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

Shapes of uncertainty in spectral graph theory

We present a flexible framework for uncertainty principles in spectral graph theory. In this framework, general filter functions modeling the spatial and spectral localization of a graph signal can be incorporated. It merges several existing uncertainty relations on graphs, among others the Landau-Pollak principle describing the joint admissibility region of two projection operators, and uncertainty relations based on spectral and spatial spreads. Using theoretical and computational aspects of the numerical range of matrices, we are able to characterize and illustrate the shapes of the uncertainty curves and to study the space-frequency localization of signals inside the admissibility regions