Analysis of Operators with Complex Gaussian Kernels over ${\mathcal{ℰ}}^{\prime }$(ℝ): Abelian Theorems

Abstract

This paper investigates Abelian theorems for operators with complex Gaussian kernels over distributions of compact support. Furthermore, our investigation encompasses an examination of the Gauss–Weierstrass semigroup, the linear canonical transform, and the Ornstein–Uhlenbeck semigroup as particular cases within the scope of our study.

1. Introduction and Preliminaries

This paper undertakes an examination of the integral operator characterized by a complex Gaussian kernel operating on a suitable complex-valued function f defined on $\mathbb{R}$ by

$$\begin{array}{c}\hfill \left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)={\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}dx,\end{array}$$

(1)

where $y\in \mathbb{R},\beta ,ϵ,\delta ,\zeta ,\gamma \in \mathbb{C}$.

The origins of this line of research can be traced back to Quantum Field Theory (refer to [1]). Operators defined by complex Gaussian kernels such as (1) play a central role, primarily due to their connection with the extended oscillator semigroup introduced by Howe [2], with further insights provided by Folland (see Chapter 5, [3]). Moreover, Lieb [4] extended the operator (1) to higher dimensions, conducting an in-depth analysis of its behavior in various Lebesgue spaces. Investigations by Negrín [5,6] explored both degenerate and non-degenerate cases of (1), where the degenerate scenario corresponds to ${(\Re \delta )}^2=(\Re \beta )(\Re ϵ)$, and the non-degenerate case satisfies ${(\Re \delta )}^2<(\Re \beta )(\Re ϵ)$. Further developments and relevant contributions are available in [7,8,9,10,11].

A significant area of study within distribution theory involves Abelian theorems for various integral transforms. This was initially explored by Zemanian [12], who focused on such theorems in the context of distributional transforms. Later, Hayek et al. [13,14] extended these ideas to the ${}_2{}^{\mathrm{}}F_1^{\mathrm{}}$-transform. Building upon this, González and Negrín [15] examined Abelian theorems for the Kontorovich–Lebedev and Mehler–Fock transforms in the distributional framework. The scope was further broadened by Maan and Prasad [16] through their study of the index Whittaker transform. More recently, Maan and Negrín [17] extended these investigations to the Laplace, Mellin, and Stieltjes transforms, all considered over compactly supported distributions and specific spaces of generalized functions. These foundational efforts culminated in the work of Maan et al. [18], who revisited Abelian theorems for the ${}_2{}^{\mathrm{}}F_1^{\mathrm{}}$-transform within the same distributional setting.

These contributions collectively offer valuable understanding regarding the asymptotic properties of integral transforms, particularly their behavior near zero and at infinity. They establish a framework to analyze how transforms behave based on the properties of the input distribution or generalized function.

Inspired by the above developments, this article is devoted to studying Abelian theorems for operators associated with complex Gaussian kernels, specifically over the space of compactly supported distributions.

Let $\mathcal{E}\left(\mathbb{R}\right)$ denote the space of all complex-valued functions on $\mathbb{R}$ that are infinitely differentiable. This space becomes a Fréchet space when endowed with the locally convex topology defined by the family of seminorms

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

where K ranges over all compact subsets of $\mathbb{R}$, and $D_x^k$ denotes the k-th derivative with respect to x. The dual space of $\mathcal{E}\left(\mathbb{R}\right)$ is denoted by ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, which coincides with the space of distributions on $\mathbb{R}$ that are compactly supported.

2. Abelian Theorems for the Operators with Complex Gaussian Kernels over ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$

Operators associated with complex Gaussian-type kernels, acting on a distribution f with compact support on $\mathbb{R}$, can be formulated via a kernel-based approach as follows:

$$\begin{array}{c}\hfill \left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)=⟨f\left(x\right),exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}⟩,\end{array}$$

(2)

where $y\in \mathbb{R}$ and the parameters $\beta ,ϵ,\delta ,\zeta ,\gamma$ are complex numbers.

In the upcoming discussion, we aim to derive Abelian theorems for the operator given in (2). To facilitate this, we begin by reviewing some essential preliminary results.

Lemma 1.

Let ${\mathcal{A}}_x$, with $x\in \mathbb{R}$, denote the differential operator defined by

$$\begin{array}{c}\hfill {\mathcal{A}}_x=D_x+2ϵx.\end{array}$$

(3)

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

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

(4)

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

Proof. 

We prove the result using mathematical induction. For the base case $k=1$, the claim holds trivially. Assume the identity holds for a fixed $k-1\in \mathbb{N}$, i.e., that

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

Then for k, we compute

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

This expression matches the desired form in (4), with the coefficients satisfying

$$P_{k,k}\left(x\right)=P_{k-1,k-1}\left(x\right)=1,P_{j,k}\left(x\right)=D_x{P}_{j,k-1}\left(x\right)+P_{j-1,k-1}\left(x\right)+2ϵx{P}_{j,k-1}\left(x\right),$$

for $1\le j\le k-1$ and

$$P_{0,k}\left(x\right)=D_x{P}_{0,k-1}\left(x\right)+2ϵx{P}_{0,k-1}\left(x\right).$$

□

Lemma 2.

For every compact subset $K\subset \mathbb{R}$ and $k\in {\mathbb{N}}_0$, define a seminorm ${\mathrm{\Gamma }}_{k,K}$ on the space $\mathcal{E}\left(\mathbb{R}\right)$ of smooth functions by

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

where ${\mathcal{A}}_x$ is defined in (3) and ${\mathcal{A}}_x^k$ denotes its k-th power. Then the family $\left\{{\mathrm{\Gamma }}_{k,K}\right\}$ defines a topology on $\mathcal{E}\left(\mathbb{R}\right)$ that coincides with the standard topology of this space.

Proof. 

From the representation of ${\mathcal{A}}_x^k$ in (4), it follows that convergence in the classical topology of $\mathcal{E}\left(\mathbb{R}\right)$ implies convergence in the topology defined by the seminorms ${\mathrm{\Gamma }}_{k,K}$. Indeed, if a sequence $\left\{{\psi }_n\right\}$ converges to zero in the usual topology, then all derivatives of ${\psi }_n$ converge uniformly to zero on compact subsets, which ensures that ${\mathrm{\Gamma }}_{k,K}\left({\psi }_n\right)\to 0$ for each k and K.

Conversely, suppose ${\mathrm{\Gamma }}_{k,K}\left({\psi }_n\right)\to 0$ as $n\to \infty$ for each k and compact K. Then, in particular, ${\psi }_n$ and ${\mathcal{A}}_x{\psi }_n$ tend uniformly to zero on every compact subset of $\mathbb{R}$. Using the identity

$$D_x{\psi }_n\left(x\right)={\mathcal{A}}_x{\psi }_n\left(x\right)-2ϵx{\psi }_n\left(x\right),$$

we conclude that $D_x{\psi }_n\left(x\right)\to 0$ uniformly on compact subsets as $n\to \infty$.

We proceed by induction: assume $D_x^m{\psi }_n\left(x\right)\to 0$ uniformly on compact sets for $m=0,1,\dots ,k-1$. From (4), we isolate

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

and since both terms on the right-hand side tend uniformly to zero, the same is true for $D_x^k{\psi }_n\left(x\right)$.

Finally, noting that both topologies (the classical one and the one generated by ${\mathrm{\Gamma }}_{k,K}$) are metrizable, and convergence with respect to all seminorms agrees in both settings, we conclude that the two topologies coincide. □

We use Lemma 2 to obtain the following result.

Lemma 3.

Assume that ${(\Re \delta )}^2=(\Re \beta )(\Re ϵ)$, where $\Re \beta \ge 0$ and $\Re ϵ\ge 0$. Let f be an element of the space ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, and consider the transform ${\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }$ defined by (2). Then, there exist a constant $M>0$ and a non-negative integer q, both depending on the distribution f, such that the following estimate holds:

$$\left|\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right|\le M{e}^{(\Re \zeta )y}\underset{0\le k\le q}{max}\left\{{\left(2\left|\delta \right|\left|y\right|+\left|\gamma \right|\right)}^k\right\},\mathrm{for}\mathrm{all}y\in \mathbb{R}.$$

(5)

Proof. 

It is well-known that the function

$$exp\left(-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right)$$

serves as an eigenfunction of the operator ${\mathcal{A}}_x$. Indeed, we have

$$\begin{array}{ccc}& & {\mathcal{A}}_x\left(exp\left(-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right)\right)\hfill \\ & & =(2\delta y+\gamma )exp\left(-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right).\hfill \end{array}$$

Consequently, for any $k\in \mathbb{N}$, it follows that

$$\begin{array}{ccc}\hfill & & {\mathcal{A}}_x^k\left(exp\left(-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right)\right)\hfill \\ & & ={(2\delta y+\gamma )}^{k}exp\left(-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right).\hfill \end{array}$$

(6)

Recalling Lemma 2, we consider the space $\mathcal{E}\left(\mathbb{R}\right)$ endowed with the topology induced by the family of seminorms ${\mathrm{\Gamma }}_{k,K}$. According to (Proposition 2, p. 97, [19]), for each f, there exist constants $C>0$, a compact subset $K\subset \mathbb{R}$, and a non-negative integer q such that the following inequality holds for all $\psi \in \mathcal{E}\left(\mathbb{R}\right)$:

$$\left|⟨f,\psi ⟩\right|\le C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\mathcal{A}}_x^k\psi \left(x\right)\right|.$$

(7)

Consider the degenerate case

$${(\Re \delta )}^2=(\Re \beta )(\Re ϵ),$$

being that $\Re \beta \ge 0$, $\Re ϵ\ge 0.$

We have two options:

$$\Re \delta =\left(\sqrt{\Re \beta }\right)\left(\sqrt{\Re ϵ}\right)$$

or

$$\Re \delta =-\left(\sqrt{\Re \beta }\right)\left(\sqrt{\Re ϵ}\right).$$

Now, observe that

$$\begin{array}{ccc}\hfill & & exp\left\{-(\Re \beta )y^2-(\Re ϵ)x^2+2(\Re \delta )xy+(\Re \zeta )y+(\Re \gamma )x\right\}\hfill \\ & & =exp\left\{-\left(\left(\Re \beta \right)y^2+\left(\Re ϵ\right)x^2-2\left(\Re \delta \right)xy\right)+\left(\Re \zeta \right)y+\left(\Re \gamma \right)x\right\},\hfill \end{array}$$

(8)

and so when $\Re \delta =\left(\sqrt{\Re \beta }\right)\left(\sqrt{\Re ϵ}\right)$, the expression (8) is equal to

$$\begin{array}{c}\hfill exp\left\{-{\left(\sqrt{\Re \beta }y-\sqrt{\Re ϵ}x\right)}^2+\left(\Re \zeta \right)y+\left(\Re \gamma \right)x\right\},\end{array}$$

(9)

and when $\Re \delta =-\left(\sqrt{\Re \beta }\right)\left(\sqrt{\Re ϵ}\right)$, the expression (8) is equal to

$$\begin{array}{c}\hfill exp\left\{-{\left(\sqrt{\Re \beta }y+\sqrt{\Re ϵ}x\right)}^2+\left(\Re \zeta \right)y+\left(\Re \gamma \right)x\right\}.\end{array}$$

(10)

In both cases, expressions (9) and (10) are bounded by

$$\begin{array}{c}\hfill exp\left\{\left(\Re \zeta \right)y+\left(\Re \gamma \right)x\right\}.\end{array}$$

(11)

Now, by means of (6) and (11), one has

$$\begin{array}{cc}\hfill & |\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)|\hfill \\ \hfill & =\left|⟨f\left(x\right),exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}⟩\right|\hfill \\ \hfill & \le C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\mathcal{A}}_x^k\left(exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}\right)\right|\hfill \\ \hfill & =C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\left(2\delta y+\gamma \right)}^{k}exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}\right|\hfill \\ \hfill & \le C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left\{{\left(2\left|\delta \right|\left|y\right|+\left|\gamma \right|\right)}^k\right.\hfill \\ \hfill & \times \left.exp\left\{-(\Re \beta )y^2-(\Re ϵ)x^2+2(\Re \delta )xy+(\Re \zeta )y+(\Re \gamma )x\right\}\right\}\hfill \\ \hfill & \le M{e}^{(\Re \zeta )y}\underset{0\le k\le q}{max}\left\{{\left(2\left|\delta \right|\left|y\right|+\left|\gamma \right|\right)}^k\right\},\mathrm{for} \mathrm{all}y\in \mathbb{R},\hfill \end{array}$$

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

The minimal integer q satisfying the inequality in (7) is called the order of the distribution f (Théorème XXIV, p. 88, [20]). Next, we proceed to prove Abelian theorems concerning the distributional transform given in (2).

Theorem 1

(Abelian theorem). Set ${(\Re \delta )}^2=(\Re \beta )(\Re ϵ)$, being $\Re \beta \ge 0$ and $\Re ϵ\ge 0.$ Let f be a member of ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ of order $q\in {\mathbb{N}}_0$, and let ${\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }$ be given by (2). Then

(i)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to 0}{lim}\left(y^{\eta }\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)=0;\end{array}$$

(ii)

For $\Re \zeta <0$, one has

$$\begin{array}{c}\hfill \underset{y\to +\infty }{lim}\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)=0;\end{array}$$

(iii)

For $\Re \zeta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to -\infty }{lim}\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)=0;\end{array}$$

(iv)

For $\Re \zeta =0$ and for any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to \pm \infty }{lim}\left(y^{-q-\eta }\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)=0.\end{array}$$

Proof. 

From Lemma 3, one has

$$\begin{array}{c}\hfill |\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)|\le M{e}^{(\Re \zeta )y}\underset{0\le k\le q}{max}\left\{{\left(2\left|\delta \right|\left|y\right|+\left|\gamma \right|\right)}^k\right\},\mathrm{for} \mathrm{all}y\in \mathbb{R},\end{array}$$

for some $M>0$, and therefore the result holds. □

Lemma 4.

Set $\Re \delta =0$. Let f be in ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, and ${\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }$ be defined by (2). Then there exist $M>0$ and a non-negative integer q, all depending on f, such that

$$\begin{array}{c}\hfill |\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)|\le M{e}^{-(\Re \beta )y^2+(\Re \zeta )y}\underset{0\le k\le q}{max}\left\{{\left(2|\Im \delta |\left|y\right|+\left|\gamma \right|\right)}^k\right\},y\in \mathbb{R}.\end{array}$$

(12)

Proof. 

From Lemma 2, we work within the space $\mathcal{E}\left(\mathbb{R}\right)$, endowed with the topology defined by the system of seminorms ${\mathrm{\Gamma }}_{k,K}$. According to (Proposition 2, p.97, [19]), for any given f, there exist a constant $C>0$, a compact subset $K\subset \mathbb{R}$, and a non-negative integer q such that

$$\begin{array}{c}\hfill |⟨f,\psi ⟩|\le C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\mathcal{A}}_x^k\psi \left(x\right)\right|,\mathrm{for}\mathrm{all}\psi \in \mathcal{E}\left(\mathbb{R}\right).\end{array}$$

(13)

Now, by means of (6) and (13), one has

$$\begin{array}{ccc}& & |\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)|\hfill \\ & =& \left|⟨f\left(x\right),exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}⟩\right|\hfill \\ & \le & C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\mathcal{A}}_x^k\left(exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}\right)\right|\hfill \\ & =& C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\left(2\delta y+\gamma \right)}^{k}exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}\right|\hfill \\ & \le & C\underset{0\le k\le q}{max}\underset{x\in K}{max}\left\{{\left(2\left|\delta \right|\left|y\right|+\left|\gamma \right|\right)}^{k}exp\left\{-(\Re \beta )y^2-(\Re ϵ)x^2+(\Re \zeta )y+(\Re \gamma )x\right\}\right\}\hfill \\ & \le & M{e}^{-(\Re \beta )y^2+(\Re \zeta )y}\underset{0\le k\le q}{max}\left\{{\left(2|\Im \delta |\left|y\right|+\left|\gamma \right|\right)}^k\right\},\mathrm{for} \mathrm{all}y\in \mathbb{R},\hfill \end{array}$$

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

Theorem 2

(Abelian theorem). Set $\Re \delta =0$. Let f be a member of ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ of order $q\in {\mathbb{N}}_0$, and let ${\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }$ be given by (2). Then

(i)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to 0}{lim}\left(y^{\eta }\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)=0;\end{array}$$

(ii)

For $\Re \beta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to \pm \infty }{lim}\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)=0;\end{array}$$

(iii)

For $\Re \beta =0,\Re \zeta <0$, one has

$$\begin{array}{c}\hfill \underset{y\to +\infty }{lim}\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)=0;\end{array}$$

(iv)

For $\Re \beta =0,\Re \zeta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to -\infty }{lim}\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)=0;\end{array}$$

(v)

For $\Re \beta =\Re \zeta =0$ and for any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to \pm \infty }{lim}\left(y^{-q-\eta }\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)=0.\end{array}$$

Proof. 

From Lemma 4, one has

$$\begin{array}{c}\hfill |\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)|\le M{e}^{-(\Re \beta )y^2+(\Re \zeta )y}\underset{0\le k\le q}{max}\left\{{\left(2|\Im \delta |\left|y\right|+\left|\gamma \right|\right)}^k\right\},\mathrm{for} \mathrm{all}y\in \mathbb{R},\end{array}$$

for some $M>0$, and therefore the proof is complete. □

Suppose f is a locally integrable function on $\mathbb{R}$ with compact support. Then, one can associate to f a regular distribution $T_f$ in the dual space ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ of order zero, defined via

$$\begin{array}{c}\hfill ⟨T_f,\psi ⟩={\int }_{-\infty }^{\infty }f\left(x\right)\psi \left(x\right)dx,\mathrm{for}\mathrm{all}\psi \in \mathcal{E}\left(\mathbb{R}\right).\end{array}$$

To confirm that $T_f$ is of order $q=0$, observe that

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

where $\mathrm{supp}\left(f\right)$ denotes the support of f. This inequality confirms that the functional $T_f$ depends only on the zeroth-order seminorm of $\psi$, and hence, $T_f$ is a distribution of order zero.

Consequently, the integral transform ${\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f$ defined as

$$\begin{array}{ccc}\hfill \left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)& =& ⟨T_f,exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}⟩\hfill \\ \hfill & =& {\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}dx,y\in \mathbb{R},\hfill \end{array}$$

can be interpreted through the distribution $T_f$ with compact support and order zero.

As a result, the statements of Theorems 1 and 2 remain valid for the function ${\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f$ expressed in (1), interpreted in the distributional framework through $T_f$.

3. Particular Cases

In this section, we apply the general Abelian theorems established in Section 2 to three prominent integral operators involving complex Gaussian kernels. These include the Gauss–Weierstrass semigroup, the linear canonical transform, and the Ornstein–Uhlenbeck semigroup. Each subsection identifies the parameter settings that specialize the general kernel to the classical operator, followed by the corresponding Abelian theorem over the space of compactly supported distributions ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$. This analysis highlights the relevance and wide applicability of the theoretical results developed earlier.

3.1. The Gauss–Weierstrass Semigroup

The Gauss–Weierstrass semigroup acting on a suitable complex-valued function f defined over $\mathbb{R}$ (refer to p. 521 in [21] and [22]) can be expressed as

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

(14)

where the complex parameter z satisfies $\Re \left(z\right)\ge 0$ and $z\ne 0$.

Ignoring the multiplicative constant ${\left(4\pi z\right)}^{-1/2}$, this integral operator coincides with the special instance of Equation (1) when the parameters take the values

$$\beta =ϵ=\delta ={\displaystyle \frac{1}{4z}},\zeta =\gamma =0.$$

Observe that

$$\begin{array}{c}\hfill \Re \beta =\Re ϵ=\Re \delta ={\displaystyle \frac{\Re z}{{4\left|z\right|}^2}},\Re z\ge 0,z\ne 0.\end{array}$$

Now, the Gauss–Weierstrass semigroup of a distribution f of compact support on $\mathbb{R}$ is defined as

$$\begin{array}{c}\hfill \left(e^{z\Delta }f\right)\left(y\right)={\left(4\pi z\right)}^{-{\displaystyle \frac{1}{2}}}⟨f\left(x\right),exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4z}}\right\}⟩,y\in \mathbb{R}.\end{array}$$

(15)

For $\Re z\ge 0$, it is $\Re \beta \ge 0$ and $\Re ϵ\ge 0$ and also it is ${(\Re \delta )}^2=(\Re \beta )(\Re ϵ).$

Now, as a consequence of Theorems 1 and 2, we have the following.

Theorem 3.

Let z be a complex number with $\Re \left(z\right)\ge 0$ and $z\ne 0$. Suppose f belongs to the space of distributions ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ and has finite order $q\in {\mathbb{N}}_0$. Consider the operator $e^{z\Delta }f$ defined as in (15). Then the following hold:

(i)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to 0}{lim}\left(y^{\eta }\left(e^{z\Delta }f\right)\left(y\right)\right)=0;\end{array}$$

(ii)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to \pm \infty }{lim}\left(y^{-q-\eta }\left(e^{z\Delta }f\right)\left(y\right)\right)=0.\end{array}$$

Proof. 

From Lemma 3, one has

$$\begin{array}{c}\hfill |\left(e^{z\Delta }f\right){\left(y\right)|\le M\left(4\pi z\right)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{\Re z}{{4\left|z\right|}^2}}{y}^2}\underset{0\le k\le q}{max}\left\{{\left({\displaystyle \frac{\left|y\right|}{2\left|z\right|}}\right)}^k\right\},\mathrm{for} \mathrm{all}y\in \mathbb{R},\end{array}$$

for some $M>0$, and therefore the result holds. □

Now, if f is a locally integrable function on $\mathbb{R}$ and f has compact support, then f gives rise to a regular member $T_f$ of ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ of order $q=0$ by means of

$$\begin{array}{ccc}\hfill & & \left(e^{z\Delta }{T}_f\right)\left(y\right)\hfill \\ \hfill & =& {\left(4\pi z\right)}^{-{\displaystyle \frac{1}{2}}}⟨T_f,exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4z}}\right\}⟩\hfill \\ \hfill & =& {\left(4\pi z\right)}^{-{\displaystyle \frac{1}{2}}}{\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{-{\displaystyle \frac{{(y-x)}^2}{4z}}\right\}dx=\left(e^{z\Delta }f\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

By using Theorem 3 for the Gauss–Weierstrass semigroup on $\mathbb{R}$ of the regular member $T_f$ of ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, we have the following result.

Corollary 1.

Suppose f is a locally integrable function on $\mathbb{R}$ with compact support. Then, for every complex number z satisfying $\Re \left(z\right)\ge 0$ and $z\ne 0$, the function $e^{z\Delta }f$ defined by (14) fulfills the following properties:

(i)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to 0}{lim}\left(y^{\eta }\left(e^{z\Delta }f\right)\left(y\right)\right)=0;\end{array}$$

(ii)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to \pm \infty }{lim}\left(y^{-\eta }\left(e^{z\Delta }f\right)\left(y\right)\right)=0.\end{array}$$

Remark 1.

For the case when $z=1$, the Gauss–Weierstrass semigroup becomes the real Weierstrass transform (see [23]). The corresponding Abelian theorems for this transform over ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ were obtained in [23].

3.2. The Linear Canonical Transform

The linear canonical transform of a suitable complex-valued function f defined on $\mathbb{R}$ (see [24,25,26,27,28,29]) is given by

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{A}f\right)\left(y\right)={\left(2\pi ib\right)}^{-{\displaystyle \frac{1}{2}}}{\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{{\displaystyle \frac{id}{2b}}{y}^2+{\displaystyle \frac{ia}{2b}}{x}^2-{\displaystyle \frac{i}{b}}xy\right\}dx,y\in \mathbb{R},\end{array}$$

(16)

where A is a $2\times 2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, $a,b,c,d\in \mathbb{R}$, $b\ne 0$, $ad-bc=1$.

Except for the factor ${\left(2\pi ib\right)}^{-{\displaystyle \frac{1}{2}}}$, this integral operator corresponds to a particular case of (1) when the parameters are given by

$$\begin{array}{c}\hfill \beta =-{\displaystyle \frac{id}{2b}},ϵ=-{\displaystyle \frac{ia}{2b}},\delta =-{\displaystyle \frac{i}{2b}},\zeta =\gamma =0.\end{array}$$

Now, the linear canonical transform of a distribution f of compact support on $\mathbb{R}$ is defined as

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{A}f\right)\left(y\right)={\left(2\pi ib\right)}^{-{\displaystyle \frac{1}{2}}}⟨f\left(x\right),exp\left\{{\displaystyle \frac{id}{2b}}{y}^2+{\displaystyle \frac{ia}{2b}}{x}^2-{\displaystyle \frac{i}{b}}xy\right\}⟩,y\in \mathbb{R}.\end{array}$$

(17)

From Theorems 1 and 2, one obtains the following.

Theorem 4

(Abelian theorem). Suppose f belongs to the space ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ and has order $q\in {\mathbb{N}}_0$. Consider the transform ${\mathcal{L}}_{A}f$ defined as in Equation (17). Then

(i)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to 0}{lim}\left(y^{\eta }\left({\mathcal{L}}_{A}f\right)\left(y\right)\right)=0;\end{array}$$

(ii)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to \pm \infty }{lim}\left(y^{-q-\eta }\left({\mathcal{L}}_{A}f\right)\left(y\right)\right)=0.\end{array}$$

Also, analogous to Corollary 1, one obtains

Corollary 2.

Suppose f is a locally integrable function defined on $\mathbb{R}$ with compact support. Then, the function ${\mathcal{L}}_{A}f$ defined by (16) fulfills the following properties:

(i)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to 0}{lim}\left(y^{\eta }\left({\mathcal{L}}_{A}f\right)\left(y\right)\right)=0;\end{array}$$

(ii)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to \pm \infty }{lim}\left(y^{-\eta }\left({\mathcal{L}}_{A}f\right)\left(y\right)\right)=0.\end{array}$$

Remark 2

(Fourier transform). Taking in (16) the values $a=d=0$, $b=-1$, $c=1$, one arrives to

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{A}f\right)\left(y\right)={(-2\pi i)}^{-{\displaystyle \frac{1}{2}}}{\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{ixy\right\}dx,y\in \mathbb{R},\end{array}$$

which, except for the factor ${(-2\pi i)}^{-{\displaystyle \frac{1}{2}}}$, corresponds to the classical Fourier transform of the function f.

Analogously, taking in (17) the same values $a=d=0$, $b=-1$, $c=1$, one arrives to

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{A}f\right)\left(y\right)={(-2\pi i)}^{-{\displaystyle \frac{1}{2}}}⟨f\left(x\right),exp\left\{ixy\right\}⟩,y\in \mathbb{R},\end{array}$$

which, except for the factor ${(-2\pi i)}^{-{\displaystyle \frac{1}{2}}}$, corresponds to the distributional Fourier transform of the distribution of compact support f.

Remark 3.

For the particular case when $\Re z=0$, $z\ne 0$, in (14) and taking in (16) the values $a=d=1$, $b=2\Im z$, $c=0$, one arrives to $e^{z\Delta }f={\mathcal{L}}_{A}f$.

3.3. The Ornstein–Uhlenbeck Semigroup

The Ornstein–Uhlenbeck semigroup acting on an appropriate complex-valued function f defined over $\mathbb{R}$ (refer to [30], Example 3.4) can be expressed as

$$\left(T_{t}f\right)\left(y\right)={\left(2\pi \right)}^{-{\displaystyle \frac{1}{2}}}{\int }_{-\infty }^{\infty }f(\sqrt{1-e^{-2t}}x+e^{-t}y)e^{-x^2/2}dx,y\in \mathbb{R},t>0,$$

(18)

which becomes

$$\begin{array}{ccc}\hfill \left(T_{t}f\right)\left(y\right)& =& {\left(2\pi (1-e^{-2t})\right)}^{-{\displaystyle \frac{1}{2}}}\hfill \\ & & {\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{-{\displaystyle \frac{e^{-2t}}{2(1-e^{-2t})}}{y}^2-{\displaystyle \frac{1}{2(1-e^{-2t})}}{x}^2+{\displaystyle \frac{e^{-t}}{(1-e^{-2t})}}xy\right\}dx,\hfill \end{array}$$

$y\in \mathbb{R},t>0.$

Apart from the multiplicative term ${\left(2\pi (1-e^{-2t})\right)}^{-1/2}$, this integral operator can be viewed as a special case of the expression (1) with the parameters specified as follows:

$$\begin{array}{c}\hfill \beta ={\displaystyle \frac{e^{-2t}}{2(1-e^{-2t})}},\epsilon ={\displaystyle \frac{1}{2(1-e^{-2t})}},\delta ={\displaystyle \frac{e^{-t}}{2(1-e^{-2t})}},\zeta =0,\gamma =0.\end{array}$$

Now, the Ornstein–Uhlenbeck semigroup of a distribution f of compact support on $\mathbb{R}$ is defined as

$$\begin{array}{ccc}\hfill & & \left(T_{t}f\right)\left(y\right)={\left(2\pi (1-e^{-2t})\right)}^{-{\displaystyle \frac{1}{2}}}\hfill \\ & & ⟨f\left(x\right),exp\left\{-{\displaystyle \frac{e^{-2t}}{2(1-e^{-2t})}}{y}^2-{\displaystyle \frac{1}{2(1-e^{-2t})}}{x}^2+{\displaystyle \frac{e^{-t}}{(1-e^{-2t})}}xy\right\}⟩,\hfill \end{array}$$

(19)

$y\in \mathbb{R},t>0.$

Since ${\delta }^2=\beta ϵ$, $\beta >0$, $ϵ>0$, one obtains as a consequence of Theorem 1 the following.

Theorem 5

(Abelian theorem). Suppose $f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ is a distribution of order $q\in {\mathbb{N}}_0$, and define the operator $T_{t}f$ as in Equation (19). Then

(i)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to 0}{lim}\left(y^{\eta }\left(T_{t}f\right)\left(y\right)\right)=0;\end{array}$$

(ii)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to \pm \infty }{lim}\left(y^{-q-\eta }\left(T_{t}f\right)\left(y\right)\right)=0.\end{array}$$

Also, analogous to Corollaries 1 and 2, one obtains the following.

Corollary 3.

Suppose that f is a locally integrable function defined on $\mathbb{R}$ with compact support. Then the function $T_{t}f$, defined by (18), obeys the following properties:

(i)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to 0}{lim}\left(y^{\eta }\left(T_{t}f\right)\left(y\right)\right)=0;\end{array}$$

(ii)

For any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{y\to \pm \infty }{lim}\left(y^{-\eta }\left(T_{t}f\right)\left(y\right)\right)=0.\end{array}$$

Remark 4

(The Ornstein–Uhlenbeck semigroup with parameter $c>0$). Analogously one can consider the Ornstein–Uhlenbeck semigroup with parameter $c>0$, given by

$$\left(T_{t}f\right)\left(y\right)={\left(2\pi c\right)}^{-1/2}{\int }_{-\infty }^{\infty }f(\sqrt{1-e^{-2t}}x+e^{-t}y)e^{-x^2/2c}dx,y\in \mathbb{R},t>0,$$

which becomes

$$\begin{array}{ccc}& & \left(T_{t}f\right)\left(y\right)={\left(2\pi (1-e^{-2t})c\right)}^{-1/2}\hfill \\ & & {\int }_{-\infty }^{\infty }f\left(x\right)exp\left\{-{\displaystyle \frac{e^{-2t}}{2(1-e^{-2t})c}}{y}^2-{\displaystyle \frac{1}{2(1-e^{-2t})c}}{x}^2+{\displaystyle \frac{e^{-t}}{(1-e^{-2t})c}}xy\right\}dx,\hfill \end{array}$$

$y\in \mathbb{R},t>0$. Apart from the multiplicative constant ${\left(2\pi (1-e^{-2t})c\right)}^{-1/2}$, the integral operator reduces to a special instance of (1), where the parameters take the following values:

$$\beta ={\displaystyle \frac{e^{-2t}}{2(1-e^{-2t})c}},\epsilon ={\displaystyle \frac{1}{2(1-e^{-2t})c}},\delta ={\displaystyle \frac{e^{-t}}{2(1-e^{-2t})c}},\zeta =\gamma =0.$$

Since again ${\delta }^2=\beta \epsilon$, $\beta >0$, $\epsilon >0$, then using Theorem 1, one obtains corresponding results to Theorem 5 and Corollary 3 for this setting.

Note that taking $c={\displaystyle \frac{1}{2}}$, then the above expression of $T_{t}f$ agrees with $e^{-tH}f$, the Hermite semigroup on $\mathbb{R}$ considered in (Equation (1.1), [22]).

Remark 5.

An extension of the Ornstein–Uhlenbeck semigroup is given in [30] by means of

$$\begin{array}{c}\hfill \left({\mathcal{F}}_{a,b}f\right)\left(y\right)={\left(2\pi \right)}^{-{\displaystyle \frac{1}{2}}}{\int }_{-\infty }^{\infty }f(ax+by)e^{-x^2/2}dx,y\in \mathbb{R},\end{array}$$

where a and b are non-zero real numbers.

Observe that

$$\begin{array}{c}\hfill \left({\mathcal{F}}_{a,b}f\right)\left(y\right)={\left(2\pi a^2\right)}^{-{\displaystyle \frac{1}{2}}}{\int }_{-\infty }^{\infty }exp\left\{-{\displaystyle \frac{b^2}{2a^2}}{y}^2-{\displaystyle \frac{1}{2a^2}}{x}^2+{\displaystyle \frac{b}{a^2}}xy\right\}dx,y\in \mathbb{R}.\end{array}$$

Apart from the multiplicative constant ${\left(2\pi a^2\right)}^{-{\displaystyle \frac{1}{2}}}$, the integral operator can be seen as a special instance of (1) with the parameters specified by

$$\begin{array}{c}\hfill \beta ={\displaystyle \frac{b^2}{2a^2}},ϵ={\displaystyle \frac{1}{2a^2}},\delta ={\displaystyle \frac{b}{2a^2}},\zeta =0,\gamma =0.\end{array}$$

Since again ${\delta }^2=\beta \epsilon$, $\beta >0$, $\epsilon >0$, then by using Theorem 1, one obtains corresponding results to Theorem 5 and Corollary 3 for this setting.

Observe that for the case when $a=b=1$, the transform ${\mathcal{F}}_{1,1}$ agrees with the inverse of the so-called Gauss transform (Corollary 2.8 (b), p. 157, [30]). Also observe that ${\mathcal{F}}_{1,1}f$ agrees with the Gauss–Weierstrass semigroup $e^{{\displaystyle \frac{1}{2}}\Delta }f$.

4. Final Observations and Conclusions

Let us define the operator ${\mathcal{A}}_x^{\prime }:=-D_x+2ϵx$, where $D_x$ denotes the distributional derivative. For any distribution $f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, it follows that ${\mathcal{A}}_x^{\prime }f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ as well, and the transform satisfies

$$\begin{array}{cc}\hfill \left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }\left({\mathcal{A}}_x^{\prime }f\right)\right)\left(y\right)& =⟨{\mathcal{A}}_x^{\prime }f,exp\left(-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right)⟩\hfill \\ \hfill & =⟨f\left(x\right),{\mathcal{A}}_x\left(exp\left(-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right)\right)⟩\hfill \\ \hfill & =\left(2\delta y+\gamma \right)⟨f\left(x\right),exp\left(-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right)⟩\hfill \\ \hfill & =\left(2\delta y+\gamma \right)\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

Consequently, for any positive integer m, the iterated operator ${\left({\mathcal{A}}_x^{\prime }\right)}^{m}f$ remains in ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, and its transform satisfies

$$\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }\left({\left({\mathcal{A}}_x^{\prime }\right)}^{m}f\right)\right)\left(y\right)={\left(2\delta y+\gamma \right)}^m\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right),y\in \mathbb{R}.$$

Therefore, also one obtains Abelian theorems for ${\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }\left({\mathcal{A}}_x^{{\prime }^m}f\right)$, $m\in \mathbb{N}$, from Theorems 1 and 2.

On the other hand, observe that for $f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, one has

$$\begin{array}{ccc}& & \left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\hfill \\ & & =⟨f\left(x\right),exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}⟩\hfill \\ & & =exp\left\{-\beta y^2+\zeta y\right\}⟨exp\left\{-ϵx^2+\gamma x\right\}f\left(x\right),exp\left\{2\delta xy\right\}⟩\hfill \\ & & =exp\left\{-\beta y^2+\zeta y\right\}\left(\mathcal{L}\left(exp\left\{-ϵx^2+\gamma x\right\}f\left(x\right)\right)\right)(-2\delta y),y\in \mathbb{R},\hfill \end{array}$$

where $\mathcal{L}$ denotes the two-sided Laplace transform considered in (Section 3.3, [31]).

Now (from Theorem 3.3-1, p. 58, [31]), the transform (2) is a smooth function on $\mathbb{R}$, and

$$\begin{array}{ccc}\hfill & & D_y^m\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)\hfill \\ & & =⟨f\left(x\right),D_y^m\left(exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}\right)⟩,\hfill \end{array}$$

(20)

$m\in \mathbb{N},y\in \mathbb{R}$.

Observe that if one denotes $B_y\equiv D_y+2\beta y$, then

$$\begin{array}{ccc}& & B_{y}exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}\hfill \\ & & =(2\delta x+\zeta )exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\},x,y\in \mathbb{R},\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill & & B_y^{m}exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}\hfill \\ & & ={(2\delta x+\zeta )}^{m}exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\},x,y\in \mathbb{R}.\hfill \end{array}$$

(21)

Now, from relation (20) one has

$$\begin{array}{ccc}& & B_y\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)\hfill \\ & & =⟨f\left(x\right),B_y\left(exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}\right)⟩,y\in \mathbb{R},\hfill \end{array}$$

which, using (21), is equal to

$$\begin{array}{ccc}& & ⟨f\left(x\right),(2\delta x+\zeta )exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}⟩\hfill \\ & & =⟨(2\delta x+\zeta )f\left(x\right),exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}⟩\hfill \\ & & =\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }\left((2\delta x+\zeta )f\left(x\right)\right)\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

Since $(2\delta x+\zeta )f\left(x\right)\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ and using (21), one has

$$\begin{array}{ccc}& & B_y^2\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)\hfill \\ & & =⟨(2\delta x+\zeta )f\left(x\right),B_y\left(exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}\right)⟩\hfill \\ & & =⟨{(2\delta x+\zeta )}^{2}f\left(x\right),exp\left\{-\beta y^2-ϵx^2+2\delta xy+\zeta y+\gamma x\right\}⟩\hfill \\ & & =\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }\left({(2\delta x+\zeta )}^{2}f\left(x\right)\right)\right)\left(y\right),y\in \mathbb{R}.\hfill \end{array}$$

And so, one arrives to

$$\begin{array}{ccc}& & B_y^m\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)\hfill \\ & & =\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }\left({(2\delta x+\zeta )}^{m}f\left(x\right)\right)\right)\left(y\right),m\in \mathbb{N},y\in \mathbb{R}.\hfill \end{array}$$

Thus, being that ${(2\delta x+\zeta )}^{m}f\left(x\right)\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, $m\in \mathbb{N}$, then one also obtains Abelian theorems for $B_y^m\left(\left({\mathcal{F}}_{\beta ,ϵ,\delta ,\zeta ,\gamma }f\right)\left(y\right)\right)$, $m\in \mathbb{N}$, from Theorems 1 and 2.

In conclusion, this research investigates Abelian theorems concerning the operators with complex Gaussian kernels over distributions of compact support. The exploration of the Gauss–Weierstrass semigroup, the linear canonical transform, and the Ornstein–Uhlenbeck semigroup over ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ within our analysis provides valuable insights. These findings significantly contribute to the understanding of various integral transform behaviors, laying down a solid foundation for future mathematical analysis in this field.

We conclude our present investigation by presenting the following open question.

Open Questions. A natural direction for further investigation is the rigorous derivation of inversion formulae for these transforms in the setting of distributions with compact support. Establishing such formulae would not only refine the spectral analysis of these transforms within the framework of distribution theory but also deepen the understanding of their invertibility in spaces of generalized functions. One may also consider a generalization of Equation (2) to higher dimensions and investigate the corresponding analytical framework. The feasibility and development of such an analysis remain open questions.