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 $\epsilon$, 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.