We study the asymptotic eigenvalue distribution of the Slepian spatiospectral concentration problem within subdomains of the $d$-dimensional unit ball $\mathbb{B}^d$. The clustering of the eigenvalues near zero and one is a well-known phenomenon. Here, we provide an analytical investigation of this phenomenon for two different notions of bandlimit: (a) multivariate polynomials, with the maximal polynomial degree determining the bandlimit, (b) basis functions that separate into radial and spherical contributions (expressed in terms of Jacobi polynomials and spherical harmonics, respectively), with separate maximal degrees for the radial and spherical contributions determining the bandlimit. In particular, we investigate the number of relevant non-zero eigenvalues (the so-called Shannon number) and obtain distinct asymptotic results for both notions of bandlimit, characterized by Jacobi weights $W_0$ and a modification $\widetilde{W_0}$, respectively. The analytic results are illustrated by numerical examples on the 3-d ball.

翻译：我们研究了$d$维单位球$\mathbb{B}^d$子区域内Slepian空间谱集中问题的渐近特征值分布。特征值聚集在零和一附近是众所周知的现象。本文针对两种不同的带限概念对该现象进行了分析研究：（a）多元多项式，以最大多项式次数确定带限；（b）将径向和球面贡献分离的基函数（分别用雅可比多项式和球谐函数表示），以径向和球面贡献的各自最大次数确定带限。具体而言，我们研究了相关非零特征值的数目（即所谓的香农数），并分别针对两种带限概念获得了由雅可比权重$W_0$及其修正形式$\widetilde{W_0}$刻画的不同渐近结果。通过三维球上的数值算例对分析结果进行了验证。