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We study the asymptotic eigenvalue distribution of the Slepian spatiospectralconcentration problem within subdomains of the $d$-dimensional unit ball$\mathbb{B}^d$. The clustering of the eigenvalues near zero and one is awell-known phenomenon. Here, we provide an analytical investigation of thisphenomenon for two different notions of bandlimit: (a) multivariatepolynomials, 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 sphericalcontributions determining the bandlimit. In particular, we investigate thenumber of relevant non-zero eigenvalues (the so-called Shannon number) andobtain distinct asymptotic results for both notions of bandlimit, characterizedby Jacobi weights $W_0$ and a modification $\widetilde{W_0}$, respectively. Theanalytic results are illustrated by numerical examples on the 3-d ball.

Slepian spatiospectral concentration problem on the $d$-dimensional ball for different notions of bandwidth