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$d$维单位球体$\mathbb{B}^d$子域内Slepian空间谱集中问题的渐近特征值分布。特征值在零和一附近的聚集是众所周知的现象。本文针对两种不同的带限概念对这一现象进行了分析研究：（a）多元多项式，其最大多项式次数决定带限；（b）分解为径向和球面贡献的基函数（分别用Jacobi多项式和球谐函数表示），径向和球面贡献的独立最大次数共同决定带限。特别地，我们考察了相关非零特征值的数量（即所谓的Shannon数），并针对两种带限概念获得了截然不同的渐近结果，分别由Jacobi权重$W_0$及其修正形式$\widetilde{W_0}$刻画。通过三维球体上的数值示例对分析结果进行了验证。