Slepian concentration problem for polynomials on the unit Ball

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.