Spectral deviation of concentration operators on reproducing kernel Hilbert spaces

TL;DR

This paper analyzes the eigenvalue distribution of concentration operators in reproducing kernel Hilbert spaces, linking spectral properties to local degrees of freedom and discretization effects.

Contribution

It provides a unified framework for estimating spectral profiles in both discrete and continuous settings, including Gabor multipliers and STFT.

Findings

Eigenvalue profiles characterize local degrees of freedom.

Discretization schemes accurately reflect continuous spectral properties.

Spectral deviation estimates hold uniformly across discretizations.

Abstract

We study the eigenvalue profile of concentration operators (multiplication by an indicator function followed by projection) acting on reproducing kernel Hilbert spaces. The spectral profile of such operators provides a useful notion of local degrees of freedom. We formalize this idea by estimating the number of eigenvalues that lie away from 0 and 1, commonly referred to as the plunge region. Our main motivation is to treat discrete and continuous settings simultaneously and uniformly, and to be able to argue that approximations arising from discretization schemes reflect, in a non-asymptotic sense, the spectral profile of their continuous counterparts. As a case in point, we show that Gabor multipliers computed on sufficiently fine grids obey spectral deviation estimates similar to those available for the short-time Fourier transform (STFT) with bounds that are uniform in the discretization step. Concretely, this means that the theoretical localization properties of the STFT are observable in practice.