Measures of localization and quantitative Nyquist densities

TL;DR

This paper refines the degrees of freedom estimate for time-frequency localized functions, showing how allowing a small energy outside a region increases the effective Nyquist density, with implications for signal analysis.

Contribution

It provides a new estimate of degrees of freedom increase when permitting energy outside a localized region, extending Landau and Pollak's results with pseudospectra techniques.

Findings

Refined degrees of freedom estimate incorporating energy outside localization region

Demonstrated increase in Nyquist density by factor (1+ε) due to allowed energy

Extended results to multivariate and Gabor localization operators

Abstract

We obtain a refinement of the degrees of freedom estimate of Landau and Pollak. More precisely, we estimate, in terms of $\epsilon$, the increase in the degrees of freedom resulting upon allowing the functions to contain a certain prescribed amount of energy $\epsilon $ outside a region delimited by a set $T$ in time and a set $\Omega $ in frequency. In this situation, the lower asymptotic Nyquist density $\vert T\vert \vert \Omega \vert /2\pi$ is increased to $(1+\epsilon)\vert T\vert \vert \Omega \vert /2\pi$. At the technical level, we prove a pseudospectra version of the classical spectral dimension result of Landau and Pollak, in the multivariate setting of Landau. Analogous results are obtained for Gabor localization operators in a compact region of the time-frequency plane.