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 $\mathrm{\Omega }$ in frequency. In this situation, the lower asymptotic Nyquist density $\mathrm{\mid }T\mathrm{\mid }\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\mathrm{/}2\pi$ is increased to $(1+ϵ)\mathrm{\mid }T\mathrm{\mid }\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\mathrm{/}2\pi$. 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.