New inversion, convolution and Titchmarsh's theorems for the half-Hartley transform

TL;DR

This paper develops new inversion, convolution, and Titchmarsh theorems for the half-Hartley transform using Mellin transform techniques, and applies these results to integral equations.

Contribution

It introduces novel inversion and convolution theorems for the half-Hartley transform and extends Titchmarsh's theorem within this context.

Findings

Established inversion theorem in L_2 space.

Derived convolution and Titchmarsh's theorems for the half-Hartley transform.

Provided solvability conditions for a class of integral equations.

Abstract

The generalized Parseval equality for the Mellin transform is employed to prove the inversion theorem in L_2 with the respective inverse operator related to the Hartley transform on the nonnegative half-axis (the half-Hartley transform). Moreover, involving the convolution method, which is based on the double Mellin-Barnes integrals, the corresponding convolution and Titchmarsh's theorems for the half-Hartley transform are established. As an application, we consider solvability conditions for a homogeneous integral equation of the second kind involving the Hartley kernel.