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 |T||Ω|/2π|T||Ω|/2π is increased to (1+ϵ)|T||Ω|/2π(1+ϵ)|T||Ω|/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.