Measures of localization and quantitative Nyquist densities

We obtain a refinement of the degrees of freedom estimate of Landau and Pollak. More precisely, we estimate, in terms of $ε$, the increase in the degrees of freedom resulting upon allowing the functions to contain a certain prescribed amount of energy $ε$ outside a region delimited by a set $T$ in time and a set $Ω$ in frequency. In this situation, the lower asymptotic Nyquist density $\vert T\vert \vert Ω\vert /2π$ is increased to $(1+ε)\vert T\vert \vert Ω\vert /2π$. 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.

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