Asymptotic Behaviours for an Index Whittaker Transform over ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$

Abstract

The objective of this research is to obtain the asymptotic behaviour of compactly supported distributions and generalized functions for a variant of the index Whittaker transform.

1. Introduction and Preliminaries

An index transform is a type of integral transform characterized by a kernel that is inherently associated with specific parameters of the corresponding special function [1]. The development of these integral transforms dates back to the twentieth century, during which time researchers in mathematics, physics, and engineering extensively employed them to address various analytical challenges. Considerable research has been conducted on the properties of integral transforms, with a primary focus on their kernels and diverse applications. These applications include solving differential and integral equations, and improving image processing techniques. The scope of research encompasses various integral transforms, such as the Fourier transform, Laplace transform, Hankel transform, Lebedev–Skalskaya transform, index Whittaker transform, Mehler–Fock transform, and Kontorovich–Lebedev transform.

First, this research introduces specific definitions and notations that will be used subsequently. The index Whittaker transform ${\mathbf{W}}_{\alpha }$ of a suitable function f is defined as [2]

$$\begin{array}{c}\hfill \left({\mathbf{W}}_{\alpha }f\right)\left(\tau \right)={\int }_0^{\infty }f\left(x\right){\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)dx,\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2,\alpha <{\displaystyle \frac{1}{2}},\end{array}$$

(1)

where ${\Delta }_{\tau }=\sqrt{\tau -{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2}$, $m_{\alpha }\left(x\right)=x^{1-4\alpha }{e}^{-{\displaystyle \frac{1}{2x^2}}}$, and

$$\begin{array}{c}\hfill {\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right):=2^{\alpha }{x}^{2\alpha }{e}^{{\displaystyle \frac{1}{4x^2}}}{W}_{\alpha ,i{\Delta }_{\tau }}\left({\displaystyle \frac{1}{2x^2}}\right),\end{array}$$

where $W_{\alpha ,i{\Delta }_{\tau }}$ is the Whittaker function [2]. Furthermore, from [2] [Theorem 3.1], the following bound holds:

$$\begin{array}{c}\hfill |{\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)|\le {\mathbf{W}}_{\alpha ,0}\left(x\right),\mathrm{for} \mathrm{all}\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2,\alpha <{\displaystyle \frac{1}{2}}.\end{array}$$

(2)

From [3] [Formula (13), (14) and (19), p. 335] and [3] [Formula (13), (14) and (21), p. 335],

$$\begin{array}{ccc}\hfill {\mathbf{W}}_{\alpha ,0}\left(x\right)& \sim & 1\mathrm{as}x\to +0,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\mathbf{W}}_{\alpha ,0}\left(x\right)& =& O\left(x^{-1+2\alpha }logx\right)\mathrm{as}x\to +\infty .\hfill \end{array}$$

(4)

The systematic investigation of an index integral transform with the Whittaker function as its kernel has been a subject of extensive research. Significant contributions in this area include the works of Wimp [4], Srivastava et al. [5,6,7], Al-Musallam et al. [8,9], Sousa et al. [2,10], Becker [11], and Maan et al. [12,13,14].

Research on Abelian theorems and asymptotic behaviours related to distributional transforms has been extensively explored across various mathematical domains. Initial investigations established Abelian theorems for distributional transforms [15]. Subsequently, this exploration expanded to include the ${}_{2}F_1$-transform, with further developments covering the distributional Kontorovich–Lebedev and Mehler–Fock transforms of general order (see [16]). The scope was further extended to examine the index Whittaker transform [12,14] and the real Weierstrass transform [17]. Recent advancements include studies on the Laplace, Mellin, Stieltjes, index Whittaker, Lebedev–Skalskaya, and real Weierstrass transforms over spaces of distributions with compact support and certain generalized function spaces (see [12,17,18]). Additionally, investigations have been conducted on Abelian theorems for the index ${}_{2}F_1$-transform over distributions with compact support and generalized functions [19]. Abelian theorems for wavelet transforms, both in the classical and distributional sense, are studied in [20]. These studies provide crucial insights into the behaviour of transforms concerning their domain variables, as well as their properties near the origin and at infinity. They establish a framework for deducing transform behaviour based on the properties of the distributions or generalized functions under analysis.

Inspired by the preceding research, this study examines the asymptotic behaviours of the index Whittaker transform ${\mathbf{W}}_{\alpha }$ over distributions with compact support and generalized functions.

The notation $\mathcal{E}\left({\mathbb{R}}_{+}\right)$ represents the vector space of complex-valued functions $\varphi$ defined on ${\mathbb{R}}_{+}=(0,\infty )$ that are infinitely differentiable. This space is endowed with the locally convex topology induced by a family of seminorms

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

where K is any compact subset of ${\mathbb{R}}_{+}$, and $D_x^k$ is the k-th derivative with respect to x-variable. It is observed that this space forms a Fréchet space. The dual space of $\mathcal{E}\left({\mathbb{R}}_{+}\right)$ is denoted by ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$, which corresponds to the space of distributions on ${\mathbb{R}}_{+}$ of compact support.

2. Asymptotic Behaviours for the Index Whittaker Transform ${\mathbf{W}}_{\alpha }$ over ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$

The index Whittaker transform ${\mathbf{W}}_{\alpha }$ of a distribution f of compact support on ${\mathbb{R}}_{+}$ is defined by the kernel method as

$$\begin{array}{c}\hfill \left({\mathbf{W}}_{\alpha }f\right)\left(\tau \right)=⟨f\left(x\right),{\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)⟩,\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2,\alpha <{\displaystyle \frac{1}{2}}.\end{array}$$

(5)

In this section, the asymptotic behaviours of the distributional index Whittaker transform ${\mathbf{W}}_{\alpha }$ given by (5) are established. To achieve this, the prior results are first presented.

 Lemma 1. 

Let ${\mathcal{L}}_{\alpha ,x}$ be the differential operator given by

$$\begin{array}{c}\hfill {\mathcal{L}}_{\alpha ,x}f\left(x\right)=-{\displaystyle \frac{m_{\alpha }\left(x\right)}{4}}\left(x^2{D}_x^2+{\displaystyle \frac{{\left(x^2{m}_{\alpha }\left(x\right)\right)}^{\prime }}{m_{\alpha }\left(x\right)}}{D}_x\right)\left({\displaystyle \frac{f\left(x\right)}{m_{\alpha }\left(x\right)}}\right).\end{array}$$

(6)

Then, for all $k\in \mathbb{N}$, there exist $C^{\infty }$-functions $P_{j,k}$ over ${\mathbb{R}}_{+}$, such that

$$\begin{array}{c}\hfill {\mathcal{L}}_{\alpha ,x}^k=\sum _{j=0}^{2k}{P}_{j,k}\left(x\right)D_x^j,\end{array}$$

(7)

where $P_{2k,k}\left(x\right)={\left(-{\displaystyle \frac{1}{4}}\right)}^k{x}^{2k}$.

 Proof. 

By induction, the result of the above lemma follows. Specifically, the base case for $k=1$ is verified, and the assumption is made that the result holds for all values up to $k-1$, after which it is proven for k. For $k\ge 2$, the following holds:

$$\begin{array}{ccc}& & {\mathcal{L}}_{\alpha ,x}^k={\mathcal{L}}_{\alpha ,x}{\mathcal{L}}_{\alpha ,x}^{k-1}\hfill \\ & & =-{\displaystyle \frac{m_{\alpha }\left(x\right)}{4}}\left(x^2{D}_x^2+{\displaystyle \frac{{\left(x^2{m}_{\alpha }\left(x\right)\right)}^{\prime }}{m_{\alpha }\left(x\right)}}{D}_x\right)\left(\sum _{j=0}^{2(k-1)}{\displaystyle \frac{P_{j,k-1}\left(x\right)D_x^j}{m_{\alpha }\left(x\right)}}\right)\hfill \\ & & =-{\displaystyle \frac{m_{\alpha }\left(x\right)}{4}}\sum _{j=0}^{2k-2}\left[x^2{D}_x^2\left({\displaystyle \frac{P_{j,k-1}\left(x\right)D_x^j}{m_{\alpha }\left(x\right)}}\right)+{\displaystyle \frac{{\left(x^2{m}_{\alpha }\left(x\right)\right)}^{\prime }}{m_{\alpha }\left(x\right)}}{D}_x\left({\displaystyle \frac{P_{j,k-1}\left(x\right)D_x^j}{m_{\alpha }\left(x\right)}}\right)\right],\hfill \end{array}$$

which readily yields to (7). This completes the inductive step, and the result follows. □

 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

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

where ${\mathcal{L}}_{\alpha ,x}$ is the differential operator given by (6), and ${\mathcal{L}}_{\alpha ,x}^k$ 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 ${\mathcal{L}}_{\alpha ,x}^k$ given in (7) yields that any sequence $\left\{{\psi }_n\right\}\subset \mathcal{E}\left({\mathbb{R}}_{+}\right)$ which tends to zero for the usual topology on $\mathcal{E}\left({\mathbb{R}}_{+}\right)$ also tends to zero 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 with respect to the topology generated by $\left\{{\Gamma }_{k,K}\right\}$. It is clear that $\left\{{\psi }_n\right\}$, $\left\{{\mathcal{L}}_{\alpha ,x}{\psi }_n\right\}$ tends to zero as $n\to \infty$, uniformly in each compact subset $K\subset {\mathbb{R}}_{+}$. Moreover, from (7),

$$\begin{array}{cc}& {\mathcal{L}}_{\alpha ,x}{\psi }_n\left(x\right)-P_{0,1}\left(x\right){\psi }_n\left(x\right)\hfill \\ =& \left(-{\displaystyle \frac{1}{4}}\right)x^2{D}_x^2{\psi }_n\left(x\right)+P_{1,1}\left(x\right)D_x{\psi }_n\left(x\right).\hfill \end{array}$$

(8)

The left-hand side of (8) tends to zero as $n\to \infty$, uniformly in each compact subset $K\subset {\mathbb{R}}_{+}$, which concludes that the right-hand side also tends to zero as $n\to \infty$, uniformly in each compact subset $K\subset {\mathbb{R}}_{+}$.

Now, dividing by $\left(-{\displaystyle \frac{x^2}{4}}\right)$ and taking integral from b to x, where $b\in {\mathbb{R}}_{+}$, $b\notin K$, one has

$$\begin{array}{ccc}& & \underset{b}{\overset{x}{\int }}{D}_t^2{\psi }_n\left(t\right)dt+\underset{b}{\overset{x}{\int }}{\displaystyle \frac{P_{1,1}\left(t\right)}{\left(-\frac{t^2}{4}\right)}}{D}_t{\psi }_n\left(t\right)dt\hfill \\ & =& D_x{\psi }_n\left(x\right)-D_x{\psi }_n\left(b\right)+{\left[{\displaystyle \frac{P_{1,1}\left(t\right)}{\left(-\frac{t^2}{4}\right)}}{\psi }_n\left(t\right)\right]}_b^x-\underset{b}{\overset{x}{\int }}{D}_t\left({\displaystyle \frac{P_{1,1}\left(t\right)}{\left(-\frac{t^2}{4}\right)}}\right){\psi }_n\left(t\right)dt,\hfill \end{array}$$

which tends to zero as $n\to \infty$, uniformly in each compact subset $K\subset {\mathbb{R}}_{+}$. Therefore,

$$\begin{array}{c}\hfill D_x{\psi }_n\left(x\right)-D_x{\psi }_n\left(b\right)\end{array}$$

(9)

tends to zero as $n\to \infty$, uniformly in each compact subset $K\subset {\mathbb{R}}_{+}$.

Integrating (9) from b to x, one has

$$\begin{array}{ccc}& & \underset{b}{\overset{x}{\int }}\left[D_t{\psi }_n\left(t\right)-D_t{\psi }_n\left(b\right)\right]dt\hfill \\ & =& {\psi }_n\left(x\right)-{\psi }_n\left(b\right)-D_t{\psi }_n\left(b\right)(x-b),\hfill \end{array}$$

which tends to zero as $n\to \infty$, uniformly on each compact subset K of ${\mathbb{R}}_{+}$. Thus, noting that $(x-b)$ is bounded away from zero for all $x\in K$, it follows that $D_t{\psi }_n\left(b\right)\to 0$ as $n\to \infty$. Consequently, from (9), $D_x{\psi }_n\left(x\right)$ and therefore from (8), $D_x^2{\psi }_n\left(x\right)$ tend to zero as $n\to \infty$, uniformly on each compact subset $K\subset {\mathbb{R}}_{+}$.

Now, assume by induction that, for $0\le m\le 2k-2$, $D_x^m{\psi }_n\left(x\right)$ tend to zero as $n\to \infty$, uniformly on each compact subset K of ${\mathbb{R}}_{+}$.

Then,

$$\begin{array}{ccc}& & {\mathcal{L}}_{\alpha ,x}^k{\psi }_n\left(x\right)-\sum _{j=0}^{2k-2}{P}_{j,k}\left(x\right)D_x^j{\psi }_n\left(x\right)\hfill \\ & =& {\left(-{\displaystyle \frac{1}{4}}\right)}^k{x}^{2k}{D}_x^{2k}{\psi }_n\left(x\right)+P_{2k-1,k}\left(x\right)D_x^{2k-1}{\psi }_n\left(x\right),\hfill \end{array}$$

which, arguing as for the case $k=1$, yields to $D_x^{2k-1}{\psi }_n\left(x\right)$, and therefore $D_x^{2k}{\psi }_n\left(x\right)$ tend 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 the ${\Gamma }_{k,K}$, are metrizable, then the conclusion follows. □

For $f\in {\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$, consider the operator given by

$$\begin{array}{c}\hfill ⟨{\mathcal{L}}_{\alpha ,x}^{*}f,\psi ⟩=⟨f,{\mathcal{L}}_{\alpha ,x}\psi ⟩,\psi \in \mathcal{E}\left({\mathbb{R}}_{+}\right).\end{array}$$

From Lemma 2, the space $\mathcal{E}\left({\mathbb{R}}_{+}\right)$ is considered with the topology arising from the family of seminorms ${\Gamma }_{k,K}$. It follows from this that ${\mathcal{L}}_{a,x}^{*}:{\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)\to {\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$.

Also, for $m\in {\mathbb{N}}_0$, one has that

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{\alpha ,x}^m\left({\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}(·)m_{\alpha }(·)\right)\right)\left(x\right)={\tau }^m{\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right),\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2,\alpha <{\displaystyle \frac{1}{2}}.\end{array}$$

(10)

Then for the ${\mathbf{W}}_{\alpha }$ transform (5), one has

$$\begin{array}{c}\hfill \left({\mathbf{W}}_{\alpha }\left({\mathcal{L}}_{\alpha ,x}^{{*}^m}f\right)\right)\left(\tau \right)={\tau }^m\left({\mathbf{W}}_{\alpha }f\right)\left(\tau \right),\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2,\alpha <{\displaystyle \frac{1}{2}},m\in {\mathbb{N}}_0.\end{array}$$

 Lemma 3. 

Let f be in ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$, and ${\mathbf{W}}_{\alpha }$ be defined by (5). Then, there exist $M>0$ and a non-negative integer q, all depending on f, such that

$$\begin{array}{c}\hfill \left|\left({\mathbf{W}}_{\alpha }\left({\mathcal{L}}_{\alpha ,x}^{{*}^m}f\right)\right)\left(\tau \right)\right|\le M\underset{0\le k\le q}{max}\left\{{\tau }^{k+m}\right\},forall\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2,\alpha <{\displaystyle \frac{1}{2}}.\end{array}$$

(11)

 Proof. 

We know from [2] that ${\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)$ is an eigenfunction of ${\mathcal{L}}_{\alpha ,x}$, as

$$\begin{array}{c}\hfill {\mathcal{L}}_{\alpha ,x}\left({\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)\right)=\tau {\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right).\end{array}$$

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 [21] [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

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

(12)

Now, one has

$$\begin{array}{ccc}& & \left|\left({\mathbf{W}}_{\alpha }\left({\mathcal{L}}_{\alpha ,x}^{{*}^m}f\right)\right)\left(\tau \right)\right|\hfill \\ & =& {\tau }^m\left|\left({\mathbf{W}}_{\alpha }f\right)\left(\tau \right)\right|\hfill \\ & \le & {\tau }^m\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\mathcal{L}}_{\alpha ,x}^k\left({\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)\right)\right|\hfill \\ & =& C{\tau }^m\underset{0\le k\le q}{max}\underset{x\in K}{max}\left|{\tau }^k\left({\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)\right)\right|\hfill \\ & \le & C{\tau }^m\underset{0\le k\le q}{max}\left\{{\tau }^k\right\}\underset{x\in K}{max}\left\{{\mathbf{W}}_{\alpha ,0}\left(x\right)x^{1-4\alpha }{e}^{-{\displaystyle \frac{1}{2x^2}}}\right\}(\mathrm{from} (2\left)\right)\hfill \\ & \le & M\underset{0\le k\le q}{max}\left\{{\tau }^{k+m}\right\},\mathrm{for} \mathrm{all}\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2,\alpha <{\displaystyle \frac{1}{2}},\hfill \end{array}$$

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

The smallest integer q which verifies that the inequality in (12) is defined as the order of the distribution f [22] [Théorème XXIV, p. 88]. Now, we establish Abelian theorems for the distributional index Whittaker transform (5).

 Theorem 1. 

(Asymptotic behaviours). Let f be a member of ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$ of order $q\in {\mathbb{N}}_0$, and let ${\mathbf{W}}_{\alpha }$ be given by (5). Then, for $m\in {\mathbb{N}}_0$ and for any $\eta >0$, one has

$$\begin{array}{c}\hfill \underset{\tau \to +\infty }{lim}\left({\tau }^{-q-m-\eta }\left({\mathbf{W}}_{\alpha }\left({\mathcal{L}}_{\alpha ,x}^{{*}^m}f\right)\right)\left(\tau \right)\right)=0.\end{array}$$

 Proof. 

From Lemma 3, one has

$$\begin{array}{c}\hfill \left|\left({\mathbf{W}}_{\alpha }\left({\mathcal{L}}_{\alpha ,x}^{{*}^m}f\right)\right)\left(\tau \right)\right|\le \underset{0\le k\le q}{max}\left\{{\tau }^{k+m}\right\},\mathrm{for} \mathrm{all}\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2,\alpha <{\displaystyle \frac{1}{2}},\end{array}$$

for some $M>0$ and, therefore, the results hold. □

Now, 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 ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$ of order zero by means of

$$\begin{array}{c}\hfill ⟨T_f,\psi ⟩={\int }_0^{\infty }f\left(x\right)\psi \left(x\right)dx,\mathrm{for} \mathrm{all}\psi \in \mathcal{E}\left({\mathbb{R}}_{+}\right).\end{array}$$

In fact, taking into account that

$$\begin{array}{ccc}\hfill |⟨T_f,\psi ⟩|& =& \left|{\int }_0^{\infty }f\left(x\right)\psi \left(x\right)dx\right|\le \underset{x\in supp\left(f\right)}{sup}\left|\psi \left(x\right)\right|{\int }_{supp\left(f\right)}\left|f\left(x\right)\right|dx\hfill \\ & =& {\Gamma }_{0,supp\left(f\right)}\left(\psi \right){\int }_{supp\left(f\right)}\left|f\left(x\right)\right|dx,\hfill \end{array}$$

where supp(f) represents the support of the function f, it follows that $T_f$ has order zero. So, for $f\in C_c^{2m}\left({\mathbb{R}}_{+}\right),m\in {\mathbb{N}}_0$, and letting

$$\begin{array}{c}\hfill {\mathcal{L}}_{\alpha ,x}^{*}f\left(x\right)=-{\displaystyle \frac{1}{4m_{\alpha }\left(x\right)}}\left[D_x^2\left(x^2{m}_{\alpha }\left(x\right)f\left(x\right)\right)-D_x\left({\left(x^2{m}_{\alpha }\left(x\right)\right)}^{\prime }f\left(x\right)\right)\right],\end{array}$$

it follows that

$$\begin{array}{cc}\hfill F_m\left(\tau \right)& :=\left({\mathbf{W}}_{\alpha }\left({\mathcal{L}}_{\alpha ,x}^{{*}^m}{T}_f\right)\right)\left(\tau \right)\hfill \\ & =⟨{\mathcal{L}}_{\alpha ,x}^{{*}^m}{T}_f,{\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)⟩\hfill \\ & =⟨T_f,{\mathcal{L}}_{\alpha ,x}^m\left({\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)\right)⟩\hfill \\ & ={\tau }^m{\int }_0^{\infty }f\left(x\right){\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)dx.\hfill \end{array}$$

(13)

By applying Theorem 1 to the index Whittaker transform of these regular members of ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$, it follows that the following holds.

 Corollary 1. 

Let $f\in C_c^{2m}\left({\mathbb{R}}_{+}\right),m\in {\mathbb{N}}_0$. Then, for any $\eta >0$, the function $F_m$ given by (13) satisfies the following:

$$\begin{array}{c}\hfill \underset{\tau \to +\infty }{lim}\left({\tau }^{-m-\eta }{F}_m\left(\tau \right)\right)=0.\end{array}$$

For the case $m=0$, in the following result, we obtain a stronger result than the one obtained in Corollary 1.

 Corollary 2. 

Let f be locally integrable function on ${\mathbb{R}}_{+}$, and such that f has compact support on ${\mathbb{R}}_{+}$. Then, for any $\eta >0$, the function $F_0$ given by (13) satisfies the following:

$$\begin{array}{c}\hfill \underset{\tau \to +\infty }{lim}\left({\tau }^{-\eta }{F}_0\left(\tau \right)\right)=0.\end{array}$$

3. Final Observations and Conclusions

The results of Section 2 can be extended from distributions of compact support to generalized functions.

Indeed, one considers the linear space $A_{\alpha ,\beta },\alpha ,\beta \in \mathbb{R}$ of all smooth complex-valued functions $\varphi$ on ${\mathbb{R}}_{+}$, such that

$$\begin{array}{c}\hfill {\gamma }_{k,\alpha ,\beta }\left(\varphi \right)=\underset{x\in {\mathbb{R}}_{+}}{sup}\left|e^{-\beta x}{\mathcal{L}}_{\alpha ,x}^k\varphi \left(x\right)\right|<\infty ,\mathrm{for} \mathrm{all}k\in {\mathbb{N}}_0,\end{array}$$

where ${\mathcal{L}}_{\alpha ,x}$ is the differential operator given by (6).

The space $A_{\alpha ,\beta }$ equipped with the topology arising from the family of seminorms ${\left\{{\gamma }_{k,\alpha ,\beta }\right\}}_{k\in {\mathbb{N}}_0}$ is a Fréchet space.

Observe that by means of relations (3), (4) and (10) the kernel ${\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)$ is a member of $A_{\alpha ,\beta }$ for $\alpha <{\displaystyle \frac{1}{2}},\beta >0,\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2$.

As is usual, the dual space of $A_{\alpha ,\beta }$ is denoted by $A_{\alpha ,\beta }^{\prime }$. Also, note that $A_{\alpha ,\beta }\subseteq \mathcal{E}\left({\mathbb{R}}_{+}\right)$, and from Lemma 2, the restriction of a member of ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$ to $A_{\alpha ,\beta }$ becomes a member of $A_{\alpha ,\beta }^{\prime }$.

So, when one defines the index Whittaker transform ${\mathbf{W}}_{\alpha }$ over the space of generalized functions $A_{\alpha ,\beta }^{\prime }$, $\alpha <{\displaystyle \frac{1}{2}},\beta >0$ by

$$\begin{array}{c}\hfill \left({\mathbf{W}}_{\alpha }f\right)\left(\tau \right)=⟨f\left(x\right),{\mathbf{W}}_{\alpha ,i{\Delta }_{\tau }}\left(x\right)m_{\alpha }\left(x\right)⟩,\tau >{\left({\displaystyle \frac{1}{2}}-\alpha \right)}^2,\end{array}$$

(14)

then (14) becomes an extension of (5) from ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$ to $A_{\alpha ,\beta }^{\prime }$, $\alpha <{\displaystyle \frac{1}{2}},\beta >0$.

The results of Section 2 remain valid for the transform (14) over $A_{\alpha ,\beta }^{\prime }$, $\alpha <{\displaystyle \frac{1}{2}},\beta >0$.

In summary, our study establishes asymptotic behaviours for the index Whittaker transform ${\mathbf{W}}_{\alpha }$. These results provide valuable insights into the behaviours of such transform over distributions of compact support and generalized functions. Our findings contribute to a deeper understanding of index-type integral transforms and establish a solid foundation for further mathematical analysis in this domain.