On the Eigenvalue Distribution of Spatio-Spectral Limiting Operators in Higher Dimensions

Abstract: Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on $\mathbb{R}$ that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval $[0, 1]$. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues. In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a \emph{spatio-spectral limiting operator} (SSLO) on $L^2(\mathbb{R}^d)$, which is an alternating product of projection operators associated to given spatial and frequency domains in $\mathbb{R}^d$. If one of the domains is a hypercube, and the other domain is a convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval $[0,1]$. To prove our results, we design an orthonormal system of wave packets in $L^2(\mathbb{R}^d)$ that are highly concentrated in the spatial and frequency domains. We show that these wave packets are ``approximate eigenfunctions'' of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for $L^2[0,1]$, and we lift the basis to higher dimensions using a tensor product.