We study the Slepian spatiospectral concentration problem for the space of multi-variate polynomials on the unit ball in $\mathbb{R}^d$. We will discuss the phenomenon of an asymptotically bimodal distribution of eigenvalues of the spatiospectral concentration operators of polynomial spaces equipped with two different notions of bandwidth: (a) the space of polynomials with a fixed maximal overall polynomial degree, (b) the space of polynomials separated into radial and spherical contributions, with fixed but separate maximal degrees for the radial and spherical contributions, respectively. In particular, we investigate the transition position of the bimodal eigenvalue distribution (the so-called Shannon number) for both setups. The analytic results are illustrated by numerical examples on the 3-D ball.

翻译：我们研究$\mathbb{R}^d$中单位球上多元多项式空间的Slepian空谱集中问题。将讨论多项式空间空谱集中算子特征值的渐近双峰分布现象，该多项式空间配备两种不同的带宽概念：(a) 固定最大整体多项式次数的多项式空间，(b) 按径向和球面贡献分离且分别具有固定最大次数的多项式空间。特别地，我们针对两种设置分别研究了双峰特征值分布的过渡位置（即Shannon数）。通过三维球上的数值算例对所获得的解析结果进行验证。