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

TL;DR

This work develops a unified framework for Slepian spatiospectral concentration inside the unit ball by introducing spectral shapes $\\Omega$ to define bandlimited spaces based on Fourier-Jacobi functions, thereby decoupling radial and spherical contributions. It provides rigorous asymptotic results for two prominent bandwidth notions: the maximal polynomial degree space $\\widehat{\\Pi}_n^d$ and the sequential radial-spherical limit $\\widetilde{\\Pi}_{m,n}^d$, revealing distinct weight functions $W$ that govern Shannon numbers and concentration behavior. The authors connect these theoretical findings to common Zernike polynomial indexing schemes, supported by numerical experiments that illustrate how spectral shape influences energy localization in interior versus near-boundary regions of the ball. By framing spectral shape as a design parameter, the paper offers practical guidance for tailoring basis functions to target specific spatial regions in geophysics, optics, and medical imaging, and it sets the stage for extensions to arbitrary shapes and efficient computational approaches.

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.

Figures (9)

• Figure 1: Illustration of spectral shapes $\Omega$ associated with notions of bandlimit \ref{['eqn:nmspace']}, \ref{['eqn:hatpi']}, and \ref{['eqn:ckeckpi']} (from left to right).
• Figure 2: Illustration of the weight functions $W=\widehat{W}$ for $\widehat{\Pi}_n^3$ and $W=\widetilde{W}$ for $\widetilde{\Pi}_{m,n}^3$.
• Figure 3: Illustration of $\widetilde{\mathcal{K}}_{m, n}(x,x)/\widetilde{{\mathcal{N}}}_{m,n}^d$ for different ratios $\kappa$ that link $n=\kappa m$. Left column: dimension $d=2$, right column: dimension $d=3$. The top, middle and bottom rows correspond to $\kappa=1$, $\kappa=\frac{1}{2}$ and $\kappa=\frac{1}{3}$, respectively. The dashed purple lines indicate the reference $\widetilde{W}$ from \ref{['eqn:wtilde1']} (which differs for dimension $d=2$ and $d=3$).
• Figure 4: Illustration of ${\widehat{\mathcal{K}}_{n}(x,x)}/\widehat{\mathcal{N}}_n^2$ (red) and ${\widecheck{\mathcal{K}}_{n}(x,x)}/\widecheck{\mathcal{N}}_n^2$ (blue).
• Figure 5: Illustration of the subdomains $D_1$ (blue) and $D_2$ (red) used in numerical experiments.

Theorems & Definitions (27)

• Proposition 3.1
• Proposition 3.2
• Remark 3.3
• Theorem 3.4
• proof
• Theorem 3.5
• Proposition 4.1
• Remark 4.2
• Proposition 4.3
• Theorem 4.4