The fractional Clifford-Fourier transform

TL;DR

This paper introduces a fractional Clifford-Fourier transform with adjustable parameters, providing series and explicit kernel expressions, and analyzing its mathematical properties for potential applications in signal processing and harmonic analysis.

Contribution

It presents the first fractional version of the Clifford-Fourier transform, including kernel series expansion and explicit Bessel function expressions for even dimensions.

Findings

Derived a series expansion for the transform's kernel.

Obtained explicit kernel expressions in terms of Bessel functions for even dimensions.

Analyzed the transform's analytic properties in detail.

Abstract

In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.