Spectral Properties of Infinitely Smooth Kernel Matrices in the Single Cluster Limit, with Applications to Multivariate Super-Resolution

TL;DR

This paper investigates the spectral behavior of smooth kernel matrices in single-cluster scenarios, revealing how node geometry influences eigenvalues and applying findings to improve super-resolution techniques.

Contribution

It provides a criterion for sampling sets that ensure proper eigenvalue scaling in multivariate Dirichlet kernel matrices and explores implications for super-resolution.

Findings

Eigenvalues scale according to node geometry.

A criterion for sampling sets guarantees eigenvalue scaling.

Results enhance understanding of super-resolution stability.

Abstract

We study the spectral properties of infinitely smooth multivariate kernel matrices when the nodes form a single cluster. We show that the geometry of the nodes plays an important role in the scaling of the eigenvalues of these kernel matrices. For the multivariate Dirichlet kernel matrix, we establish a criterion for the sampling set ensuring precise scaling of eigenvalues. Additionally, we identify specific sampling sets that satisfy this criterion. Finally, we discuss the implications of these results for the problem of super-resolution, i.e. stable recovery of sparse measures from bandlimited Fourier measurements.