Ball Prolate Spheroidal Wave Functions In Arbitrary Dimensions

TL;DR

This paper introduces a new class of prolate spheroidal wave functions on the unit ball in arbitrary dimensions, generalizing existing functions and establishing connections with Fourier and Hankel transforms, along with an efficient computational algorithm.

Contribution

It defines ball PSWFs as a generalization of orthogonal ball polynomials, linking them to Fourier and Hankel transforms, and provides an efficient algorithm for their computation.

Findings

New class of PSWFs on the unit ball in arbitrary dimensions.

Efficient algorithm for computing ball PSWFs and eigenvalues.

Connections established between ball PSWFs, Fourier, and Hankel transforms.

Abstract

In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $\alpha>-1$ on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both a weighted concentration integral operator, and a Sturm-Liouville differential operator. Different from existing works on multi-dimensional PSWFs, the ball PSWFs are defined as a generalisation of orthogonal {\em ball polynomials} in primitive variables with a tuning parameter $c>0$, through a "perturbation" of the Sturm-Liouville equation of the ball polynomials. From this perspective, we can explore some interesting intrinsic connections between the ball PSWFs and the finite Fourier and Hankel transforms. We provide an efficient and accurate algorithm for computing the ball PSWFs and the associated eigenvalues, and present various numerical results to illustrate the efficiency of the method. Under this uniform framework, we can recover the existing PSWFs by suitable variable substitutions.