Slice Fourier transform and convolutions

TL;DR

This paper introduces the slice Fourier transform for slice monogenic functions, constructs its kernel explicitly, and explores associated convolutions, revealing their interrelation and foundational properties.

Contribution

It presents the first explicit construction of the slice Fourier transform kernel and analyzes two types of convolutions, connecting them within the framework of slice monogenic functions.

Findings

Explicit kernel expression for the slice Fourier transform

Analysis of Mustard and generalized translation convolutions

Demonstration of the connection between the two convolutions

Abstract

Recently the construction of various integral transforms for slice monogenic functions has gained a lot of attention. In line with these developments, the article at hand introduces the slice Fourier transform. In the first part, the kernel function of this integral transform is constructed using the Mehler formula. An explicit expression for the integral transform is obtained and allows for the study of its properties. In the second part, two kinds of corresponding convolutions are examined: Mustard convolutions and convolutions based on generalised translation operators. The paper finishes by demonstrating the connection between both.