On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, II

Abstract

Let $F$, $S$ be bounded measurable sets in ${\mathbb{R}}^d$. Let $P_F:L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)$ be the orthogonal projection on the subspace of functions with compact support on $F$, and let $B_S:L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)$ be the orthogonal projection on the subspace of functions with Fourier transforms having compact support on $S$. In this paper, we derive improved distributional estimates on the eigenvalue sequence $1\ge {\lambda }_1(F,S)\ge {\lambda }_2(F,S)\ge \cdots >0$ of the \emph{spatio-spectral limiting operator} $B_S{P}_F{B}_S:L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)$. The significance of such estimates lies in their diverse applications in medical imaging, signal processing, geophysics and astronomy. Our proof is based on the decomposition techniques developed in \cite{MaRoSp23}. The novelty of our approach is in the use of a two-stage dyadic decomposition with respect to both the spatial and frequency domains, and the application of the results in \cite{ArieAzita23} on the eigenvalues of spatio-spectral limiting operators associated to cubical domains.

18 pages