Gabor representation of generalized functions

Abstract

This paper deals with Gabor representation of (generalized) functions. By this we mean that a given function is to be expanded in a series involving Gabor functions that are located in the points of a given lattice in the time-frequency plane. We shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1. In 1946, Gabor used these expansions for the simultaneous analysis of signals in time and frequency (cf. 181). Gabor stated that the above mentioned expansions exist for every reasonable signal, and that the coefficients in the expansion are uniquely determined by the signal. This statement is true when interpreted carefully, and the aim of this paper is to find out what kind of (generalized) functions are sufficiently well-behaved to allow a development in a Gabor series. We shall also consider the question of the uniqueness of the coefficients in the expansions, as well as questions concerning convergence. It turns out that the uniqueness questions can be handled with the aid of the main results of [IO] (in particular 2.12 and 2.13); the questions on the existence of expansions are harder to answer, and require special care depending on the particular function. We consider Lpfunctions (1 <p < 2) in detail, and show existence of Gabor representation for tempered distribution in general.