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

2024/07/15 by Nuha Diab, Diab, Nuha, Dmitry Batenkov +1 · 1 citation
Computer Science · Mathematics · #15A18 #47A55 #47A75 #47B34 #65F20 #65T40 #Blind Source Separation Techniques #FOS: Mathematics #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Numerical Analysis (math.NA) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2407.10600

openalex publication_date 2024/07/15 · openalex created_date 2024/07/17 · openalex updated_date 2026/07/28

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.

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