Expansion into Clifford Prolate Spheroidal Wave Functions

TL;DR

This paper studies the properties of Clifford prolate spheroidal wave functions, establishing decay rates of expansion coefficients and connecting them to eigenfunctions of the finite Hankel transform, with numerical validation.

Contribution

It introduces new spectral decay bounds for CPSWFs and links their properties to finite Hankel transform eigenfunctions, advancing understanding of Clifford-valued function approximations.

Findings

Expansion coefficients decay super-exponentially with order and homogeneity.

Explicit bounds on eigenvalues of the finite Hankel transform in the Clifford setting.

Numerical experiments confirm the accuracy and efficiency of the CPSWF-based approximations.

Abstract

In this paper, we investigate the properties of Clifford prolate spheroidal wave functions (CPSWFs) through their associated eigenvalues. We prove that the expansion coefficients in CPSWFs series decay as both the order and the homogeneity degree increase. By establishing a precise connection between the radial CPSWFs and the eigenfunctions of the finite Hankel transform, we derive explicit and non-asymptotic bounds on the corresponding eigenvalues and transfer the spectral decay estimates to the Clifford setting. Consequently, we obtain super-exponential decay rates for the CPSWF expansion coefficients of band-limited Clifford-valued functions. Numerical experiments illustrate both the accuracy and the efficiency of these approximations.