Spatiospectral localization within the ball -- studies on the influence of the spectral shape

TL;DR

This paper explores how different spectral shapes affect the eigenvalue distribution in the Slepian spatiospectral localization problem within the d-dimensional ball, with implications for inverse problems and optical applications.

Contribution

It introduces a framework for analyzing the impact of spectral shape on eigenvalues, providing asymptotic results and numerical illustrations for various spectral configurations.

Findings

Asymptotic eigenvalue distributions depend on spectral shape.

Different spectral shapes influence the notion of bandlimit.

Numerical results illustrate the effects of spectral shape variations.

Abstract

We investigate the Slepian spatiospectral localization problem within subdomains of the $d$-dimensional ball. Opposed to the more classical setups of the Euclidean space or the sphere, the ball lacks a standard or universally accepted definition of bandwidth. Here, we consider a Fourier-Jacobi function system, decoupling the spherical and radial contributions via spherical harmonics and Jacobi polynomials. Special cases of this setup are of interest for various inverse problems in geophysics and medical imaging, since they relate to the underlying non-uniqueness, as well as in optics, where they represent the widely used Zernike polynomials. Bandwidth can be prescribed separately for the spherical and the radial contributions, where the particular choice of coupling between the two contributions determines the spectral shape, i.e., the overall notion of bandlimit. Understanding the effects of the spectral shape on the eigenvalue distribution of the Slepian spatiospectral localization problem can provide hints on particularly suitable notions of bandwidth for different applications. We provide rigorous asymptotic results for the spectral shape being defined via the overall polynomial degree as well as for being defined via sequential limits for the spherical and radial contributions. For various other spectral shapes, we provide numerical illustrations of the asymptotic eigenvalue distribution. Furthermore, we demonstrate a direct connection of the spectral shape to common indexing schemes for Zernike polynomials.