  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.
