Clifford Prolate Spheroidal wave Functions

TL;DR

This paper introduces multidimensional Clifford prolate spheroidal wave functions (CPSWFs), explores their properties as eigenfunctions of Clifford differential operators and Fourier transforms, and examines their role in spectral concentration problems.

Contribution

It presents a new class of CPSWFs defined on the unit ball, along with a Galerkin method for their computation and analysis of their spectral properties.

Findings

CPSWFs are eigenfunctions of a Clifford differential operator.

They serve as eigenfunctions of the truncated Fourier transform.

The eigenvalues exhibit spectral accumulation properties.

Abstract

In the present paper, we introduce the multidimensional Clifford prolate spheroidal wave functions (CPSWFs) defined on the unit ball as eigenfunctions of a Clifford differential operator and provide a Galerkin method for their computation as linear combinations of Clifford-Legendre polynomials. We show that these functions are eigenfunctions of the truncated Fourier transformation. Then we investigate the role of the CPSWFs in the spectral concentration problem associated with balls in the space and frequency domains, the behaviour of the eigenvalues of the time-frequency limiting operator and their spectral accumulation property.