Relaxed Gabor expansion at critical density and a `certainty principle'

Abstract

A version of Gabor expansion over a lattice of critical density is shown to converge to an arbitrary function that belongs to domain of the oscillator operator. This expansion is used for approximation of an arbitrary function concentrated (in the sense of the Gabor transform) in a bounded domain D of the phase plane. It is shown that a reasonable approximation can be done by means of Gabor functions, whose number is equivalent to the simplectic area of D.