Parseval–Goldstein Identities and Abelian Theorems for the Hartley Transform over Distributions of Compact Support and Generalized Functions

Abstract

This work develops an analytical framework for the Hartley transform, which, unlike the Fourier transform, converts real-valued functions into real-valued functions. We derive Parseval–Goldstein-type identities for suitable classes of functions and establish Abelian theorems in the setting of compactly supported distributions and generalized functions. Moreover, a finite formulation of the Hartley transform is constructed and analyzed within the spaces of distributions of compact support and generalized functions.

1. Introduction and Preliminaries

Integral transforms constitute an essential tool in modern mathematical analysis and continue to play a central role in the study of generalized functions and spaces of distributions. The systematic extension of classical integral transforms to distribution spaces was developed in the pioneering work of Zemanian [1,2], where Abelian-type relations for distributional Hankel and related transforms were established within a rigorous functional–analytic framework. These developments are closely connected with the structural theory of distributions presented in the classical monograph of Horváth [3]. Standard references, such as those of Sneddon [4], Churchill [5], and Bracewell [6], further illustrate the importance of transform techniques in harmonic analysis, applied mathematics, engineering problems and signal processing.

Abelian and Tauberian theorems form an important component of asymptotic transform theory since they describe how the local behaviour of functions or distributions influences the asymptotic structure of their transforms. Significant contributions in this direction include the Abelian theorems for distributional Hankel and related transforms obtained by Zemanian [1], together with the extensions to generalized hypergeometric transforms studied by Hayek and González [7]. More recently, Abelian-type results for several classical transforms over compactly supported distributions and generalized function spaces were established by Maan and Prasad [8] and by Maan and Negrín [9]. Related Abelian and Tauberian relations connecting the local behaviour of functions with the asymptotic properties of their Fourier transforms were investigated earlier by Cline [10]. Additional progress in this direction includes recent distributional Abelian and Tauberian results for the fractional Fourier transform and the short-time Fourier transform obtained by Atanasova, Maksimović and Pilipović [11].

Alongside these developments, Parseval–Goldstein-type identities have been widely studied because of their usefulness in establishing transform-domain energy relations and structural representations. Such identities were obtained for several transform classes, including the $L_4$-, $P_4$-, $L_{s,2}$- and $F_{c,2}$-transforms, by Dernek, Srivastava and Yürekli [12,13]. Related transform identities involving the Hankel and Glasser transforms were later investigated by Dernek and coauthors [14,15]. More recently, Parseval–Goldstein-type theorems have been extended to the Dunkl transform [16], the Sturm–Liouville transform [17], and various generalized integral transforms considered by Albayrak [18] and by Karataş, Albayrak and Uçar [19]. These results clearly demonstrate the continuing relevance of Parseval-type identities in modern transform analysis.

Within this broad framework, the Hartley transform occupies a distinctive position as a real-valued counterpart of the Fourier transform. Introduced originally by Hartley [20] as a symmetric alternative to complex exponential representations in spectral analysis, the transform preserves many structural features of the Fourier transform while avoiding complex-valued kernels. Its theoretical development and applications were presented systematically in the classical monograph of Bracewell [21]. Because of its symmetry and computational efficiency, the Hartley transform has found important applications in signal processing, numerical harmonic analysis and spectral estimation.

Due to its close relationship with the Fourier transform, it is natural to compare asymptotic-transform properties of the Hartley transform with their Fourier-transform counterparts. Although Abelian and Tauberian theorems for the Fourier transform have been extensively studied in the literature, corresponding results for the Hartley transform require independent investigation because of differences in kernel symmetry, involution structure and derivative interaction properties. Recent progress in this direction includes Titchmarsh-type results for the Hartley transform obtained by Bouzeffour [22], which further illustrate the growing analytical interest in Hartley-transform-based techniques.

Despite these advances, distributional extensions of the Hartley transform, together with associated Abelian-type and final value theorems, remain comparatively less explored, particularly in the framework of compactly supported distributions and generalized function spaces. The present work is therefore devoted to developing Parseval–Goldstein-type identities, Abelian-type results and final-value theorems for the Hartley transform in suitable distributional settings. The results obtained here extend classical asymptotic-transform correspondences to the Hartley transform and provide useful analytical tools for transform-domain analysis, spectral methods and signal-processing applications.

Among classical integral transforms, the Hartley transform deserves particular attention. For a function $f\in L^1\left(\mathbb{R}\right)$, it is defined by [21]

$$\left(\mathcal{H}f\right)\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{\infty }f\left(t\right)cas\left(xt\right)dt,x\in \mathbb{R},$$

where

$$cas\left(v\right)=cos\left(v\right)+sin\left(v\right),v\in \mathbb{R}.$$

The Hartley transform retains many structural similarities with the Fourier transform while remaining entirely real-valued, which simplifies both theoretical analysis and computational implementation in applications involving real signals. Moreover, it is an involutory transform satisfying the inversion relation [21]

$$f\left(t\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{\infty }\left(\mathcal{H}f\right)\left(x\right)cas\left(xt\right)dx,t\in \mathbb{R},$$

so that

$$\mathcal{H}\left(\mathcal{H}f\right)=f.$$

A key ingredient in the analysis is the behaviour of derivatives of the kernels $cos\left(xt\right)$ and $sin\left(xt\right)$. For $k\in \mathbb{N}\cup \left\{0\right\}$, one has

$$\begin{array}{cccc}\hfill D_t^{2k}cos\left(xt\right)& ={(-1)}^k{x}^{2k}cos\left(xt\right),\hfill & \hfill D_t^{2k}sin\left(xt\right)& ={(-1)}^k{x}^{2k}sin\left(xt\right),\hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill D_t^{2k+1}cos\left(xt\right)& ={(-1)}^{k+1}{x}^{2k+1}sin\left(xt\right),\hfill & \hfill D_t^{2k+1}sin\left(xt\right)& ={(-1)}^k{x}^{2k+1}cos\left(xt\right).\hfill \end{array}$$

(2)

From these relations, the derivatives of the Hartley kernel follow as

$$D_t^{2k}\left(cos\left(xt\right)+sin\left(xt\right)\right)={(-1)}^k{x}^{2k}\left(cos\left(xt\right)+sin\left(xt\right)\right),$$

$$D_t^{2k+1}\left(cos\left(xt\right)+sin\left(xt\right)\right)={(-1)}^k{x}^{2k+1}\left(cos\left(xt\right)-sin\left(xt\right)\right).$$

Using integration by parts and these identities, one obtains intertwining properties between differentiation in the t-variable and polynomial weights in the transform variable. In particular, for $f\in C_c^{2k}\left(\mathbb{R}\right)$ and $x\in \mathbb{R}$,

$$\left(\mathcal{H}\left(D_t^{2k}f\right)\right)\left(x\right)={(-1)}^k{x}^{2k}\left(\mathcal{H}f\right)\left(x\right).$$

and more generally,

$$\left(\mathcal{H}\left(D_t^{2k+1}f\right)\right)\left(x\right)={(-1)}^k{x}^{2k+1}\left({\mathcal{H}}^{\perp }f\right)\left(x\right),$$

where

$$\left({\mathcal{H}}^{\perp }f\right)\left(x\right):={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{\infty }f\left(t\right)cas(-xt)dt.$$

Such relations are fundamental in deriving Parseval–Goldstein-type identities and Abelian theorems associated with the Hartley transform.

For distributional formulations, it is convenient to work in the test function space $\mathcal{E}\left(\mathbb{R}\right)$, consisting of smooth complex-valued functions with the locally convex topology generated by seminorms

$${\rho }_{k,K}\left(\sigma \right)=\underset{t\in K}{sup}\left|D_t^k\sigma \left(t\right)\right|,$$

where $k\in \mathbb{N}\cup \left\{0\right\}$ and $K\subset \mathbb{R}$ is compact. Its continuous dual ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ consists of all compactly supported distributions. This framework allows one to rigorously extend the Hartley transforms and to establish limit theorems in the distributional sense.

The structure of the paper is as follows. Section 1 introduces the preliminaries and notations. Section 2 establishes Parseval–Goldstein-type identities for the Hartley transform. Section 3 presents Abelian theorems in the framework of distributions with compact support. Section 4 extends these results to spaces of generalized functions. Section 5 analyzes final-value theorems for the finite Hartley transform over distributions of compact support and generalized functions, and Section 6 concludes with a summary and possible directions for further research.

2. Parseval–Goldstein-Type Relations for the Hartley Transform

2.1. The Hartley Transform on $L^1\left(\mathbb{R}\right)$

Recall that the Hartley transform is defined by

$$\left(\mathcal{H}f\right)\left(x\right):={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{\infty }f\left(t\right)[cos\left(xt\right)+sin\left(xt\right)]dt,x\in \mathbb{R},$$

for $f\in L^1\left(\mathbb{R}\right)$.

Proposition 1.

The operator $\mathcal{H}:L^1\left(\mathbb{R}\right)\to L^{\infty }\left(\mathbb{R}\right)$ is linear and bounded. In fact, one has

$${\parallel \mathcal{H}f\parallel }_{L^{\infty }\left(\mathbb{R}\right)}\le \sqrt{{\displaystyle \frac{2}{\pi }}}{\parallel f\parallel }_{L^1\left(\mathbb{R}\right)}.$$

Furthermore, $\mathcal{H}f$ is continuous on $\mathbb{R}$.

Proof. 

Linearity follows immediately from the integral definition. For the norm estimate, note that for every real u,

$$|cosu+sinu|\le |cosu|+|sinu|\le 2.$$

Hence, for each $x\in \mathbb{R}$,

$$\left|\left(\mathcal{H}f\right)\left(x\right)\right|={\displaystyle \frac{1}{\sqrt{2\pi }}}\left|{\int }_{\mathbb{R}}f\left(t\right)[cos\left(xt\right)+sin\left(xt\right)]dt\right|\le {\displaystyle \frac{2}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}\left|f\left(t\right)\right|dt=\sqrt{{\displaystyle \frac{2}{\pi }}}{\parallel f\parallel }_1,$$

which proves boundedness (up to the constant factor).

To show continuity, fix $f\in L^1\left(\mathbb{R}\right)$ and let $x_n\to x$. For each fixed t, we have $[cos\left(x_{n}t\right)+sin\left(x_{n}t\right)]\to [cos\left(xt\right)+sin\left(xt\right)]$. Moreover,

$$|f\left(t\right)[cos\left(x_{n}t\right)+sin\left(x_{n}t\right)]|\le 2|f\left(t\right)|,$$

and since $f\in L^1\left(\mathbb{R}\right)$, the dominated convergence theorem implies that

$$\underset{n\to \infty }{lim}\left(\mathcal{H}f\right)\left(x_n\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}f\left(t\right)[cos\left(xt\right)+sin\left(xt\right)]dt=\left(\mathcal{H}f\right)\left(x\right).$$

Hence, $\mathcal{H}f$ is continuous on $\mathbb{R}$. □

Proposition 2.

Let $0<q<\infty$ and let $w\in L^1\left(\mathbb{R}\right)$, $w>0$ almost everywhere on $\mathbb{R}$. Then, for any $f\in L^1\left(\mathbb{R}\right)$,

$${\parallel \mathcal{H}f\parallel }_{L^q\left(wdx\right)}\le \sqrt{{\displaystyle \frac{2}{\pi }}}{\parallel f\parallel }_{L^1\left(\mathbb{R}\right)}{\parallel w\parallel }_{L^1\left(\mathbb{R}\right)}^{1/q}.$$

Proof. 

By Proposition 1, we have $|\left(\mathcal{H}f\right)\left(x\right)|\le \sqrt{{\displaystyle \frac{2}{\pi }}}{\parallel f\parallel }_1$ for all $x\in \mathbb{R}$. Hence,

$${\int }_{\mathbb{R}}|\left(\mathcal{H}f\right){\left(x\right)|}^{q}w\left(x\right)dx\le {\left(\sqrt{{\displaystyle \frac{2}{\pi }}}{\parallel f\parallel }_1\right)}^q{\int }_{\mathbb{R}}w\left(x\right)dx={\left(\sqrt{{\displaystyle \frac{2}{\pi }}}{\parallel f\parallel }_1\right)}^q{\parallel w\parallel }_1,$$

and taking the q-th root yields the desired result. □

2.2. Parseval–Goldstein-Type Relations

The proofs below follow the standard route of Fubini’s theorem and integration by parts.

Theorem 1.

For $f,g\in L^1\left(\mathbb{R}\right)$, the (unweighted) Parseval–Goldstein identity for the Hartley transform holds:

$${\int }_{\mathbb{R}}\left(\mathcal{H}f\right)\left(x\right)g\left(x\right)dx={\int }_{\mathbb{R}}f\left(t\right)\left(\mathcal{H}g\right)\left(t\right)dt.$$

Proof. 

Set

$$I:={\int \int }_{{\mathbb{R}}^2}\left|f\left(t\right)\right|\left|g\left(x\right)\right||cos\left(xt\right)+sin\left(xt\right)|dxdt.$$

Since $|cos(xt)+sin(xt\left)\right|\le 2$, we have

$$I\le {2\parallel f\parallel }_1{\parallel g\parallel }_1<\infty ,$$

so Fubini’s theorem applies and the order of integration may be interchanged. Therefore,

$$\begin{array}{cc}\hfill {\int }_{\mathbb{R}}\left(\mathcal{H}f\right)\left(x\right)g\left(x\right)dx& ={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}\left({\int }_{\mathbb{R}}f\left(t\right)[cos\left(xt\right)+sin\left(xt\right)]dt\right)g\left(x\right)dx\hfill \\ & ={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}f\left(t\right)g\left(x\right)[cos\left(xt\right)+sin\left(xt\right)]dtdx\hfill \\ & ={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}f\left(t\right)\left({\int }_{\mathbb{R}}g\left(x\right)[cos\left(tx\right)+sin\left(tx\right)]dx\right)dt\hfill \\ & ={\int }_{\mathbb{R}}f\left(t\right)\left(\mathcal{H}g\right)\left(t\right)dt,\hfill \end{array}$$

as required. □

Example 1.

We verify Theorem 1 for the Gaussian functions

$$f\left(t\right)=e^{a{t}^2},g\left(t\right)=e^{b{t}^2},a<0,b<0,t\in \mathbb{R}.$$

Since both functions belong to $L^1\left(\mathbb{R}\right)$, their Hartley transforms exist and are given by

$$\left(\mathcal{H}f\right)\left(x\right)={\displaystyle \frac{e^{x^2/4a}}{\sqrt{-2a}}},\left(\mathcal{H}g\right)\left(x\right)={\displaystyle \frac{e^{x^2/4b}}{\sqrt{-2b}}},x\in \mathbb{R}.$$

Hence,

$${\int }_{\mathbb{R}}\left(\mathcal{H}f\right)\left(x\right)g\left(x\right)dx={\displaystyle \frac{1}{\sqrt{-2a}}}{\int }_{\mathbb{R}}exp\left({\displaystyle \frac{1+4ab}{4a}}{x}^2\right)dx=\sqrt{{\displaystyle \frac{2\pi }{1+4ab}}}.$$

Analogously,

$${\int }_{\mathbb{R}}f\left(x\right)\left(\mathcal{H}g\right)\left(x\right)dx=\sqrt{{\displaystyle \frac{2\pi }{1+4ab}}}.$$

Therefore,

$${\int }_{\mathbb{R}}\left(\mathcal{H}f\right)\left(x\right)g\left(x\right)dx={\int }_{\mathbb{R}}f\left(x\right)\left(\mathcal{H}g\right)\left(x\right)dx,$$

which verifies the Parseval–Goldstein relation of Theorem 1 for these Gaussian functions.

Lemma 1.

Let $K(x,t)=cos\left(xt\right)+sin\left(xt\right)$ and define the auxiliary kernel

$$K^{\perp }(x,t):=cos\left(xt\right)-sin\left(xt\right).$$

Then, for every $k\in {\mathbb{N}}_0$ and all $x,t\in \mathbb{R}$,

$$D_t^{2k}K(x,t)={(-1)}^k{x}^{2k}K(x,t),D_t^{2k+1}K(x,t)={(-1)}^k{x}^{2k+1}{K}^{\perp }(x,t).$$

Proof. 

The derivative relations for $cos\left(xt\right)$ and $sin\left(xt\right)$ are classical:

$$D_t^{2k}cos\left(xt\right)={(-1)}^k{x}^{2k}cos\left(xt\right),D_t^{2k+1}cos\left(xt\right)={(-1)}^{k+1}{x}^{2k+1}sin\left(xt\right),$$

and similar for $sin\left(xt\right)$. Applying these termwise to $K(x,t)=cos\left(xt\right)+sin\left(xt\right)$ yields

$$D_t^{2k}K(x,t)={(-1)}^k{x}^{2k}[cos\left(xt\right)+sin\left(xt\right)],$$

and for odd derivatives,

$$D_t^{2k+1}K(x,t)={(-1)}^k{x}^{2k+1}[cos\left(xt\right)-sin\left(xt\right)]={(-1)}^k{x}^{2k+1}{K}^{\perp }(x,t),$$

as claimed. □

Proposition 3.

Let $k\in {\mathbb{N}}_0$ and $f\in C_c^{2k}\left(\mathbb{R}\right)$. Then, for every $x\in \mathbb{R}$,

$$\left(\mathcal{H}\left(D_t^{2k}f\right)\right)\left(x\right)={(-1)}^k{x}^{2k}\left(\mathcal{H}f\right)\left(x\right).$$

and

$$\left(\mathcal{H}\left(D_t^{2k+1}f\right)\right)\left(x\right)={(-1)}^k{x}^{2k+1}\left({\mathcal{H}}^{\perp }f\right)\left(x\right),$$

where

$$\left({\mathcal{H}}^{\perp }f\right)\left(x\right):={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}f\left(t\right)cas(-xt)dt.$$

Proof. 

For the even-order identity, write

$$\left(\mathcal{H}\left(D_t^{2k}f\right)\right)\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}{D}_t^{2k}f\left(t\right)[cos\left(xt\right)+sin\left(xt\right)]dt.$$

Since f has compact support, integration by parts can be applied $2k$ times (all boundary terms vanish). Thus,

$${\int }_{\mathbb{R}}{D}_t^{2k}f\left(t\right)[cos\left(xt\right)+sin\left(xt\right)]dt={\int }_{\mathbb{R}}f\left(t\right)D_t^{2k}[cos\left(xt\right)+sin\left(xt\right)]dt.$$

By Lemma 1, $D_t^{2k}[cos\left(xt\right)+sin\left(xt\right)]={(-1)}^k{x}^{2k}[cos\left(xt\right)+sin\left(xt\right)]$; hence,

$$\left(\mathcal{H}\left(D_t^{2k}f\right)\right)\left(x\right)={(-1)}^k{x}^{2k}{\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}f\left(t\right)[cos\left(xt\right)+sin\left(xt\right)]dt={(-1)}^k{x}^{2k}\left(\mathcal{H}f\right)\left(x\right),$$

as required.

For the odd-order case, integrating by parts $2k+1$ times and using Lemma 1, one has

$$\begin{array}{ccc}\hfill \left(\mathcal{H}\left(D_t^{2k+1}f\right)\right)\left(x\right)& =& {(-1)}^k{x}^{2k+1}{\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}f\left(t\right)[cos\left(xt\right)-sin\left(xt\right)]dt\hfill \\ & =& {(-1)}^k{x}^{2k+1}\left({\mathcal{H}}^{\perp }f\right)\left(x\right).\hfill \end{array}$$

□

Theorem 2

(Weighted Parseval–Goldstein Identities). Let $k\in {\mathbb{N}}_0$. If $f\in C_c^{2k}\left(\mathbb{R}\right)$ and $g\in L^1\left(\mathbb{R}\right)$, then

$${(-1)}^k{\int }_{\mathbb{R}}\left(\mathcal{H}f\right)\left(x\right)g\left(x\right)x^{2k}dx={\int }_{\mathbb{R}}\left(D_t^{2k}f\right)\left(t\right)\left(\mathcal{H}g\right)\left(t\right)dt.$$

and

$${(-1)}^k{\int }_{\mathbb{R}}\left(\mathcal{H}f\right)\left(x\right)g\left(x\right)x^{2k+1}dx={\int }_{\mathbb{R}}\left(D_t^{2k+1}f\right)\left(t\right)\left({\mathcal{H}}^{\perp }g\right)\left(t\right)dt.$$

Proof. 

Applying Theorem 1 to the pair $(D^{2k}f,g)$; since $D^{2k}f\in L^1\left(\mathbb{R}\right)$ (compact support), we have

$${\int }_{\mathbb{R}}\left(\mathcal{H}\left(D^{2k}f\right)\right)\left(x\right)g\left(x\right)dx={\int }_{\mathbb{R}}\left(D^{2k}f\right)\left(t\right)\left(\mathcal{H}g\right)\left(t\right)dt.$$

Using Proposition 3, replacing $\left(\mathcal{H}\left(D^{2k}f\right)\right)\left(x\right)$ by ${(-1)}^k{x}^{2k}\left(\mathcal{H}f\right)\left(x\right)$, one obtains the even-order identity.

The odd-order identity follows similarly by applying Theorem 1 to $(D^{2k+1}f,g)$ and using the intertwining relation $\mathcal{H}\left(D_t^{2k+1}f\right)={(-1)}^k{x}^{2k+1}{\mathcal{H}}^{\perp }f$; then, interchanging the roles of f and g, we get the desired form with ${\mathcal{H}}^{\perp }g$. □

3. Abelian Theorems for the Distributional Hartley Transform

The results in the previous section hold under the compact support assumption on f. In the framework of distributions, these identities extend by duality: the integration by parts steps remain valid in ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$, and the Hartley transform extends naturally to spaces of compactly supported distributions and generalized functions. In such cases, the smoothness assumptions on f are replaced by distributional regularity and moment conditions, and the integrals are interpreted as duality pairings.

We work with compactly supported distributions $f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$. For such f, define the (distributional) Hartley transform

$$\left(\mathcal{H}f\right)\left(x\right):=\langle f\left(t\right),cas\left(xt\right)\rangle ,x\in \mathbb{R},cas\left(u\right):=cosu+sinu,u\in \mathbb{R}.$$

Lemma 2.

Let $f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$. Then there exist constants $C>0$ and $r\in {\mathbb{N}}_0$ (the order of f) such that

$$\left|\left(\mathcal{H}f\right)\left(x\right)\right|\le C{(1+|x\left|\right)}^r,x\in \mathbb{R}.$$

Proof. 

By definition of ${\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$, there exist a compact set $K\subset \mathbb{R}$, an integer $r\ge 0$ (the order of f), and a constant $C_0>0$, such that for every test function, $\phi \in \mathcal{E}\left(\mathbb{R}\right)$,

$$\left|\langle f,\phi \rangle \right|\le C_0\underset{0\le j\le r}{max}\underset{t\in K}{max}\left|D_t^j\phi \left(t\right)\right|.$$

Observe that for each $j\ge 0$, one has

$$D_t^{j}cas\left(xt\right)=x^j\left({\alpha }_{j}cos\left(xt\right)+{\beta }_{j}sin\left(xt\right)\right),$$

where ${\alpha }_j,{\beta }_j\in \{1,-1\}$ depend only on the parity of j. Hence,

$$|D_t^{j}cas\left(xt\right)|\le 2{\left|x\right|}^j.$$

Since K is compact, the supremum over $t\in K$ of the trigonometric factors is bounded by 1. Therefore,

$$\underset{0\le j\le r}{max}\underset{t\in K}{max}|D_t^{j}cas\left(xt\right)|\le 2\underset{0\le j\le r}{max}{\left|x\right|}^j\le 2{(1+|x\left|\right)}^r.$$

Combining this with the continuity estimate for f gives

$$\left|\left(\mathcal{H}f\right)\left(x\right)\right|=\left|\langle f,cas\left(xt\right)\rangle \right|\le 2C_0{(1+|x\left|\right)}^r.$$

Take $C:=2C_0$ to conclude the proof. □

Theorem 3

(Abelian theorems). Let $f\in {\mathcal{E}}^{\prime }\left(\mathbb{R}\right)$ be of order r. Then for every $\alpha >0$, the following hold:

$$\underset{x\to 0}{lim}\left\{x^{\alpha }\left(\mathcal{H}f\right)\left(x\right)\right\}=0,\underset{\left|x\right|\to \infty }{lim}\left\{{\left|x\right|}^{-r-\alpha }\left(\mathcal{H}f\right)\left(x\right)\right\}=0.$$

Proof. 

By Lemma 2, there exist constants $C>0$, such that

$$\left|\left(\mathcal{H}f\right)\left(x\right)\right|\le C{(1+|x\left|\right)}^r,x\in \mathbb{R}.$$

Limit as $x\to 0$,

$$\left|x^{\alpha }\left(\mathcal{H}f\right)\left(x\right)\right|\le {C\left|x\right|}^{\alpha }{(1+|x\left|\right)}^r.$$

Since $\alpha >0$, we have ${\left|x\right|}^{\alpha }\to 0$ as $x\to 0$, while ${(1+|x\left|\right)}^r\to 1$. Hence, the right-hand side tends to 0, proving ${\displaystyle \underset{x\to 0}{lim}\left\{x^{\alpha }\left(\mathcal{H}f\right)\left(x\right)\right\}=0.}$

Limit as $\left|x\right|\to \infty$,

$${\left|\right|x|}^{-r-\alpha }\left(\mathcal{H}f\right)\left(x\right)|\le {C\left|x\right|}^{-r-\alpha }{(1+|x\left|\right)}^r=C{\left|x\right|}^{-\alpha }{\left({\displaystyle \frac{1+\left|x\right|}{\left|x\right|}}\right)}^r.$$

For $\left|x\right|\ge 1$, we have ${\left(\frac{1+\left|x\right|}{\left|x\right|}\right)}^r\le 2^r$. Thus,

$${\left|\right|x|}^{-r-\alpha }\left(\mathcal{H}f\right)\left(x\right)|\le C{2}^r{\left|x\right|}^{-\alpha },$$

and since $\alpha >0$, the right-hand side tends to 0 as $\left|x\right|\to \infty$.

Therefore, ${\displaystyle \underset{\left|x\right|\to \infty }{lim}\left\{{\left|x\right|}^{-r-\alpha }\left(\mathcal{H}f\right)\left(x\right)\right\}=0,}$ as required. □

Example 2.

Consider the compactly supported distribution

$$f\left(t\right)=\delta \left(t\right),$$

the Dirac delta distribution at the origin. Then, its Hartley transform is

$$\left(\mathcal{H}f\right)\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}\langle \delta \left(t\right),cas\left(xt\right)\rangle ={\displaystyle \frac{1}{\sqrt{2\pi }}}cas\left(0\right)={\displaystyle \frac{1}{\sqrt{2\pi }}},x\in \mathbb{R}.$$

For $\alpha >0$, we obtain

$$\underset{x\to 0}{lim}\left\{x^{\alpha }\left(\mathcal{H}f\right)\left(x\right)\right\}=0,$$

and

$$\underset{\left|x\right|\to \infty }{lim}{\left\{\right|x|}^{-\alpha }\left(\mathcal{H}f\right)\left(x\right)\}=0.$$

Thus, taking into account that f is of order 0, the conclusions of Theorem 3 hold for this distribution.

Remark 1.

The same statements hold for regular distributions given by compactly supported $L^1$-functions. Indeed, if $f\in L^1\left(\mathbb{R}\right)$ has compact support, then the functional $T_f$ given by

$$\begin{array}{c}\hfill 〈T_f,\varphi 〉={\int }_0^{\infty }f\left(t\right)\varphi \left(t\right)dt,\varphi \in \mathcal{E}\left({\mathbb{R}}_{+}\right)\end{array}$$

is a member of ${\mathcal{E}}^{\prime }\left({\mathbb{R}}_{+}\right)$ of order 0.

Now, Theorem 3 yields to

$$\underset{x\to 0}{lim}\left\{x^{\alpha }\left(\mathcal{H}{T}_f\right)\left(x\right)\right\}=0,\underset{\left|x\right|\to \infty }{lim}\left\{{\left|x\right|}^{-\alpha }\left(\mathcal{H}{T}_f\right)\left(x\right)\right\}=0,for every\alpha >0.$$

4. Abelian Theorems for the Hartley Transform on ${\mathcal{L}}_{0,0}^{\prime }$

Let ${\mathcal{L}}_{0,0}$ denote Zemanian’s space of all $\sigma \in C^{\infty }\left(\mathbb{R}\right)$, such that

$${\gamma }_k\left(\sigma \right):=\underset{t\in \mathbb{R}}{sup}\left|D_t^k\sigma \left(t\right)\right|<\infty (k=0,1,2,\dots ).$$

It is a Fréchet space with seminorms ${\gamma }_k$ and its strong dual is denoted by ${\mathcal{L}}_{0,0}^{\prime }$.

For $f\in {\mathcal{L}}_{0,0}^{\prime }$ the distributional Hartley transform is defined by

$$\left(\mathcal{H}f\right)\left(x\right):=\langle f\left(t\right),cas\left(xt\right)\rangle ,x\in \mathbb{R},$$

where $cas\left(u\right):=cosu+sinu,u\in \mathbb{R}$.

Lemma 3.

There exist constants $C>0$ and $r\in {\mathbb{N}}_0$, such that

$$\left|\left(\mathcal{H}f\right)\left(x\right)\right|\le C{(1+|x\left|\right)}^r,x\in \mathbb{R},$$

for every $f\in {\mathcal{L}}_{0,0}^{\prime }$.

Proof. 

Since $f\in {\mathcal{L}}_{0,0}^{\prime }$ is continuous, there exist a constant $C_0>0$ and an integer $r\ge 0$ (called the order of f), such that

$$\left|\langle f,\sigma \rangle \right|\le C_0\underset{0\le k\le r}{max}{\gamma }_k\left(\sigma \right)\mathrm{for}\mathrm{all}\sigma \in {\mathcal{L}}_{0,0}.$$

Applying this to $\sigma \left(t\right)=cas\left(xt\right)=cos\left(xt\right)+sin\left(xt\right)$, for each $k\ge 0$, we have

$$D_t^{k}cas\left(xt\right)=x^k\left({\alpha }_{k}cos\left(xt\right)+{\beta }_{k}sin\left(xt\right)\right),$$

where ${\alpha }_k,{\beta }_k\in \{1,-1\}$. Hence

$$\underset{t\in \mathbb{R}}{sup}|D_t^{k}cas\left(xt\right)|\le 2{\left|x\right|}^k.$$

Therefore,

$$\left|\left(\mathcal{H}f\right)\left(x\right)\right|=\left|\langle f,cas\left(xt\right)\rangle \right|\le C_0\underset{0\le k\le r}{max}{\gamma }_k(cas\left(xt\right))\le 2C_0\underset{0\le k\le r}{max}{\left|x\right|}^k.$$

Finally,

$$\underset{0\le k\le r}{max}{\left|x\right|}^k\le {(1+|x\left|\right)}^r,$$

so with $C=2C_0$, the desired inequality follows. □

Theorem 4

(Abelian theorems for the Hartley transform on ${\mathcal{L}}_{0,0}^{\prime }$). Let $f\in {\mathcal{L}}_{0,0}^{\prime }$ be of order r. Then, for every $\alpha >0$,

$$\underset{x\to 0}{lim}\left\{x^{\alpha }\left(\mathcal{H}f\right)\left(x\right)\right\}=0,\underset{\left|x\right|\to \infty }{lim}\left\{{\left|x\right|}^{-r-\alpha }\left(\mathcal{H}f\right)\left(x\right)\right\}=0.$$

Proof. 

By Lemma 3, there exists $C>0$, such that

$$\left|\left(\mathcal{H}f\right)\left(x\right)\right|\le C{(1+|x\left|\right)}^r,x\in \mathbb{R}.$$

Limit as $x\to 0$. Fix $\alpha >0$. Then,

$$\left|x^{\alpha }\left(\mathcal{H}f\right)\left(x\right)\right|\le {C\left|x\right|}^{\alpha }{(1+|x\left|\right)}^r.$$

As $x\to 0$, ${\left|x\right|}^{\alpha }\to 0$ and ${(1+|x\left|\right)}^r\to 1$, so ${lim}_{x\to 0}\left\{x^{\alpha }\left(\mathcal{H}f\right)\left(x\right)\right\}=0$.

Limit as $\left|x\right|\to \infty$. For $\left|x\right|\ge 1$, ${(1+|x\left|\right)}^r\le 2^r{\left|x\right|}^r$; hence,

$${\left|\right|x|}^{-r-\alpha }\left(\mathcal{H}f\right)\left(x\right)|\le {C\left|x\right|}^{-r-\alpha }{(1+|x\left|\right)}^r\le C{2}^r{\left|x\right|}^{-\alpha }.$$

Since $\alpha >0$, this tends to 0 as $\left|x\right|\to \infty$. □

5. Final-Value Theorems for the Finite Hartley Transform over Distributions of Compact Support and Generalized Functions

In this section, and according to [4] (pp. 28, 425–426), we consider the finite version of the Hartley transform, which plays a role analogous to the finite Fourier, Fourier sine and Fourier cosine transforms, with the kernel:

$$cas\left(\frac{m\pi }{T}t\right)=cos\left(\frac{m\pi }{T}t\right)+sin\left(\frac{m\pi }{T}t\right),t\in (-T,T),T>0,m\in \mathbb{Z}.$$

Definition 1.

Let $T>0$. For $f\in L^1\left((-T,T)\right)$, the finite Hartley transform is defined by

$$\left({\mathcal{H}}_{T}f\right)\left(m\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-T}^{T}f\left(t\right)cas\left(\frac{m\pi }{T}t\right)dt,m\in \mathbb{Z},$$

(3)

where $cas\left(x\right)=cosx+sinx,x\in \mathbb{R}$.

For distributions $f\in {\mathcal{E}}^{\prime }\left((-T,T)\right)$, the transform is extended in the distributional sense as

$$\left({\mathcal{H}}_{T}f\right)\left(m\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}〈f\left(t\right),cas\left(\frac{m\pi }{T}t\right)〉,m\in \mathbb{Z}.$$

Lemma 4.

Let $m\in \mathbb{Z}$ and $k\ge 0$. Then

$$D_t^{2k}cas\left(\frac{m\pi }{T}t\right)={\left(\frac{m\pi }{T}\right)}^{2k}{(-1)}^{k}cas\left(\frac{m\pi }{T}t\right),$$

$$D_t^{2k+1}cas\left(\frac{m\pi }{T}t\right)={\left(\frac{m\pi }{T}\right)}^{2k+1}{(-1)}^k\left[cos\left(\frac{m\pi }{T}t\right)-sin\left(\frac{m\pi }{T}t\right)\right].$$

Proof. 

This follows from repeated differentiation of $cas\left(x\right)=cosx+sinx$. Every second derivative introduces a factor of $-{\left(\frac{m\pi }{T}\right)}^2$, while odd derivatives yield the alternating combination $cos-sin$ or $sin-cos$, with signs determined by parity. □

Lemma 5.

Let $f\in {\mathcal{E}}^{\prime }\left((-T,T)\right)$ be of order r. Then there exists $C>0$, such that

$$|\left({\mathcal{H}}_{T}f\right)\left(m\right)|\le C{\left(1+\frac{\left|m\right|\pi }{T}\right)}^r,m\in \mathbb{Z}.$$

Proof. 

By standard properties of compactly supported distributions (see, e.g., [3] (Prop. 2, p. 97)), there exist constants $C_0>0$ and $r\ge 0$, such that for every $\phi \in C^{\infty }\left((-T,T)\right)$,

$$\left|\langle f,\phi \rangle \right|\le C_0\underset{0\le j\le r}{max}\underset{t\in (-T,T)}{sup}\left|D^j\phi \left(t\right)\right|.$$

From Lemma 4, for each $0\le j\le r$,

$$\underset{t\in (-T,T)}{sup}|D^{j}cas\left(\frac{m\pi }{T}t\right)|\le 2{\left(\frac{\left|m\right|\pi }{T}\right)}^j,$$

Hence

$$\left|\left({\mathcal{H}}_{T}f\right)\left(m\right)\right|\le \sqrt{{\displaystyle \frac{2}{\pi }}}{C}_0\underset{0\le j\le r}{max}{\left(\frac{\left|m\right|\pi }{T}\right)}^j\le C\underset{0\le j\le r}{max}{\left(\frac{\left|m\right|\pi }{T}\right)}^j.$$

Finally, ${max}_{0\le j\le r}{\left(\frac{\left|m\right|\pi }{T}\right)}^j\le {(1+\frac{\left|m\right|\pi }{T})}^r$, proving the Lemma. □

5.1. Final-Value Theorem on ${\mathcal{E}}^{\prime }\left((-T,T)\right)$

Theorem 5

(Final-Value Theorem on ${\mathcal{E}}^{\prime }\left((-T,T)\right)$). Let $f\in {\mathcal{E}}^{\prime }\left((-T,T)\right)$ be of order r. Then for every $\alpha >0$,

$$\underset{m\to \pm \infty }{lim}\left\{m^{-r-\alpha }\left({\mathcal{H}}_{T}f\right)\left(m\right)\right\}=0.$$

Proof. 

From Lemma 5, for $f\in {\mathcal{E}}^{\prime }\left((-T,T)\right)$ of order r, there exists $C>0$, such that

$$|\left({\mathcal{H}}_{T}f\right)\left(m\right)|\le C{\left(1+\frac{\left|m\right|\pi }{T}\right)}^r,m\in \mathbb{Z}.$$

Hence, for every $\alpha >0$,

$$|m^{-r-\alpha }\left({\mathcal{H}}_{T}f\right)\left(m\right)|\le {C\left|m\right|}^{-r-\alpha }{\left(1+\frac{\left|m\right|\pi }{T}\right)}^r=C{\left({\displaystyle \frac{1+\frac{\left|m\right|\pi }{T}}{\left|m\right|}}\right)}^r{\left|m\right|}^{-\alpha }.$$

Since ${\displaystyle \frac{1+\frac{\left|m\right|\pi }{T}}{\left|m\right|}}={\displaystyle \frac{1}{\left|m\right|}}+{\displaystyle \frac{\pi }{T}}\to {\displaystyle \frac{\pi }{T}}$ as $m\to \infty$, the factor ${\left({\displaystyle \frac{1+\frac{\left|m\right|\pi }{T}}{\left|m\right|}}\right)}^r$ remains bounded. Therefore the right-hand side is $O\left(\right|m{|}^{-\alpha })$, and since $\alpha >0$, it follows that

$$\underset{m\to \pm \infty }{lim}\left\{m^{-r-\alpha }\left({\mathcal{H}}_{T}f\right)\left(m\right)\right\}=0.$$

□

Example 3.

Consider the delta functional ${\delta }_T$ concentrated at the origin, defined by

$$\langle {\delta }_T\left(t\right),\varphi \left(t\right)\rangle =\varphi \left(0\right),\mathrm{for}\mathrm{all}\varphi \in \mathcal{E}\left((-T,T)\right),T>0.$$

Clearly, ${\delta }_T\in {\mathcal{E}}^{\prime }\left((-T,T)\right)$ and is a distribution of order 0.

Its finite Hartley transform is given by

$$\left({\mathcal{H}}_T{\delta }_T\right)\left(m\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}〈{\delta }_T\left(t\right),cas\left({\displaystyle \frac{m\pi }{T}}t\right)〉={\displaystyle \frac{1}{\sqrt{2\pi }}},m\in \mathbb{Z}.$$

Thus, for $\alpha >0$, we obtain

$$\underset{m\to \pm \infty }{lim}\left\{m^{-\alpha }\left({\mathcal{H}}_T{\delta }_T\right)\left(m\right)\right\}=0.$$

The conclusions of Theorem 5 hold for this distribution.

5.2. Final Value Theorem on ${\mathcal{B}}_T^{\prime }$

Let ${\mathcal{B}}_T$ denote the Fréchet space

$${\mathcal{B}}_T=\left\{\phi \in C^{\infty }\left((-T,T)\right):{\beta }_T^k\left(\phi \right)=\underset{t\in (-T,T)}{sup}\left|D^k\phi \left(t\right)\right|<\infty ,k\ge 0\right\},$$

with strong dual ${\mathcal{B}}_T^{\prime }$.

Lemma 6.

Let $f\in {\mathcal{B}}_T^{\prime }$ be of order r. Then, there exists $C>0$, such that

$$|\left({\mathcal{H}}_{T}f\right)\left(m\right)|\le C{\left(1+\frac{\left|m\right|\pi }{T}\right)}^r,m\in \mathbb{Z}.$$

Proof. 

The continuity of $f\in {\mathcal{B}}_T^{\prime }$ implies the existence of $C_0>0$ and $r\ge 0$, such that

$$\left|\langle f,\phi \rangle \right|\le C_0\underset{0\le k\le r}{max}{\beta }_T^k\left(\phi \right),\phi \in {\mathcal{B}}_T.$$

Apply this to $\phi \left(t\right)=cas\left(\frac{m\pi }{T}t\right)$. Since $cas\left(x\right)=cosx+sinx$ and its derivatives are bounded,

$$\underset{t\in (-T,T)}{sup}|D^{k}cas\left(\frac{m\pi }{T}t\right)|\le 2{\left(\frac{\left|m\right|\pi }{T}\right)}^k.$$

Hence

$$|\left({\mathcal{H}}_{T}f\right)\left(m\right)|\le C{\left(1+\frac{\left|m\right|\pi }{T}\right)}^r,$$

as claimed. □

Theorem 6

(Final-Value Theorem on ${\mathcal{B}}_T^{\prime }$). Let $f\in {\mathcal{B}}_T^{\prime }$ be of order r. Then for every $\alpha >0$,

$$\underset{m\to \pm \infty }{lim}\left\{m^{-r-\alpha }\left({\mathcal{H}}_{T}f\right)\left(m\right)\right\}=0.$$

Proof. 

By Lemma 6, $\left({\mathcal{H}}_{T}f\right)\left(m\right)$ satisfies the same polynomial growth bound as in the distributional case. Following the same argument as in Theorem 5 gives the result. □

6. Conclusions and Perspectives

In this paper, we have investigated the Hartley transform defined by

$$\left(\mathcal{H}f\right)\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{\infty }f\left(t\right)cas\left(xt\right)dt,x\in \mathbb{R},$$

where $cas\left(xt\right)=cos\left(xt\right)+sin\left(xt\right)$. This transform arises as the real-valued counterpart of the Fourier transform.

Our contributions include the derivation of Parseval–Goldstein-type relations for the Hartley transform, together with Abelian theorems formulated in the context of compactly supported distributions and generalized functions. Furthermore, we introduced and analyzed a finite version of the Hartley transform on the interval $(-T,T)$, extending the analysis to distribution spaces of compact support and their duals. These results confirm that the asymptotic properties of distributions are faithfully reflected in the growth behaviour of their Hartley transforms, thereby generalizing known results from the Fourier setting to the real-valued domain.

The present study opens several directions for future research. One natural extension is the development of Tauberian theorems for the Hartley transform, which would establish converse results linking the asymptotic behaviour of a function or distribution with that of its transform. Another promising direction is the exploration of fractional and multidimensional analogues of the Hartley transform, where scaling properties and higher-dimensional symmetries could yield deeper theoretical insights. Weighted distribution spaces and ultradistribution frameworks also deserve attention, as they may clarify the role of growth restrictions on the transform side.

From an applied perspective, the Hartley transform, being entirely real-valued, offers computational advantages in signal and image processing, spectral methods for partial differential equations, and discrete data analysis. The finite Hartley transform may further find applications in numerical algorithms where symmetric and efficient real-valued transforms are advantageous.

In conclusion, the Hartley transform provides a strong alternative to the classical Fourier analysis, extending its utility to generalized function spaces and modern computational frameworks while maintaining the simplicity of a real-valued formulation.