Abelian Theorems for the Real Weierstrass Transform over Compactly Supported Distributions

Abstract

This paper explores Abelian theorems associated with the real Weierstrass transform over distributions of compact support. This study contributes to both mathematical analysis and distribution theory by offering new insights into the interaction between integral transforms and compactly supported distributions.

1. Introduction and Preliminaries

The Weierstrass transform, often referred to as the Gauss–Weierstrass transform, is a classical integral transform that plays an essential role in smoothing operations and the study of differential equations [1]. This transform, closely related to the heat equation, provides a fundamental tool in harmonic analysis and potential theory due to its regularizing properties [2,3].

The Weierstrass transform ([4], Section 7.1, (1)) of a suitable function f defined on $\mathbb{R}$ is given by

$$\left(\mathcal{W}f\right)\left(y\right)={\displaystyle \frac{1}{\sqrt{4\pi }}}{\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}dx,y\in \mathbb{C}.$$

(1)

Like the real Laplace transform ([4], Section 8.5), the real Weierstrass transform of a suitable function $f\in L^1\left(\mathbb{R}\right)$ is defined by

$$\left(\mathcal{W}f\right)\left(y\right)={\displaystyle \frac{1}{\sqrt{4\pi }}}{\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}dx,y\in \mathbb{R}.$$

(2)

Observe that locally integrable functions with potential or exponential behaviors in $\pm \infty$ make the integral in (2) convergent. This operation convolves the function f with a Gaussian kernel, leading to the smoothing of f and its approximation by infinitely differentiable functions. The Gaussian kernel in the integrand is a solution to the heat equation, which links the Weierstrass transform to the diffusion of heat in physical systems [5].

The study of distributional transforms has been the subject of considerable research, particularly in relation to Abelian theorems in various mathematical frameworks. Zemanian [6] initiated this inquiry by focusing on Abelian theorems for distributional transforms. González and Negrín [7] contributed further by examining distributional versions of the Kontorovich–Lebedev and Mehler–Fock transforms. Maan and Prasad [8] extended the scope to include the index Whittaker transform, and Maan and Negrín [9] broadened the study to encompass Abelian theorems for Laplace, Mellin, and Stieltjes transforms over distributions of compact support, as well as certain spaces of generalized functions. Building on these foundations, Maan et al. [10] investigated Abelian theorems for the ₂$F_1$-transform, particularly focusing on distributions of compact support and generalized functions. These works have significantly advanced the understanding of how Abelian theorems apply to various integral transforms within the framework of distributions, enriching the mathematical analysis of such transforms.

Through these investigations, we acquire significant insights into the behavior of these transforms with respect to their domains of variables, particularly their properties near the origin and at infinity. This research study provides a framework for analyzing the behavior of transforms based on the characteristics of the underlying distribution or generalized function, including aspects such as continuity, differentiability, and compact support. By examining how these transforms interact with various classes of functions, we can enhance our understanding of their mathematical properties and potential applications.

The present study extends the framework of Abelian theorems to the real Weierstrass transform applied to compactly supported distributions, a context not covered in [6]. This approach provides new insights into the regularizing effects of the Weierstrass transform on such distributions, thereby generalizing classical results. Additionally, it establishes a foundation for applying these findings to other integral transforms, broadening the scope of existing research. Furthermore, the real Weierstrass transform exhibits different integral properties and regularizing behavior compared with the distributional Hankel and K-transforms discussed in [6].

Let $\mathcal{E}\left(\mathbb{R}\right)$ be the vector space of all infinitely differentiable complex-valued functions $\psi$ defined on $\mathbb{R}$. This space is equipped with a locally convex topology, which is derived from a collection of seminorms that capture the behavior of functions and their derivatives at various points as

$${\rho }_{k,K}\left(\psi \right)=\underset{x\in K}{max}\left|D_x^k\psi \left(x\right)\right|,\mathrm{for} \mathrm{all}k\in {\mathbb{N}}_0,$$

and compact subsets K of $\mathbb{R}$, and $D_x^k$ is the k-th derivative with respect to variable x. $\mathcal{E}\left(\mathbb{R}\right)$ becomes a Fréchet space. As usual, we denote by ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ the dual space of $\mathcal{E}\left(\mathbb{R}\right)$. This ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ agrees with the space of distributions on $\mathbb{R}$ of compact support in $\mathbb{R}$.

This paper is organized into four sections. Section 1 provides an introduction to the real Weierstrass transform and its background, along with a review of existing Abelian theorems for different transforms. It also includes definitions for the space $\mathcal{E}\left(\mathbb{R}\right)$. Section 2 analyzes the distributional real Weierstrass transform and proves Abelian theorems in the setting of distributions of compact support. Section 3 focuses on these theorems about regular distributions with compact support. Section 4 offers concluding remarks and suggests directions for future research, particularly in applying Abelian theorems to other integral transforms.

2. Abelian Theorems for the Real Weierstrass Transform over ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$

The real Weierstrass transform of a distribution f of compact support on $\mathbb{R}$ is defined by the kernel method as

$$\left(\mathcal{W}f\right)\left(y\right)=〈f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}〉,$$

(3)

where $y\in \mathbb{R}$.

Observe that from ([4], p. 209, II) and ([4], Theorem 7.3.2), the transform in (3) is a smooth function on $\mathbb{R}$, and

$$D_y^m\left(\left(\mathcal{W}f\right)\left(y\right)\right)=〈f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}{D}_y^m\left(exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right)〉,m\in \mathbb{N},y\in \mathbb{R}.$$

(4)

In this section, we establish Abelian theorems for (3). To do so, we demonstrate some previous outcomes.

Lemma 1.

Let ${\mathcal{A}}_x$, $x\in \mathbb{R}$, be the differential operator given by

$${\mathcal{A}}_x=D_x+{\displaystyle \frac{1}{2}}x.$$

(5)

Then, for all $k\in \mathbb{N}$, there exist polynomials $P_{j,k}\left(x\right)$ such that

$${\mathcal{A}}_x^k=\sum _{j=0}^k{P}_{j,k}\left(x\right)D_x^j,$$

(6)

where $P_{k,k}\left(x\right)=1$.

Proof. 

By induction, the result of the above lemma follows. Specifically, we check the base case for $k=1$, assuming that the result holds for all values up to $k-1$; then, we prove it for k. In fact, for $k\ge 2$,

$$\begin{array}{c}{\mathcal{A}}_x^k={\mathcal{A}}_x{\mathcal{A}}_x^{k-1}=\left(D_x+{\displaystyle \frac{1}{2}}x\right)\left(\sum _{j=0}^{k-1}{P}_{j,k-1}\left(x\right)D_x^j\right)\hfill \\ =\sum _{j=0}^{k-1}\left[\left(D_x{P}_{j,k-1}\left(x\right)\right)D_x^j+P_{j,k-1}\left(x\right)D_x^{j+1}+{\displaystyle \frac{1}{2}}x{P}_{j,k-1}\left(x\right)D_x^j\right],\hfill \end{array}$$

which agrees with (6), where $P_{k,k}\left(x\right)=P_{k-1,k-1}\left(x\right)=1$, $P_{j,k}\left(x\right)=D_x{P}_{j,k-1}\left(x\right)+P_{j-1,k-1}\left(x\right)+{\displaystyle \frac{1}{2}}x{P}_{j,k-1}\left(x\right)$, $1\le j\le k-1$, and $P_{0,k}\left(x\right)=D_x{P}_{0,k-1}\left(x\right)+{\displaystyle \frac{1}{2}}x{P}_{0,k-1}\left(x\right)$. □

Lemma 2.

For each compact set $K\subset \mathbb{R}$ and each $k\in {\mathbb{N}}_0$, let ${\Gamma }_{k,K}$ be the seminorm on $\mathcal{E}\left(\mathbb{R}\right)$ given by

$${\Gamma }_{k,K}\left(\psi \right)=\underset{x\in K}{sup}\left|{\mathcal{A}}_x^k\psi \left(x\right)\right|,\psi \in \mathcal{E}\left(\mathbb{R}\right),$$

where ${\mathcal{A}}_x$ is the differential operator given by (5) and ${\mathcal{A}}_x^k$ is the k-th iteration. Then, ${\Gamma }_{k,K}$ generates a topology on $\mathcal{E}\left(\mathbb{R}\right)$, which agrees with the usual topology of this space.

Proof. 

The expression for ${\mathcal{A}}_x^k$ given in (6) yields that any sequence $\left\{{\psi }_n\right\}\subset \mathcal{E}\left(\mathbb{R}\right)$ which tends to zero as $n\to \infty$ for the usual topology on $\mathcal{E}\left(\mathbb{R}\right)$ also tends to zero as $n\to \infty$ for the topology generated by the family of seminorms $\left\{{\Gamma }_{k,K}\right\}$. Conversely, let $\left\{{\psi }_n\right\}$ be a sequence on $\mathcal{E}\left(\mathbb{R}\right)$ which tends to zero as $n\to \infty$ with respect to the topology generated by $\left\{{\Gamma }_{k,K}\right\}$. It is clear that $\left\{{\psi }_n\right\}$, $\left\{{\mathcal{A}}_x{\psi }_n\right\}$ tends to zero as $n\to \infty$, uniformly in each compact subset $K\subset \mathbb{R}$. Moreover, from (5),

$${\mathcal{A}}_x{\psi }_n\left(x\right)-{\displaystyle \frac{1}{2}}x{\psi }_n\left(x\right)=D_x{\psi }_n\left(x\right).$$

The left-hand side of the above equation tends to zero as $n\to \infty$, uniformly in each compact subset $K\subset \mathbb{R}$, which means that $D_x{\psi }_n\left(x\right)$ tends to zero as $n\to \infty$, uniformly in each compact subset $K\subset \mathbb{R}$.

Now assume by induction that, for $0\le m\le k-1$, $D_x^m{\psi }_n\left(x\right)$ tend to zero as $n\to \infty$, uniformly on each compact subset K of $\mathbb{R}$.

Then,

$${\mathcal{A}}_x^k{\psi }_n\left(x\right)-\sum _{j=0}^{k-1}{P}_{j,k}\left(x\right)D_x^j{\psi }_n\left(x\right)=D_x^k{\psi }_n\left(x\right),$$

which, arguing as for the case $k=1$, indicates that $D_x^k{\psi }_n\left(x\right)$ tends to zero as $n\to \infty$, uniformly on each compact subset $K\subset \mathbb{R}$.

Finally, taking into account that the topologies on $\mathcal{E}\left(\mathbb{R}\right)$ for both families of seminorms, the usual and ${\Gamma }_{k,K}$ are metrizable, and the conclusion follows. □

We use Lemma 2 to obtain the following result.

Lemma 3.

Let f be in ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ and $\mathcal{W}$ be defined by (3). Then, there exist $M>0$ and a non-negative integer q, all depending on f, such that

$$|\left(\mathcal{W}f\right)\left(y\right)|\le M{e}^{-{\displaystyle \frac{y^2}{4}}}\underset{0\le k\le q}{max}\left\{{\left({\displaystyle \frac{\left|y\right|}{2}}\right)}^k\right\},for ally\in \mathbb{R}.$$

(7)

Proof. 

We know that ${\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}$ is an eigenfunction of ${\mathcal{A}}_x$, as

$${\mathcal{A}}_x\left({\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right)={\displaystyle \frac{y}{2}}{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\},$$

(8)

and that thus for $k\in \mathbb{N}$, one has

$${\mathcal{A}}_x^k\left({\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right)={\left({\displaystyle \frac{y}{2}}\right)}^k{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\},$$

From Lemma 2 above, we may consider the space $\mathcal{E}\left(\mathbb{R}\right)$ equipped with the topology arising from the family of seminorms ${\Gamma }_{k,K}$. From ([11], Proposition 2, p.97), there exist $C>0$, a compact set $K\subset \mathbb{R}$, and a non-negative integer q, all depending on f, such that

$$|〈f,\psi 〉|\le C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\mathcal{A}}_x^k\psi \left(x\right)\right|,\mathrm{for} \mathrm{all}\psi \in \mathcal{E}\left(\mathbb{R}\right).$$

(9)

Now, by means of (8) and (9), one has

$$\begin{array}{cc}\hfill & |\left(\mathcal{W}f\right)\left(y\right)|\hfill \\ \hfill & =\left|〈f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}〉\right|\hfill \\ \hfill & \le C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\displaystyle \frac{1}{\sqrt{4\pi }}}{\mathcal{A}}_x^k\left(exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right)\right|\hfill \\ \hfill & =C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\left({\displaystyle \frac{y}{2}}\right)}^k{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right|\hfill \\ \hfill & \le M{e}^{-{\displaystyle \frac{y^2}{4}}}\underset{0\le k\le q}{max}\left\{{\left({\displaystyle \frac{\left|y\right|}{2}}\right)}^k\right\},\mathrm{for} \mathrm{all}y\in \mathbb{R}.\hfill \end{array}$$

for certain $M>0$, since x ranges on the compact set $K\subset \mathbb{R}$. □

Theorem 1

(Abelian theorem). Let f be a member of ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, and let $\mathcal{W}$ be given by (3). Then,

(i)

for any $\eta >0$, one has

$$\underset{y\to 0}{lim}\left(y^{\eta }\left(\mathcal{W}f\right)\left(y\right)\right)=0,$$

and

(ii)

$$\underset{y\to \pm \infty }{lim}\left(\left(\mathcal{W}f\right)\left(y\right)\right)=0.$$

Proof. 

From Lemma 3, one has

$$|\left(\mathcal{W}f\right)\left(y\right)|\le M{e}^{-{\displaystyle \frac{y^2}{4}}}\underset{0\le k\le q}{max}\left\{{\left({\displaystyle \frac{\left|y\right|}{2}}\right)}^k\right\},\mathrm{for} \mathrm{all}y\in \mathbb{R},$$

for some $M>0$ and some non-negative integer q. Now, let y tend to zero or y tend to $\pm \infty$; then, one obtains $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$, respectively. □

If one denotes ${\mathcal{A}}_x^{\prime }=-D_x+{\displaystyle \frac{1}{2}}x$, where $D_x$ is the distributional derivative, then for $f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, one obtains ${\mathcal{A}}_x^{\prime }f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ and

$$\begin{array}{cc}\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime }f\right)\right)\left(y\right)& =〈{\mathcal{A}}_x^{\prime }f,{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}〉\hfill \\ & =〈f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}{\mathcal{A}}_x\left(exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right)〉\hfill \\ & ={\displaystyle \frac{y}{2}}〈f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}〉\hfill \\ & ={\displaystyle \frac{y}{2}}\left(\mathcal{W}f\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

Thus, for $k\in \mathbb{N}$, one has ${\mathcal{A}}_x^{\prime k}f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ and

$$\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}f\right)\right)\left(y\right)={\left({\displaystyle \frac{y}{2}}\right)}^k\left(\mathcal{W}f\right)\left(y\right),y\in \mathbb{R}.$$

(10)

Therefore, from Theorem 1 and relation (10), one has

Corollary 1.

Let f be a member of ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, and let $\mathcal{W}$ be given by (3). Set ${\mathcal{A}}_x^{\prime }=-D_x+{\displaystyle \frac{1}{2}}x$, where $D_x$ denotes the distributional derivative. Then, for $k\in \mathbb{N}$,

(i)

for any $\eta >0$, one has

$$\underset{y\to 0}{lim}\left(y^{\eta }\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}f\right)\right)\left(y\right)\right)=0,$$

and

(ii)

$$\underset{y\to \pm \infty }{lim}\left(\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}f\right)\right)\left(y\right)\right)=0.$$

From relation (4), one obtains

$${\mathcal{A}}_y\left(\left(\mathcal{W}f\right)\left(y\right)\right)=〈f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}{\mathcal{A}}_y\left(exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right)〉,y\in \mathbb{R},$$

which, using (8), is equal to

$$\begin{array}{c}〈f\left(x\right),{\displaystyle \frac{x}{2}}{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}〉\hfill \\ =〈{\displaystyle \frac{x}{2}}f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}〉\hfill \\ =\left(\mathcal{W}\left({\displaystyle \frac{x}{2}}f\left(x\right)\right)\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

Since ${\displaystyle \frac{x}{2}}f\left(x\right)\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ and using again (4) and (8), one has

$$\begin{array}{c}{\mathcal{A}}_y^2\left(\left(\mathcal{W}f\right)\left(y\right)\right)\hfill \\ =〈{\displaystyle \frac{x}{2}}f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}{\mathcal{A}}_y\left(exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right)〉\hfill \\ =〈{\left({\displaystyle \frac{x}{2}}\right)}^{2}f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}〉\hfill \\ =\left(\mathcal{W}\left({\left({\displaystyle \frac{x}{2}}\right)}^{2}f\left(x\right)\right)\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

And so, one arrives to

$${\mathcal{A}}_y^k\left(\left(\mathcal{W}f\right)\left(y\right)\right)=\left(\mathcal{W}\left({\left({\displaystyle \frac{x}{2}}\right)}^{k}f\left(x\right)\right)\right)\left(y\right),k\in \mathbb{N},y\in \mathbb{R}.$$

(11)

Therefore, from Theorem 1 and relation (11), one has the following.

Corollary 2.

Let f be a member of ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, and let $\mathcal{W}$ be given by (3). Then, for $k\in \mathbb{N}$,

(i)

for any $\eta >0$, one has

$$\underset{y\to 0}{lim}\left(y^{\eta }{\mathcal{A}}_y^k\left(\left(\mathcal{W}f\right)\left(y\right)\right)\right)=0,$$

and

(ii)

$$\underset{y\to \pm \infty }{lim}\left({\mathcal{A}}_y^k\left(\left(\mathcal{W}f\right)\left(y\right)\right)\right)=0.$$

3. Regular Distributions of Compact Support Versus Abelian Theorems

If f is a locally integrable function on $\mathbb{R}$ and f has compact support on $\mathbb{R}$, then f gives rise to a regular member $T_f$ of ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ by means of

$$〈T_f,\psi 〉={\int }_{-\infty }^{\infty }f\left(x\right)\psi \left(x\right)dx,\mathrm{for} \mathrm{all}\psi \in \mathcal{E}\left(\mathbb{R}\right).$$

In fact, taking into account that

$$\begin{array}{cc}|〈T_f,\psi 〉|& =\left|{\int }_{-\infty }^{\infty }f\left(x\right)\psi \left(x\right)dx\right|\le \underset{x\in supp\left(f\right)}{sup}\left|\psi \left(x\right)\right|{\int }_{supp\left(f\right)}\left|f\left(x\right)\right|dx\hfill \\ & ={\Gamma }_{0,supp\left(f\right)}\left(\psi \right){\int }_{supp\left(f\right)}\left|f\left(x\right)\right|dx,\hfill \end{array}$$

where supp(f) represents the support of the function f, we have

$$\begin{array}{cc}\left(\mathcal{W}{T}_f\right)\left(y\right)& =〈T_f\left(x\right),{\displaystyle \frac{1}{\sqrt{4\pi }}}exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}〉\hfill \\ & ={\displaystyle \frac{1}{\sqrt{4\pi }}}{\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}dx=\left(\mathcal{W}f\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

Thus, the real Weierstrass transform in (3) of the member $T_f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ agrees with the classical real Weierstrass transform in (2) of the function f.

Also, from (4) and (12), the real Weierstrass transform in (2) is a smooth function on $\mathbb{R}$, and

$$D_y^m\left(\left(\mathcal{W}f\right)\left(y\right)\right)={\displaystyle \frac{1}{\sqrt{4\pi }}}{\int }_{-\infty }^{\infty }f\left(x\right)D_y^m\left(exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right)dx,m\in \mathbb{N},y\in \mathbb{R}.$$

From Section 2, one obtains the next results.

Lemma 4.

Let f be a locally integrable function on $\mathbb{R}$ such that f has compact support. Then, for the real Weierstrass transform $\mathcal{W}$ given by (2), there exist $M>0$ and a non-negative integer q, all depending on f, such that

$$|\left(\mathcal{W}f\right)\left(y\right)|\le M{e}^{-{\displaystyle \frac{y^2}{4}}}\underset{0\le k\le q}{max}\left\{{\left({\displaystyle \frac{\left|y\right|}{2}}\right)}^k\right\},for ally\in \mathbb{R}.$$

Proof. 

Here, we make use of Lemma 3 and the fact that

$$\left(\mathcal{W}f\right)\left(y\right)=\left(\mathcal{W}{T}_f\right)\left(y\right)\mathrm{for} \mathrm{all}y\in \mathbb{R}.$$

□

Theorem 2

(Abelian theorem). Let f be a locally integrable function on $\mathbb{R}$ such that f has compact support. Then, for the real Weierstrass transform $\mathcal{W}$ given by (2), one has that

(i)

for any $\eta >0$,

$$\underset{y\to 0}{lim}\left(y^{\eta }\left(\mathcal{W}f\right)\left(y\right)\right)=0,$$

and

(ii)

$$\underset{y\to \pm \infty }{lim}\left(\left(\mathcal{W}f\right)\left(y\right)\right)=0.$$

Proof. 

Here, we make use of Theorem 1 and the fact that

$$\left(\mathcal{W}f\right)\left(y\right)=\left(\mathcal{W}{T}_f\right)\left(y\right)\mathrm{for} \mathrm{all}y\in \mathbb{R}.$$

□

We denote $C_c^k\left(\mathbb{R}\right)$, $k\in \mathbb{N}$, the space of compactly supported functions on $\mathbb{R}$ which are k-times differentiable functions with continuity.

Observe that for $f\in C_c^k\left(\mathbb{R}\right)$, $k\in \mathbb{N}$, ${\mathcal{A}}_x\equiv D_x+{\displaystyle \frac{1}{2}}x$, and ${\mathcal{A}}_x^{\prime }\equiv -D_x+{\displaystyle \frac{1}{2}}x$, where $D_x$ is the ordinary derivative, and by making use of integration by parts, one has

$$\begin{array}{cc}\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}f\right)\right)\left(y\right)& ={\displaystyle \frac{1}{\sqrt{4\pi }}}{\int }_{-\infty }^{\infty }\left({\mathcal{A}}_x^{\prime k}f\right)\left(x\right)exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}dx\hfill \\ & ={\displaystyle \frac{1}{\sqrt{4\pi }}}{\int }_{-\infty }^{\infty }f\left(x\right){\mathcal{A}}_x^k\left(exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4}}\right\}\right)dx\hfill \\ & ={\left({\displaystyle \frac{y}{2}}\right)}^k\left(\mathcal{W}f\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

On the other hand, from (10) and for $k\in \mathbb{N}$,

$$\begin{array}{cc}\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}{T}_f\right)\right)\left(y\right)& ={\left({\displaystyle \frac{y}{2}}\right)}^k\left(\mathcal{W}{T}_f\right)\left(y\right)\hfill \\ & ={\left({\displaystyle \frac{y}{2}}\right)}^k\left(\mathcal{W}f\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

Thus, for $f\in C_c^k\left(\mathbb{R}\right)$, $k\in \mathbb{N}$, one has

$$\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}f\right)\right)\left(y\right)=\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}{T}_f\right)\right)\left(y\right),y\in \mathbb{R}.$$

Corollary 3.

Let $f\in C_c^k\left(\mathbb{R}\right)$, $k\in \mathbb{N}$, and ${\mathcal{A}}_x^{\prime }=-D_x+{\displaystyle \frac{1}{2}}x$, where $D_x$ denotes an ordinary derivative. Then, for the real Weierstrass transform $\mathcal{W}$ given by (2), one has that

(i)

for any $\eta >0$,

$$\underset{y\to 0}{lim}\left(y^{\eta }\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}f\right)\right)\left(y\right)\right)=0,k\in \mathbb{N},$$

and

(ii)

$$\underset{y\to \pm \infty }{lim}\left(\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}f\right)\right)\left(y\right)\right)=0,k\in \mathbb{N}.$$

Proof. 

Here, we make use of Corollary 1 and the fact that

$$\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}f\right)\right)\left(y\right)=\left(\mathcal{W}\left({\mathcal{A}}_x^{\prime k}{T}_f\right)\right)\left(y\right),$$

for $k\in \mathbb{N}$ and $y\in \mathbb{R}$. □

Also, one has the following.

Corollary 4.

Let f be a locally integrable function on $\mathbb{R}$ such that f has compact support. Then, for the real Weierstrass transform $\mathcal{W}$ given by (2), one has that

(i)

for any $\eta >0$,

$$\underset{y\to 0}{lim}\left(y^{\eta }{\mathcal{A}}_y^k\left(\left(\mathcal{W}f\right)\left(y\right)\right)\right)=0,k\in \mathbb{N},$$

and

(ii)

$$\underset{y\to \pm \infty }{lim}\left({\mathcal{A}}_y^k\left(\left(\mathcal{W}f\right)\left(y\right)\right)\right)=0,k\in \mathbb{N}.$$

Proof. 

Here, we make use of Corollary 2 and the fact that

$$\left(\mathcal{W}f\right)\left(y\right)=\left(\mathcal{W}{T}_f\right)\left(y\right),y\in \mathbb{R}.$$

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4. Conclusions

In conclusion, this research study established Abelian theorems for the real Weierstrass transform (2) acting on distributions with compact support. Our analysis focused on the connection between the real Weierstrass transform and the original distributions, specifically examining how the transform behaves within the compact support setting. These results generalize classical Abelian theorems to the framework of the real Weierstrass transform, emphasizing its regularizing effect on distributions. This work deepens our understanding of the real Weierstrass transform in the context of distribution theory and functional analysis. These findings provide a solid foundation for advancing the mathematical analysis of integral transform behaviours and their applications.