Endowed with this topology, ${\mathcal{S}}_k$ becomes a complete locally convex space.

Moreover, the classical space of test functions $\mathcal{D}=C_c^{\infty }\left({\mathbb{R}}^n\right)$ is dense in ${\mathcal{S}}_k$; see ([8] p. 419). As it is usual, $\mathcal{D}$ denotes the vector space of all functions defined on ${\mathbb{R}}^n$, and whose partial derivatives of all orders exist and are continuous, and whose support is contained in some compact subset of ${\mathbb{R}}^n$.

According to ([8], Proposition 2, p. 97), for every $f\in {\mathcal{S}}_k^{\prime }$, there exist constants $C>0$ and an integer $m\ge 0$, depending on f, such that

$$\left|\langle f,\varphi \rangle \right|\le C\underset{\left|p\right|\le m}{max}{q}_{k,p}\left(\varphi \right)\mathrm{for}\mathrm{all}\varphi \in {\mathcal{S}}_k,$$

(3)

where $\left|p\right|=p_1+\cdots +p_n$ is the total order of the multi-index $p=(p_1,\dots ,p_n)$.

The smallest such integer m for which inequality (3) holds is called the order of the distribution f; see ([9], Théorème XXIV, p. 88).

For $k\in \mathbb{Z}$, $k<0$, one has that for each $y\in {\mathbb{R}}^n$ the function $x\mapsto e^{ixy}$ is a member of the space ${\mathcal{S}}_k$. Thereby, it was proved in ([10], Theorem 2.1) that the function

$$\begin{array}{c}\hfill y\mapsto 〈T\left(x\right),e^{ixy}〉,T\in {\mathcal{S}}_k^{\prime },y\in {\mathbb{R}}^n,\end{array}$$

represents the usual distributional Fourier transform ([9] Chap. 7, §6, p. 248) of $T\in {\mathcal{S}}_k^{\prime }$ when it acts over functions on $\mathcal{S}$, namely

$$\begin{array}{c}\hfill 〈T,\widehat{\varphi }〉={\int }_{{\mathbb{R}}^n}〈T\left(x\right),e^{ixy}〉\varphi \left(y\right)dy,\mathrm{for}\mathrm{all}\varphi \in \mathcal{S},\end{array}$$

where

$$\begin{array}{c}\hfill \widehat{\varphi }\left(y\right)={\int }_{{\mathbb{R}}^n}\varphi \left(x\right)e^{ixy}dx,y\in {\mathbb{R}}^n.\end{array}$$

The space $\mathcal{S}$ of Schwartz test functions on ${\mathbb{R}}^n$ consists of all complex-valued functions $\varphi \in C^{\infty }\left({\mathbb{R}}^n\right)$ that, along with all their partial derivatives, decay at infinity faster than the reciprocal of any polynomial. More precisely, a function $\varphi \in C^{\infty }\left({\mathbb{R}}^n\right)$ belongs to $\mathcal{S}$ if for every pair of multi-indices $\alpha ,\beta \in {\mathbb{N}}^n$, the following supremum is finite:

$$\underset{x\in {\mathbb{R}}^n}{sup}\left|x^{\alpha }{\partial }^{\beta }\varphi \left(x\right)\right|<\infty .$$

This condition implies that both the function and all its derivatives vanish infinity faster than any inverse power of $\left|x\right|$, making $\mathcal{S}$ a natural space for Fourier analysis and distribution theory.

The topology of $\mathcal{S}$ is defined by the countable family of seminorms:

$$p_{\alpha ,\beta }\left(\varphi \right)=\underset{x\in {\mathbb{R}}^n}{sup}\left|x^{\alpha }{\partial }^{\beta }\varphi \left(x\right)\right|,\mathrm{for}\mathrm{all}\alpha ,\beta \in {\mathbb{N}}^n.$$

(4)

The space $\mathcal{S}$ is equipped with the locally convex topology defined by the family $\left(p_{\alpha ,\beta }\right)$ of seminorms, where $\left(p_{\alpha ,\beta }\right)$ is defined by (4), $\alpha$ and $\beta$ running through ${\mathbb{N}}^n$.

Another equivalent family of seminorms on $\mathcal{S}$ is given by

$$q_{k,p}\left(\varphi \right)=\underset{x\in {\mathbb{R}}^n}{sup}\left|{(1+|x|}^2{)}^k{\partial }^p\varphi \left(x\right)\right|,$$

(5)

where $p\in {\mathbb{N}}^n$ and $k\in \mathbb{Z}$.

The importance of the class $\mathcal{S}$ is due to the following result, where we use the notation $D_j=-i{\partial }_j$, $D_j\mathcal{S}\subset \mathcal{S}$, $x_j\mathcal{S}\subset \mathcal{S}$, and $\mathcal{S}\subset L^1$.

Then, the Schwartz space is given by the intersection:

$$\mathcal{S}=\bigcap _{k\in \mathbb{Z}}{\mathcal{S}}_k.$$

3. A Representation of the Fourier Transform over ${\mathcal{S}}_{\mathit{k}}^{\mathbf{\prime }}$

The following Lemma gives structures for members of ${\mathcal{S}}_k^{\prime }$, $k\in \mathbb{Z}$. In this context, the functional $T\in {\mathcal{S}}_k^{\prime }$ is expressed as a finite sum of derivatives of continuous functions that exhibit specific growth properties. This formulation is essential for the results that we will present in this paper regarding the distributional Fourier transform.

Lemma 1.

Let $k\in \mathbb{Z}$, and $T\in {\mathcal{S}}_k^{\prime }$ of order m, then there exist ${\left(g_p\right)}_{\left|p\right|\le m+2n}$ continuous functions, such that

$$\begin{array}{c}\hfill T=\sum _{\left|p\right|\le m+2n}{\partial }^p{g}_p,over\mathcal{S},\end{array}$$

where $\left|g_p\left(x\right)\right|\le M_p·{\left(1+{\left|x\right|}^2\right)}^k{\left|x\right|}^n$, for all $x\in {\mathbb{R}}^n$, being $M_p>0$ and $\left|p\right|\le m+2n$.

Proof. 

Using the techniques of Trèves ([11], p. 272), it was proved in ([12], Lemma, p. 364) that there exists $l\in \mathbb{N}$ and ${\left(h_p\right)}_{\left|p\right|\le l}$ measurable functions, such that for all $p\in {\mathbb{N}}^n$, $\left|p\right|\le l$, one has ${\left(1+{\left|x\right|}^2\right)}^{-k}{h}_p\left(x\right)\in L^{\infty }\left({\mathbb{R}}^n\right)$ and

$$\begin{array}{c}\hfill T=\sum _{\left|p\right|\le l}{\partial }^p{h}_p,\mathrm{over}\mathcal{S}.\end{array}$$

(6)

If one analyzes the proof of ([12], Lemma, p. 364) one observes that $l=m+n$.

Now, for all $p\in {\mathbb{N}}^n$ with $\left|p\right|\le m+n$ and all $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ set

$$\begin{array}{c}\hfill {\tilde{g}}_p\left(x\right)={\int }_0^{x_1}d{t}_1{\int }_0^{x_2}d{t}_2\cdots {\int }_0^{x_n}{\left(1+{\left|t\right|}^2\right)}^{-k}{h}_p\left(t\right)d{t}_n,t=(t_1,\dots ,t_n).\end{array}$$

Note that ${\left(1+{\left|t\right|}^2\right)}^{-k}{h}_p\left(t\right)\in L^{\infty }\left({\mathbb{R}}^n\right)$, and so ${\left(1+{\left|t\right|}^2\right)}^{-k}{h}_p\left(t\right)\in L_{loc}^1\left({\mathbb{R}}^n\right)$. Thus, ${\tilde{g}}_p$ are continuous functions on ${\mathbb{R}}^n$. Moreover, for $\beta =(1,1,\dots ,1)\in {\mathbb{N}}^n$, one has

$$\begin{array}{c}\hfill {\partial }^{\beta }{\tilde{g}}_p\left(x\right)={\left(1+{\left|x\right|}^2\right)}^{-k}{h}_p\left(x\right),\mathrm{almost} \mathrm{everywhere} \mathrm{on}{\mathbb{R}}^n.\end{array}$$

Clearly

$$\begin{array}{c}\hfill \left|{\tilde{g}}_p\right(x\left)\right|\le |x_1\cdots x_n|\parallel {\left(1+{\left|t\right|}^2\right)}^{-k}{h}_p{\left(t\right)\parallel }_{\infty },\end{array}$$

(7)

for all $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ and $p\in {\mathbb{N}}^n$ with $\left|p\right|\le m+n$.

From ([11], Eq. (24.2), p. 260), one obtains that

$$\begin{array}{c}\hfill {\left(1+{\left|x\right|}^2\right)}^k{\partial }^{\beta }{\tilde{g}}_p\left(x\right)=\sum _{\alpha \le \beta }{(-1)}^{\left|\alpha \right|}{\partial }^{\beta -\alpha }\left[\left\{{\partial }^{\alpha }{\left(1+{\left|x\right|}^2\right)}^k\right\}{\tilde{g}}_p\left(x\right)\right].\end{array}$$

Furthermore, from ([8], Eq. (3), p. 102) and all $\alpha \in {\mathbb{N}}^n$, it follows

$$\begin{array}{c}\hfill {\partial }^{\alpha }{\left(1+{\left|x\right|}^2\right)}^k={\left(1+{\left|x\right|}^2\right)}^{k-\left|\alpha \right|}·\left(\sum _{\left|\gamma \right|\le \left|\alpha \right|}{M}_{\alpha ,\gamma }{x}^{\gamma }\right),\end{array}$$

where $M_{\alpha ,\gamma }$ are suitable real constants.

Thus, we have

$$\begin{array}{ccc}& & T=\sum _{\left|p\right|\le m+n}{\partial }^p{h}_p=\sum _{\left|p\right|\le m+n}{\partial }^p\left\{{\left(1+{\left|x\right|}^2\right)}^k{\partial }^{\beta }{\tilde{g}}_p\left(x\right)\right\}\hfill \\ & =& \sum _{\left|p\right|\le m+n}{\partial }^p\left\{\sum _{\alpha \le \beta }{(-1)}^{\left|\alpha \right|}{\partial }^{\beta -\alpha }\left[\left\{{\partial }^{\alpha }{\left(1+{\left|x\right|}^2\right)}^k\right\}{\tilde{g}}_p\left(x\right)\right]\right\}\hfill \\ & =& \sum _{\left|p\right|\le m+n}\sum _{\alpha \le \beta }{\partial }^p{\partial }^{\beta -\alpha }\left\{\left(\sum _{\left|\gamma \right|\le \left|\alpha \right|}{(-1)}^{\left|\alpha \right|}{M}_{\alpha ,\gamma }{x}^{\gamma }\right){\left(1+{\left|x\right|}^2\right)}^{k-\left|\alpha \right|}{\tilde{g}}_p\left(x\right)\right\},\hfill \end{array}$$

which yields

$$\begin{array}{c}\hfill T=\sum _{\left|p\right|\le m+2n}{\partial }^p{g}_p,\mathrm{over}\mathcal{S},\end{array}$$

where

$$\begin{array}{c}\hfill g_p\left(x\right)=\left(\sum _{\left|\gamma \right|\le \left|\alpha \right|}{(-1)}^{\left|\alpha \right|}{M}_{\alpha ,\gamma }{x}^{\gamma }\right){\left(1+{\left|x\right|}^2\right)}^{k-\left|\alpha \right|}{\tilde{g}}_q\left(x\right),\end{array}$$

for $p=q+\beta -\alpha$, whenever $\left|q\right|\le m+n$, $\beta =(1,1,\dots ,1)\in {\mathbb{N}}^n$ and $\alpha \le \beta$, and $g_p\left(x\right)=0$ otherwise.

From the estimation (7) and taking into account that $|x_j|\le |x|$, for all $1\le j\le n$, $x=(x_1,\dots ,x_n)$, one obtains the bound $|g_p\left(x\right)|\le M_p·{\left(1+{\left|x\right|}^2\right)}^k{\left|x\right|}^n$, for all $x\in {\mathbb{R}}^n$, being $M_p>0$ and $\left|p\right|\le m+2n$. □

The next theorem is the main result of this paper. Taking into account that the above functions $g_p$ possess classical Fourier transforms, it establishes the representation of the distributional Fourier transform in terms of classical Fourier transforms.

Theorem 1.

Let $T\in {\mathcal{S}}_k^{\prime }$ of order m with $k\in \mathbb{Z}$, $k<-n$. Then,

$$\begin{array}{c}\hfill 〈T\left(x\right),e^{ixy}〉=\sum _{\left|p\right|\le m+2n}{(-iy)}^p{\widehat{g}}_p\left(y\right),y\in {\mathbb{R}}^n,\end{array}$$

(8)

where the symbol ^{^} denotes the classical Fourier transform, i.e.,

$$\begin{array}{c}\hfill {\widehat{g}}_p\left(y\right)={\int }_{{\mathbb{R}}^n}{g}_p\left(x\right)e^{ixy}dx,y\in {\mathbb{R}}^n,\end{array}$$

(9)

and

$$\begin{array}{c}\hfill T=\sum _{\left|p\right|\le m+2n}{\partial }^p{g}_p,over\mathcal{S},\end{array}$$

where $g_p$ are given by the above Lemma 1.

Proof. 

First note that ${\widehat{g}}_p\left(y\right)$ exits since $|g_p\left(x\right)|\le M_p·{\left(1+{\left|x\right|}^2\right)}^k{\left|x\right|}^n$, for all $x\in {\mathbb{R}}^n$ and $k<-n$. In fact, for all $y\in {\mathbb{R}}^n$

$$\begin{array}{ccc}& & \left|{\widehat{g}}_p\left(y\right)\right|\hfill \\ & \le & {\int }_{{\mathbb{R}}^n}\left|g_p\left(x\right)\right|dx\hfill \\ & \le & M_p{\int }_{{\mathbb{R}}^n}{\left(1+{\left|x\right|}^2\right)}^k{\left|x\right|}^{n}dx.\hfill \end{array}$$

Observe that for the case when $n=1$, the integral ${\int }_{\mathbb{R}}{\left(1+x^2\right)}^k\left|x\right|dx$ converges for $k<-1.$

For the case when $n=2$ and making the change to polar coordinates, one has

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}{\left(1+{\left|x\right|}^2\right)}^k{\left|x\right|}^{2}dx={\int }_0^{2\pi }d\theta {\int }_0^{\infty }{(1+r^2)}^k{r}^{3}dr,\end{array}$$

which converges for $k<-2.$

For the case when $n\ge 3$ and making the change to sspherical coordinates, one has

$$\begin{array}{ccc}\hfill {\int }_{{\mathbb{R}}^n}{\left(1+{\left|x\right|}^2\right)}^k{\left|x\right|}^{n}dx& =& {\int }_0^{\pi }|{sin}^{n-2}{\theta }_1|d{\theta }_1{\int }_0^{\pi }\left|{sin}^{n-3}{\theta }_2\right|d{\theta }_2\cdots \hfill \\ & & \cdots {\int }_0^{\pi }|sin{\theta }_{n-2}|d{\theta }_{n-2}{\int }_0^{2\pi }d{\theta }_{n-1}{\int }_0^{\infty }{(1+r^2)}^k{r}^{2n-1}dr,\hfill \end{array}$$

which converges for $k<-n.$

Thus, in all the cases, the integral ${\int }_{{\mathbb{R}}^n}{\left(1+{\left|x\right|}^2\right)}^k{\left|x\right|}^{n}dx$ converges for $k<-n.$

Now, by Theorem 2.1 in [10], the equality

$$\begin{array}{c}\hfill 〈T,\widehat{\varphi }〉={\int }_{{\mathbb{R}}^n}〈T\left(x\right),e^{ixy}〉\varphi \left(y\right)dy,\end{array}$$

(10)

holds for all $\varphi \in \mathcal{S}$.

Next, denoting by ${\varphi }_p\left(y\right)={(-iy)}^p\varphi \left(y\right)$, one obtains

$$\begin{array}{ccc}& & 〈T,\widehat{\varphi }〉=\sum _{\left|p\right|\le m+2n}〈g_p\left(y\right),{\widehat{\varphi }}_p\left(y\right)〉=\sum _{\left|p\right|\le m+2n}〈{\widehat{g}}_p\left(y\right),{\varphi }_p\left(y\right)〉\hfill \\ & =& \sum _{\left|p\right|\le m+2n}〈{\widehat{g}}_p\left(y\right),{(-iy)}^p\varphi \left(y\right)〉=\sum _{\left|p\right|\le m+2n}〈{(-iy)}^p{\widehat{g}}_p\left(y\right),\varphi \left(y\right)〉.\hfill \end{array}$$

Thus, using this fact and the equality (10), one has

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}\sum _{\left|p\right|\le m+2n}{(-iy)}^p{\widehat{g}}_p\left(y\right)\varphi \left(y\right)dy={\int }_{{\mathbb{R}}^n}〈T\left(x\right),e^{ixy}〉\varphi \left(y\right)dy,\end{array}$$

(11)

for all $\varphi \in \mathcal{S}$.

Finally, since $〈T\left(x\right),e^{ixy}〉$ and ${\sum }_{\left|p\right|\le m+2n}{(-iy)}^p{\widehat{g}}_p\left(y\right)$ are continuous functions (see [10], Proposition 2.1 and [4], Theorem 9.6, p. 182, respectively), the use of (11) and ([4], Theorem 1.39, p. 30), yields to the next equality

$$\begin{array}{c}\hfill 〈T\left(x\right),e^{ixy}〉=\sum _{\left|p\right|\le m+2n}{(-iy)}^p{\widehat{g}}_p\left(y\right),y\in {\mathbb{R}}^n.\end{array}$$

□

In Proposition 2.1 of [13], it was proved that for f being a measurable function defined on ${\mathbb{R}}^n$, such that ${\left(1+{\left|x\right|}^2\right)}^{-k}f\left(x\right)$ is integrable on ${\mathbb{R}}^n$ for some $k\in \mathbb{Z}$, $k<0$, then the linear functional $T_f$ over ${\mathcal{S}}_k^{\prime }$, given by

$$\begin{array}{c}\hfill 〈T_f,\varphi 〉={\int }_{{\mathbb{R}}^n}f\left(x\right)\varphi \left(x\right)dx,\varphi \in {\mathcal{S}}_k,\end{array}$$

is a member of ${\mathcal{S}}_k^{\prime }$ of order zero. Moreover, the distributional Fourier transform of $T_f$ agrees with the classical Fourier transform of the function f.

Then, according to this result and Theorem 1 above, one obtains

Corollary 1.

Let f be a measurable function defined on ${\mathbb{R}}^n$ such that ${\left(1+{\left|x\right|}^2\right)}^{-k}f\left(x\right)$ be integrable on ${\mathbb{R}}^n$. Assume that $k<-n$. Then,

$$\begin{array}{c}\hfill \widehat{f}\left(y\right)=\sum _{\left|p\right|\le 2n}{(-iy)}^p{\widehat{g}}_p\left(y\right),y\in {\mathbb{R}}^n,\end{array}$$

where the symbol ^{^} denotes the classical Fourier transform (9) and

$$\begin{array}{c}\hfill T_f=\sum _{\left|p\right|\le 2n}{\partial }^p{g}_p,over\mathcal{S},\end{array}$$

with $g_p$ given by the above Lemma 1.

4. A Representation of the Fourier Convolution over ${\mathcal{S}}_{\mathit{k}}^{\mathbf{\prime }}$

The Fourier convolution $T_1\ast T_2$ of two members $T_1,T_2\in {\mathcal{S}}_k^{\prime }$, $k\in \mathbb{Z}$, $k<0$, is the member of ${\mathcal{S}}_k^{\prime }$ given in [14] by means of

$$\begin{array}{c}\hfill 〈T_1\ast T_2,\varphi 〉=〈T_1\left(x\right),〈T_2\left(y\right),\varphi (x+y)〉〉,\varphi \in {\mathcal{S}}_k.\end{array}$$

Denoting $\left(\mathcal{F}T\right)\left(y\right)=〈T\left(x\right),e^{ixy}〉$, $y\in {\mathbb{R}}^n$, $T\in {\mathcal{S}}_k^{\prime }$, $k\in \mathbb{Z}$, $k<0$, one obtains from ([10], Proposition 4.1)

$$\begin{array}{c}\hfill \left(\mathcal{F}(T_1\ast T_2)\right)\left(y\right)=\left(\mathcal{F}{T}_1\right)\left(y\right)·\left(\mathcal{F}{T}_2\right)\left(y\right),y\in {\mathbb{R}}^n.\end{array}$$

Now, using the inversion formula obtained in ([15], Theorem 2.2), one has

$$\begin{array}{cc}\hfill 〈T_1\ast T_2,\varphi 〉& ={\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\underset{a\to 0^{+}}{lim}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\left(\mathcal{F}{T}_1\right)\left(y\right)·\left(\mathcal{F}{T}_2\right)\left(y\right)·e^{-ity}\hfill \\ \hfill & ·e^{-a^2{\parallel y\parallel }^2}dy\varphi \left(t\right)dt,\varphi \in \mathcal{S}.\hfill \end{array}$$

(12)

Now, from (12) and using Theorem 1 above, one arrives at the next representation for the convolution $T_1\ast T_2$.

Proposition 1.

Let $T_1,T_2\in {\mathcal{S}}_k^{\prime }$, $k\in \mathbb{Z}$, $k<-n$, of order $m_1$ and $m_2$, respectively, then

$$\begin{array}{ccc}\hfill 〈T_1\ast T_2,\varphi 〉& =& {\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\underset{a\to 0^{+}}{lim}\sum _{\begin{array}{c}\left|p\right|\le m_1+2n\\ \left|q\right|\le m_2+2n\end{array}}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{(-iy)}^{p+q}{\widehat{g}}_{1,p}\left(y\right)·{\widehat{g}}_{2,p}\left(y\right)\hfill \\ & & e^{-ity}{e}^{-a^2{\parallel y\parallel }^2}dy\varphi \left(t\right)dt,\varphi \in \mathcal{S},\hfill \end{array}$$

(13)

where the symbol ^{^} denotes the classical Fourier transform and where

$$T_1=\sum _{\left|p\right|\le m_1+2n}{\partial }^p{g}_{1,p}\left(x\right),over\mathcal{S},$$

and

$$T_2=\sum _{\left|q\right|\le m_2+2n}{\partial }^q{g}_{2,q}\left(x\right),over\mathcal{S},$$

with $g_{1,p}$ and $g_{2,p}$ given by the above Lemma 1.

Observe that from ([16], Proposition 4), one has that for f and g being measurable functions on ${\mathbb{R}}^n$ such that ${\left(1+{\left|x\right|}^2\right)}^{-k}f\left(x\right)$ and ${\left(1+{\left|x\right|}^2\right)}^{-k}g\left(x\right)$ are Lebesgue integrable on ${\mathbb{R}}^n$ for some $k\in \mathbb{Z}$, $k<0$, then $T_{f\ast g}=T_f\ast T_g$ where $f\ast g$ denotes the classical convolution of the functions f and g.

Now, taking into account that $T_f$ and $T_g$ have order zero, then from Proposition 1, one obtains the next result.

Corollary 2.

Let f and g be measurable functions on ${\mathbb{R}}^n$, such that ${(1+|x|}^2{)}^{-k}f\left(x\right)$ and ${(1+|x|}^2{)}^{-k}g\left(x\right)$ are Lebesgue integrable over ${\mathbb{R}}^n$ for some $k\in \mathbb{Z}$ with $k<0$, then

$$\begin{array}{ccc}\hfill 〈T_{f\ast g},\varphi 〉& =& {\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\underset{a\to 0^{+}}{lim}\sum _{\begin{array}{c}\left|p\right|\le 2n\\ \left|q\right|\le 2n\end{array}}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{(-iy)}^{p+q}{\widehat{g}}_{1,p}\left(y\right)·{\widehat{g}}_{2,p}\left(y\right)\hfill \\ & & e^{-ity}{e}^{-a^2{\parallel y\parallel }^2}dy\varphi \left(t\right)dt,\varphi \in \mathcal{S},\hfill \end{array}$$

(14)

where $f\ast g$ denotes the classical convolution of functions f and g, symbol ^{^} denotes the classical Fourier transform, and where

$$T_f=\sum _{\left|p\right|\le 2n}{\partial }^p{g}_{1,p}\left(x\right),over\mathcal{S},$$

and

$$T_g=\sum _{\left|q\right|\le 2n}{\partial }^q{g}_{2,q}\left(x\right),over\mathcal{S},$$

with $g_{1,p}$ and $g_{2,p}$ given by the above Lemma 1.

Remark 1.

Observe that from ([15], Corollary 2.4) and for $\varphi \in \mathcal{S}$, such that $\varphi \left(t\right)={\varphi }_1\left(t_1\right)\cdots {\varphi }_n\left(t_n\right)$, $t=(t_1,t_2,\dots ,t_n)\in {\mathbb{R}}^n$, where ${\varphi }_1,{\varphi }_2,\dots ,{\varphi }_n\in \mathcal{S}\left(\mathbb{R}\right)$, the representations (13) and (14) become, respectively,

$$\begin{array}{ccc}\hfill 〈T_1\ast T_2,\varphi 〉& =& {\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\underset{Y\to +\infty }{lim}\sum _{\begin{array}{c}\left|p\right|\le m_1+2n\\ \left|q\right|\le m_2+2n\end{array}}{\int }_{{\mathbb{R}}^n}{\int }_{C[0;Y]}{(-iy)}^q{\widehat{g}}_{1,p}\left(y\right)·{\widehat{g}}_{2,p}\left(y\right)\hfill \\ & & e^{-ity}dy\varphi \left(t\right)dt,\varphi \in \mathcal{S},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill 〈T_{f\ast g},\varphi 〉& =& {\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\underset{Y\to +\infty }{lim}\sum _{\begin{array}{c}\left|p\right|\le 2n\\ \left|q\right|\le 2n\end{array}}{\int }_{{\mathbb{R}}^n}{\int }_{C[0;Y]}{(-iy)}^q{\widehat{g}}_{1,p}\left(y\right)·{\widehat{g}}_{2,p}\left(y\right)\hfill \\ & & e^{-ity}dy\varphi \left(t\right)dt,\hfill \end{array}$$

where $C[0;Y]$ is the n-cube $[-Y,Y]\times \cdots \times [-Y,Y]\subseteq {\mathbb{R}}^n$, $Y>0.$

Remark 2.

According to [17,18], we consider linear partial differential equations with constant coefficients as

$$P(\partial )u=\nu ,$$

(15)

where, as it is usual, P is a complex-valued polynomial in ${\mathbb{R}}^n$ and $P(\partial )$ denotes the corresponding polynomial differential operator given by

$$P(\partial )=\sum _{\left|\alpha \right|\le \sigma }{a}_{\alpha }{\partial }^{\alpha },\alpha \in {\mathbb{N}}^n,a_{\alpha }\in \mathbb{C},\sigma \in \mathbb{N},$$

and ν is a member of ${\mathcal{S}}_k^{\prime }$, $k\in \mathbb{Z},k<0$.

Observe that, since

$$P(\partial )u=\left(P\right(\partial \left)\delta \right)\ast u,$$

where δ denotes the Dirac delta distribution, then Equation (15) can be written as a convolution equation.

Since

$$\left(\mathcal{F}\left[\left(P(\partial )\delta \right)\right]\right)\left(y\right)=P(-iy),y\in {\mathbb{R}}^n,$$

and using ([15], Corollary 2.5), then if P has no zeros of type $\alpha i$, where $\alpha \in {\mathbb{R}}^n$, then there exists a unique solution $u\in {\mathcal{S}}_k^{\prime }$ of the Equation (15).

Also, one obtains the next representation over $\mathcal{S}$:

$$\langle u,\varphi \rangle ={\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\underset{a\to 0^{+}}{lim}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\left(\mathcal{F}\nu \right)\left(y\right)}{P(-iy)}}{e}^{-ity}{e}^{-a^2{\parallel y\parallel }^2}dy\varphi \left(t\right)dt,\mathit{for}\mathit{all}\varphi \in \mathcal{S}.$$

Now, using Theorem 1 above, and $k\in \mathbb{Z},k<-n$, then we obtain the next representation of u:

$$u={\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\underset{a\to 0^{+}}{lim}\sum _{\left|p\right|\le m+2n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{(-iy)}^p{\widehat{g}}_p\left(y\right)}{P(-iy)}}{e}^{-ity}{e}^{-a^2{\parallel y\parallel }^2}dy\varphi \left(t\right)dt,over\mathcal{S},$$

where m denotes the order of ν, and the symbol ^{^} denotes the classical Fourier transform, i.e.,

$${\widehat{g}}_p\left(y\right)=\int g_p\left(x\right)e^{ixy}dx,y\in {\mathbb{R}}^n,$$

and

$$\nu =\sum _{\left|p\right|\le m}{\partial }^p{g}_p,\mathrm{over}\mathcal{S},$$

where the $g_p$ are given by the above Lemma 1.

Using again ([15], Corollary 2.5), the considerations in Remark 2 can be extended to convolution equations on ${\mathcal{S}}_k^{\prime }$, $k\in \mathbb{Z}$, $k<-n$. Indeed, let $h,\nu \in {\mathcal{S}}_k^{\prime }$, $k\in \mathbb{Z}$, $k<-n$, and assume that $\mathcal{F}h$ has no zeros in ${\mathbb{R}}^n$. Suppose that $\mathcal{F}h\in C^{-2k+2n}\left({\mathbb{R}}^n\right)$ and there exists a polynomial P, such that

$$\left|{\partial }^p\left({\displaystyle \frac{1}{\left(\mathcal{F}h\right)\left(y\right)}}\right)\right|\le P\left(\right|y\left|\right),y\in {\mathbb{R}}^n,p\in {\mathbb{N}}^n,\left|p\right|\le -2k+2n.$$

Then, the convolution equation

$$h\ast u=\nu$$

has a unique solution $u\in {\mathcal{S}}_k^{\prime }$, and this solution has the next representation over $\mathcal{S}$:

$$u={\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\underset{a\to 0^{+}}{lim}\sum _{\left|p\right|\le m+2n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{(-iy)}^p\widehat{g_p}\left(y\right)}{\left(\mathcal{F}h\right)\left(y\right)}}{e}^{-ity}{e}^{-a^2{\parallel y\parallel }^2}dy\varphi \left(t\right)dt,$$

where m denotes the order of $\nu$, the symbol ^{^} denotes the classical Fourier transform, and

$$\nu =\sum _{\left|p\right|\le m+2n}{\partial }^p{g}_p\mathrm{over}\mathcal{S},$$

where the $g_p$ are given by the above Lemma 1.

5. Conclusions

In ([19], p. 37), Friedman asserts the following:

“One of the most interesting and important problems in the theory of generalized functions is the problem of finding the structure of generalized functions by expressing them in terms of differential operators acting on functions or on measures.”

The results showcased in our study invite further exploration into similar properties across diverse integral transforms within the realms of spaces of distributional or generalized functions, thereby providing possibilites for exploration within this field.