Ball Prolate Spheroidal Wave Functions In Arbitrary Dimensions
For researchers in signal processing and applied mathematics, this work offers a unified framework for PSWFs in arbitrary dimensions, but the contribution is incremental as it extends known concepts.
This paper introduces ball prolate spheroidal wave functions (PSWFs) in arbitrary dimensions, generalizing existing multi-dimensional PSWFs via a perturbation of ball polynomials. They provide an efficient algorithm for computation and demonstrate its accuracy with numerical results.
In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $α>-1$ 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.