On The Generalisation of Henstock-Kurzweil Fourier Transform

TL;DR

This paper introduces a generalized Laplace integral that extends the Henstock-Kurzweil integral, and uses it to define a Fourier transform with established properties, broadening the scope of integral transforms.

Contribution

It presents a new generalized integral called the Laplace integral and demonstrates its advantages over the Henstock-Kurzweil integral for defining Fourier transforms.

Findings

Laplace integral is more general than Henstock-Kurzweil integral.

Fourier transform defined via Laplace integral retains key properties.

Necessary and sufficient conditions for differentiation under the integral sign are provided.

Abstract

In this paper, a generalised integral called the Laplace integral is defined on unbounded intervals, and some of its properties, including necessary and sufficient condition for differentiating under the integral sign, are discussed. It is also shown that this integral is more general than the Henstock-Kurzweil integral. Finally, the Fourier transform is defined using the Laplace integral, and its well-known properties are established.