The Clifford-Fourier transform $\mathcal{F}_O$ and monogenic extensions

TL;DR

This paper introduces a Clifford-Fourier transform based on Clifford algebra's geometric properties, providing operational calculus, a method for monogenic extensions, and versions of Paley-Wiener theorems.

Contribution

It presents a new Clifford-Fourier transform and develops associated operational calculus and monogenic extension techniques, expanding the analytical tools in Clifford analysis.

Findings

Development of a Clifford-Fourier transform using geometric properties

Operational calculus results for the transform

A technique for monogenic extensions and Paley-Wiener theorems

Abstract

Several versions of the Fourier transform have been formulated in the framework of Clifford algebra. We present a (Clifford-Fourier) transform, constructed using the geometric properties of Clifford algebra. We show the corresponding results of operational calculus. We obtain a technique to construct monogenic extensions of a certain type of continuous functions, and versions of the Paley-Wiener theorems are formulated.