The Generalized Mehler–Fock Transform over Lebesgue Spaces

Abstract

This paper focuses on establishing boundedness properties and Parseval–Goldstein-type relations for the generalized Mehler–Fock transform initially introduced by B. L. J. Braaksma and B. M. Meulenbeld (Compositio Math., 18(3):235–287, 1967). Also, we derive an inversion formula for this transform over Lebesgue spaces.

1. Introduction and Preliminaries

The concept of the Mehler–Fock transforms has its roots in the groundbreaking contributions of F. G. Mehler [1] and V. A. Fock [2]. Evolving from their seminal work, it emerged as a distinct integral transform, finding applications across a broad spectrum of mathematical physics problems. It is used in numerous applications, such as finding solutions to the electrostatic potential of two overlapping conductive spheres, determining the polarizability of intersecting metallic spheres, and addressing physical problems in continuum mechanics and electromagnetic wave diffraction in conical regions. It is also useful in signal and image processing, as well as solving boundary value problems. Additionally, the Mehler–Fock integral transform naturally arises when solving harmonic boundary value problems defined by the surface of intersecting spheres or the revolution of hyperboloid sheets. It can also be employed to solve boundary value problems in potential theory, which are important in the mathematical theory of elasticity, especially in analyzing stress near external cracks. Furthermore, the Mehler–Fock inversion theorem has intriguing applications in finding solutions to dual integral equations with trigonometrical kernels (see [3,4,5,6,7]).

Integral transforms are powerful tools because they can turn complex problems in one domain into simpler ones in another domain. This often makes mathematical manipulation and finding solutions easier. They are useful in solving differential and integral equations and in tackling issues in signal and image processing, among other mathematical and physical problems. Choosing the right integral transform depends on the nature of the problem and the desired outcomes. Each transform has its own benefits and limitations, so the selection is based on the specific needs of the problem and the characteristics of the original function. Studying integral transforms from the perspective of mathematical analysis reveals their core properties and theoretical foundations, which is crucial for a thorough mathematical understanding. This deeper insight goes beyond practical application and helps drive innovation and progress in mathematical theory. Various scholars have conducted extensive research on its properties and applications. For an in-depth exploration, interested readers can consult the following references, among others [8,9,10,11,12,13,14,15]. In this work, we offer an in-depth examination of the generalized Mehler–Fock transform from a functional analytic viewpoint, focusing on their mathematical properties.

In 1989, Yürekli introduced a Parseval–Goldstein-type theorem, which clarified the relationship between Laplace and Stieltjes transforms. He further explored the implications of this theorem. By 1992, Yürekli expanded his research to include the generalized Stieltjes transform. Building on his work, various researchers have investigated similar connections among different integral transforms, using Parseval–Goldstein-type theorems. The importance of Parseval’s and Plancherel’s theorems in mathematics lies in their ability to establish crucial links between functions and their transforms, demonstrating the preservation of energy or inner products during these transformations (see [16,17,18,19,20,21,22,23]).

We consider the generalized Mehler–Fock transform of a suitable complex-valued function f on ${\mathbb{R}}_{+}$ analyzed by B. L. J. Braaksma and B. M. Meulenbeld [14] and R. S. Pathak [15], Chapter 11, given by

$$\begin{array}{c}\hfill ({\mathcal{B}}_{m,n}f)(\tau )={\int }_0^{\infty }f(x)P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh xdx,\tau >0,\end{array}$$

(1)

where $m,n$ are complex numbers with $\Re m<{\displaystyle \frac{1}{2}}$ and $P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)$ is the associated Legendre function of the first kind [24], Chapter 3, defined by

$$\begin{array}{ccc}\hfill P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)& =& {\displaystyle \frac{{(1+\cosh x)}^{\frac{n}{2}}}{\Gamma (1-m){(\cosh x-1)}^{\frac{m}{2}}}}\hfill \\ & \times & { }_2{F}_1\left({\displaystyle \frac{1}{2}}+i\tau +{\displaystyle \frac{n-m}{2}},{\displaystyle \frac{1}{2}}-i\tau +{\displaystyle \frac{n-m}{2}};1-m;{\displaystyle \frac{1-\cosh x}{2}}\right),\hfill \end{array}$$

where ${ }_2{F}_1$ is the Gauss hypergeometric function [24], p. 57.

A corresponding inversion formula of (1) for suitable f is given by

$$\begin{array}{c}\hfill f(x)={\int }_0^{\infty }\chi (\tau )P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)({\mathcal{B}}_{m,n}f)(\tau )d\tau ,x>0,\end{array}$$

(2)

where

$$\begin{array}{ccc}\hfill \chi (\tau )& =& \Gamma \left({\displaystyle \frac{1-m+n}{2}}+i\tau \right)\Gamma \left({\displaystyle \frac{1-m+n}{2}}-i\tau \right)\Gamma \left({\displaystyle \frac{1-m-n}{2}}+i\tau \right)\hfill \\ & \times & \Gamma \left({\displaystyle \frac{1-m-n}{2}}-i\tau \right){\left[\Gamma (2i\tau )\Gamma (-2i\tau )\pi 2^{n-m+2}\right]}^{-1}.\hfill \end{array}$$

(3)

The conditions of validity of (1) and (2) are provided by Braaksma and Meulenbeld [14].

Notice that for $m=n=0$, one obtains the Mehler–Fock transform of the zeroth order [3], §7-6, p. 390 and [14], Theorem 7, p. 247,

$$\begin{array}{c}\hfill F(\tau )={\int }_0^{\infty }f(x)P_{-{\displaystyle \frac{1}{2}}+i\tau }(\cosh x)\sinh xdx,\tau >0.\end{array}$$

(4)

and a corresponding inversion formula given by

$$\begin{array}{c}\hfill f(x)={\int }_0^{\infty }\tau \tanh (\pi \tau )P_{-{\displaystyle \frac{1}{2}}+i\tau }(\cosh x)F(\tau )d\tau ,x>0.\end{array}$$

(5)

The case $m=n$, $\Re m<{\displaystyle \frac{1}{2}}$ is considered in [14], Theorem 7, p. 247:

$$\begin{array}{c}\hfill F_m(\tau )={\int }_0^{\infty }f(x)P_{-{\displaystyle \frac{1}{2}}+i\tau }^m(\cosh x)\sinh xdx,\tau >0.\end{array}$$

(6)

and a corresponding inversion formula given by

$$\begin{array}{ccc}\hfill f(x)& =& {\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }\tau \sinh (\pi \tau )\Gamma \left({\displaystyle \frac{1}{2}}-m+i\tau \right)\Gamma \left({\displaystyle \frac{1}{2}}-m-i\tau \right)\hfill \\ & \times & P_{-{\displaystyle \frac{1}{2}}+i\tau }^m(\cosh x)F_m(\tau )d\tau ,x>0.\hfill \end{array}$$

(7)

From [15], Formula (11.1.8), p. 345,

$$\begin{array}{c}\hfill P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)=O(x^{-\Re m})\mathrm{as}x\to 0^{+}.\end{array}$$

(8)

From [15], Formula (11.1.9), p. 345,

$$\begin{array}{c}\hfill P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)=O(e^{-{\displaystyle \frac{x}{2}}})\mathrm{as}x\to +\infty .\end{array}$$

(9)

Thus one has

$$\begin{array}{ccc}& & P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x=O(x^{1-\Re m})\mathrm{as}x\to 0^{+},\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x=O(e^{{\displaystyle \frac{x}{2}}})\mathrm{as}x\to +\infty .\hfill \end{array}$$

(11)

From [15], Formula (11.3.2), p. 347,

$$\begin{array}{c}\hfill \left|P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\right|\le C\sqrt{{\displaystyle \frac{\pi }{2}}}\Gamma \left({\displaystyle \frac{1}{2}}-\Re m\right)P_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x),\end{array}$$

(12)

where C is a constant independent of x and $\tau$, $\Re m<{\displaystyle \frac{1}{2}}$, $x>0$, $\tau >0$.

Now, observe that from (11), the convergence of the integral

$$\begin{array}{c}\hfill {\int }_0^{\infty }{P}_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x)\sinh xdx\end{array}$$

is not ensured in ∞.

For $\gamma \in \mathbb{R}$, we consider the vector space ${\mathcal{E}}_{\gamma }$ consisting of all complex-valued measurable functions f on ${\mathbb{R}}_{+}$ such that $e^{\gamma x}f(x)\in L^{\infty }({\mathbb{R}}_{+})$. In their recent study [25], Maan and Negrín made extensive use of spaces of type ${\mathcal{E}}_{\gamma }$.

A norm ${\parallel ·\parallel }_{\gamma }$ on ${\mathcal{E}}_{\gamma }$ is given by

$$\begin{array}{c}\hfill {\parallel f\parallel }_{\gamma }={\parallel e^{\gamma x}f(x)\parallel }_{L^{\infty }({\mathbb{R}}_{+})}.\end{array}$$

With this norm, the map

$$\begin{array}{c}\hfill T_{\gamma }:{\mathcal{E}}_{\gamma }\to L^{\infty }({\mathbb{R}}_{+})\end{array}$$

where, for any $f\in {\mathcal{E}}_{\gamma }$,

$$\begin{array}{c}\hfill \left(T_{\gamma }f\right)(x)=e^{\gamma x}f(x),x\in {\mathbb{R}}_{+},\end{array}$$

is an isometric isomorphism from ${\mathcal{E}}_{\gamma }$ to $L^{\infty }({\mathbb{R}}_{+})$. Thus, since $L^{\infty }({\mathbb{R}}_{+})$ is complete, then the space ${\mathcal{E}}_{\gamma }$ becomes a Banach space.

The $C_c^k({\mathbb{R}}_{+}),k\in \mathbb{N}$, denotes, as is usual, the space of compactly supported functions on ${\mathbb{R}}_{+}$ which are k-times differentiable with continuity.

The content of this paper is as follows: Section 1 is concerned with the definitions and useful results that are used throughout. Section 2 provides an inversion formula for the generalized Mehler–Fock transform given by (1). Section 3 deals with the continuity features over Lebesgue spaces and Parseval–Goldstein-type relations for the generalized Mehler–Fock transform given by (1). Section 4 gives concluding remarks.

2. An Inversion Formula over the Spaces ${\mathcal{E}}_{\gamma }$

By means of [14], Theorem 5, p.245 one obtains the next inversion formula for the transform (1) over the spaces ${\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$. As a consequence, we obtain the injectivity of the transform (1) over a subset of the space ${\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$.

Theorem 1

(Inversion formula). Assume $f\in {\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$, and let $m,n$ be complex numbers with $|\Re n|<1-\Re m$ and $-{\displaystyle \frac{3}{2}}<\Re m<{\displaystyle \frac{1}{2}}$. Suppose that the function f is continuous at the point x and of bounded variation in the neighborhood of the point x. Then.

$$\begin{array}{c}\hfill {\int }_0^{\infty }\chi (\tau )P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)({\mathcal{B}}_{m,n}f)(\tau )d\tau =f(x),\end{array}$$

where $\chi (\tau )$ is given by (3).

Proof. 

Observe that when $f\in {\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$, the function $g(t)=f(\log (t+\sqrt{t^2-1})),t>1$, satisfies the condition (0.6) of p. 236 of [14].

Also, the function g is continuous at the point $\cosh x$ and of bounded variation in the neighborhood of the point $\cosh x$.

Thus, from Theorem 5 of [14], one has

$$\begin{array}{c}\hfill {\int }_0^{\infty }\chi (\tau )P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x){\int }_1^{\infty }g(t)P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(t)dtd\tau =g(\cosh x).\end{array}$$

(13)

Now, taking in (13) $t=\cosh u,u>0$, and taking into account that $g(\cosh u)=f(u)$, one obtains

$$\begin{array}{c}\hfill {\int }_0^{\infty }\chi (\tau )P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x){\int }_0^{\infty }f(u)P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh u)\sinh udud\tau =f(x).\end{array}$$

□

As a consequence of Theorem 1, one obtains the next result.

Corollary 1

(Injectivity). Assume that the functions $f,g\in {\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$, and let $m,n$ be complex numbers with $|\Re n|<1-\Re m$ and $-{\displaystyle \frac{3}{2}}<\Re m<{\displaystyle \frac{1}{2}}$. Suppose that the functions f and g are continuous on ${\mathbb{R}}_{+}$ and of bounded variation in the neighborhood of each point of ${\mathbb{R}}_{+}$. Then, if ${\mathcal{B}}_{m,n}f={\mathcal{B}}_{m,n}g$ on ${\mathbb{R}}_{+}$, it follows that $f=g$ on ${\mathbb{R}}_{+}$.

3. Boundedness Properties and Parseval–Goldstein-Type Relations over the Spaces ${\mathcal{E}}_{\gamma }$

In this section, we investigate the generalized Mehler–Fock transform given by (1) over the spaces ${\mathcal{E}}_{\gamma }$. This exploration yields a Parseval–Goldstein-type relation as a significant outcome.

Observe that for $m,n$, being complex numbers with $\Re m<{\displaystyle \frac{1}{2}}$, $f\in {\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$, $\tau >0$, one obtains from (12)

$$\begin{array}{ccc}\hfill & & |({\mathcal{B}}_{m,n}f)(\tau )|\hfill \\ \hfill & \le & {\parallel f\parallel }_{\gamma }{\int }_0^{\infty }{e}^{-\gamma x}f(x)|P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)|\sinh xdx\hfill \\ & \le & {\parallel f\parallel }_{\gamma }C\sqrt{{\displaystyle \frac{\pi }{2}}}\Gamma \left({\displaystyle \frac{1}{2}}-\Re m\right){\int }_0^{\infty }{e}^{-\gamma x}{P}_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x)\sinh xdx.\hfill \end{array}$$

(14)

Now, from (10) and (11), the integral

$$\begin{array}{c}\hfill {\int }_0^{\infty }{e}^{-\gamma x}{P}_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x)\sinh xdx\end{array}$$

converges for $\gamma >{\displaystyle \frac{1}{2}}$.

Thus, the next result holds.

Proposition 1.

Set $m,n$ as complex numbers with $\Re m<{\displaystyle \frac{1}{2}}$. The generalized Mehler–Fock transform ${\mathcal{B}}_{m,n}$ given by (1) is a bounded linear operator from ${\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$, into $L^{\infty }({\mathbb{R}}_{+})$. If $f\in {\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$, then

$$\begin{array}{c}\hfill \parallel {\mathcal{B}}_{m,n}{f\parallel }_{L^{\infty }({\mathbb{R}}_{+})}\le M{\parallel f\parallel }_{\gamma },for someM>0,\end{array}$$

and ${\mathcal{B}}_{m,n}f$ is a continuous function on ${\mathbb{R}}_{+}$. Moreover, the generalized Mehler–Fock transform ${\mathcal{B}}_{m,n}$ is a continuous map from ${\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$, to the Banach space of bounded continuous functions on ${\mathbb{R}}_{+}$.

Proof. 

Let ${\tau }_0>0$ be arbitrary. Since the map

$$\begin{array}{c}\hfill \tau \to P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x\end{array}$$

is continuous for each fixed $x>0$, we have

$$\begin{array}{c}\hfill P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x\to P_{-{\displaystyle \frac{1}{2}}+i{\tau }_0}^{m,n}(\cosh x)\sinh xas\tau \to {\tau }_0.\end{array}$$

Further, we have that

$$\begin{array}{c}\hfill \left|P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x-P_{-{\displaystyle \frac{1}{2}}+i{\tau }_0}^{m,n}(\cosh x)\sinh x\right||f(x)|\end{array}$$

is dominated by the integrable function

$$\begin{array}{c}\hfill 2C\sqrt{{\displaystyle \frac{\pi }{2}}}\Gamma \left({\displaystyle \frac{1}{2}}-\Re m\right)P_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x)\sinh x|f(x)|.\end{array}$$

Therefore, by using dominated convergence theorem, we obtain

$$\begin{array}{ccc}& & \left|({\mathcal{B}}_{m,n}f)(\tau )-({\mathcal{B}}_{m,n}f)({\tau }_0)\right|\hfill \\ & & \le {\int }_0^{\infty }\left|P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x-P_{-{\displaystyle \frac{1}{2}}+i{\tau }_0}^{m,n}(\cosh x)\sinh x\right||f(x)|dx,\hfill \end{array}$$

which tends to 0 as $\tau \to {\tau }_0$.

Thus, ${\mathcal{B}}_{m,n}f$ is a continuous function on ${\mathbb{R}}_{+}$.

Since for each $\tau >0$ and from (14) one has

$$\begin{array}{ccc}\hfill & & |({\mathcal{B}}_{m,n}f)(\tau )|\hfill \\ \hfill & \le & {\parallel f\parallel }_{\gamma }C\sqrt{{\displaystyle \frac{\pi }{2}}}\Gamma \left({\displaystyle \frac{1}{2}}-\Re m\right){\int }_0^{\infty }{e}^{-\gamma x}{P}_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x)\sinh xdx\hfill \\ & =& {M\parallel f\parallel }_{\gamma },\mathrm{for} \mathrm{some}M>0,\hfill \end{array}$$

(15)

and then ${\mathcal{B}}_{m,n}f$ is a bounded function.

The linearity of the integral operator implies that the ${\mathcal{B}}_{m,n}$ transform is linear. Also, from (15), we obtain that

$$\begin{array}{c}\hfill \parallel {\mathcal{B}}_{m,n}{f\parallel }_{L^{\infty }({\mathbb{R}}_{+})}\le M{\parallel f\parallel }_{\gamma }\end{array}$$

and hence

$$\begin{array}{c}\hfill {\mathcal{B}}_{m,n}:{\mathcal{E}}_{\gamma }\to L^{\infty }({\mathbb{R}}_{+}),\gamma >{\displaystyle \frac{1}{2}},\end{array}$$

is a continuous linear map. □

Also, the next result holds.

Proposition 2.

Let $m,n$ be complex numbers with $\Re m<{\displaystyle \frac{1}{2}}$, $\gamma >{\displaystyle \frac{1}{2}}$, w be a measurable function on ${\mathbb{R}}_{+}$ such that $w>0$ a.e. on ${\mathbb{R}}_{+}$ and ${\int }_0^{\infty }w(x)dx<\infty$. Then, for $0<q<\infty$,

$$\begin{array}{c}\hfill {\mathcal{B}}_{m,n}:{\mathcal{E}}_{\gamma }\to L^q({\mathbb{R}}_{+},w(x)dx)\end{array}$$

is a bounded linear operator.

Example 1.

Examples of weights w for Proposition 2 are:

$$\begin{array}{ccc}& & (\mathrm{i})w(x)={(1+x)}^r,for r<-1.\hfill \\ & & (\mathrm{ii})w(x)=e^{rx},for r<0.\hfill \end{array}$$

For a suitable function g we denote

$$\begin{array}{c}\hfill ({\mathcal{B}}_{m,n}^{\ast }g)(x)=\sinh x{\int }_0^{\infty }g(\tau )P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)d\tau ,x>0,\end{array}$$

(16)

where m and n are complex numbers with $\Re m<{\displaystyle \frac{1}{2}}$.

The ${\mathcal{B}}_{m,n}^{\ast }g$ exists for $g\in L^1\left({\mathbb{R}}_{+}\right)$.

In fact, from (12)

$$\begin{array}{ccc}\hfill & & |({\mathcal{B}}_{m,n}^{\ast }g)(x)|\hfill \\ \hfill & \le & \sinh x{\int }_0^{\infty }|g(\tau )||P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)|d\tau \hfill \\ & \le & \sinh xC\sqrt{{\displaystyle \frac{\pi }{2}}}\Gamma \left({\displaystyle \frac{1}{2}}-\Re m\right)P_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x){\int }_0^{\infty }|g(\tau )|d\tau <\infty ,\hfill \end{array}$$

(17)

for each $x\in {\mathbb{R}}_{+}$.

Now, observe that for $\gamma \le -{\displaystyle \frac{1}{2}}$ and taking into account (10) and (11) the expression

$$\begin{array}{c}\hfill e^{\gamma x}{P}_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x)\sinh x\le A,\mathrm{for} \mathrm{all}x>0\mathrm{and} \mathrm{some}A>0.\end{array}$$

Thus, for $g\in L^1({\mathbb{R}}_{+})$, one has

$$\begin{array}{c}\hfill \parallel {\mathcal{B}}_{m,n}^{\ast }{g\parallel }_{\gamma }\le A·{\parallel g\parallel }_{L^1({\mathbb{R}}_{+})},\gamma \le -{\displaystyle \frac{1}{2}}.\end{array}$$

So one obtains the next result.

Proposition 3.

Let $m,n$ be complex numbers with $\Re m<{\displaystyle \frac{1}{2}}$. The linear operator ${\mathcal{B}}_{m,n}^{\ast }$ given by (16) is a bounded linear operator from $L^1({\mathbb{R}}_{+})$ into ${\mathcal{E}}_{\gamma }$, $\gamma \le -{\displaystyle \frac{1}{2}}$.

The next result holds.

Proposition 4.

Let $m,n$ be complex numbers with $\Re m<{\displaystyle \frac{1}{2}}$ and let w be a measurable function on ${\mathbb{R}}_{+}$ such that $w>0$ a.e. on ${\mathbb{R}}_{+}$ and $0<q<\infty$. If ${\int }_0^{\infty }{\left(P_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x)\sinh x\right)}^{q}w(x)dx<\infty$. Then,

$$\begin{array}{c}\hfill {\mathcal{B}}_{m,n}^{\ast }:L^1({\mathbb{R}}_{+})\to L^q({\mathbb{R}}_{+},w(x)dx)\end{array}$$

is a bounded linear operator.

Proof. 

From (17), one has

$$\begin{array}{ccc}& & {\left({\int }_0^{\infty }{|({\mathcal{B}}_{m,n}^{\ast }g)(x)|}^{q}w(x)dx\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & \le M·{\parallel g\parallel }_{L^1({\mathbb{R}}_{+})}·{\left({\int }_0^{\infty }{\left(P_{-{\displaystyle \frac{1}{2}}}^{\Re m,0}(\cosh x)\sinh x\right)}^{q}w(x)dx\right)}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

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

Example 2.

Examples of weights w for Proposition 4 are

$$\begin{array}{ccc}& & (\mathrm{i})w(x)=e^{rx},for r<-{\displaystyle \frac{q}{2}}.\hfill \\ & & (\mathrm{ii})w(x)=e^{r{x}^2},for r<0.\hfill \end{array}$$

The next Parseval–Goldstein-type relation holds.

Theorem 2.

Let $m,n$ be complex numbers with $\Re m<{\displaystyle \frac{1}{2}}$, $f\in {\mathcal{E}}_{\gamma }$, $\gamma >{\displaystyle \frac{1}{2}}$ and $g\in L^1({\mathbb{R}}_{+})$; then, the following Parseval–Goldstein-type relation holds

$$\begin{array}{c}\hfill {\int }_0^{\infty }({\mathcal{B}}_{m,n}f)(x)g(x)dx={\int }_0^{\infty }f(x)({\mathcal{B}}_{m,n}^{\ast }g)(x)dx.\end{array}$$

(18)

Proof. 

Applying Fubini’s theorem in the following, we obtain

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }\left({\mathcal{B}}_{m,n}f\right)(\tau )g(\tau )d\tau & & ={\int }_0^{\infty }{\int }_0^{\infty }f(x)P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh xdxg(\tau )d\tau \hfill \\ & & ={\int }_0^{\infty }\sinh x{\int }_0^{\infty }g(\tau )P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)d\tau f(x)dx\hfill \\ & & ={\int }_0^{\infty }f(x)\left({\mathcal{B}}_{m,n}^{\ast }g\right)(x)dx.\hfill \end{array}$$

Observe that from (17), the ${\mathcal{B}}_{m,n}^{\ast }g$ exists for $g\in L^1\left({\mathbb{R}}_{+}\right)$. □

Consider the differential operator

$$\begin{array}{c}\hfill L_{m,n,x}=D_x\sinh x{D}_x{(\sinh x)}^{-1}+\left[{\displaystyle \frac{m^2}{2(1-\cosh x)}}+{\displaystyle \frac{n^2}{2(1+\cosh x)}}\right].\end{array}$$

From [15], Formula (11.2.4), p. 346

$$\begin{array}{c}\hfill L_{m,n,x}\left(P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x\right)=-\left({\tau }^2+{\displaystyle \frac{1}{4}}\right)P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x,\end{array}$$

(19)

and then for $k\in \mathbb{N}$,

$$\begin{array}{c}\hfill L_{m,n,x}^k\left(P_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x\right)={(-1)}^k{\left({\tau }^2+{\displaystyle \frac{1}{4}}\right)}^k{P}_{-{\displaystyle \frac{1}{2}}+i\tau }^{m,n}(\cosh x)\sinh x.\end{array}$$

(20)

Denote

$$\begin{array}{c}\hfill L_{m,n,x}^{\ast }={(\sinh x)}^{-1}{D}_x\sinh x{D}_x+\left[{\displaystyle \frac{m^2}{2(1-\cosh x)}}+{\displaystyle \frac{n^2}{2(1+\cosh x)}}\right].\end{array}$$

From (20) and $f\in C_c^{2k}({\mathbb{R}}_{+})$, $k\in \mathbb{N}$, it follows that

$$\begin{array}{c}\hfill \left({\mathcal{B}}_{m,n}\left(L_{m,n,x}^{{\ast }^k}f\right)\right)(\tau )={(-1)}^k{\left({\tau }^2+{\displaystyle \frac{1}{4}}\right)}^k({\mathcal{B}}_{m,n}f)(\tau ),\tau >0.\end{array}$$

(21)

Then, from Theorem 2, one has the next result.

Theorem 3.

If $f\in C_c^{2k}({\mathbb{R}}_{+})$, $k\in \mathbb{N}$, $m,n$ are complex numbers with $\Re m<{\displaystyle \frac{1}{2}}$, and $g\in L^1({\mathbb{R}}_{+})$; then, the following Parseval–Goldstein-type relation holds

$$\begin{array}{c}\hfill {(-1)}^k{\int }_0^{\infty }({\mathcal{B}}_{m,n}f)(x)g(x){\left(x^2+{\displaystyle \frac{1}{4}}\right)}^{k}dx={\int }_0^{\infty }(L_{m,n,x}^{{\ast }^k}f)(x)({\mathcal{B}}_{m,n}^{\ast }g)(x)dx.\end{array}$$

(22)

4. Final Observations and Conclusions

This paper has been dedicated to the establishment of boundedness properties and Parseval–Goldstein-type relations for the generalized Mehler–Fock transform, as first introduced by B. L. J. Braaksma and B. M. Meulenbeld in 1967 and subsequently by R. S. Pathak in 1997. This study highlights the importance of the Parseval–Goldstein relations, revealing how the generalized Mehler–Fock transform preserves energy and maintains consistency across different domains. Moreover, we have derived an inversion formula for this transform across Lebesgue spaces. These results not only contribute to a deeper understanding of the generalized Mehler–Fock transform but also pave the way for future investigations into analogous properties within the spaces of type ${\mathcal{E}}_{\gamma }$ for diverse integral transforms.