The Uniformity of Non-Uniform Gabor Bases

There have been extensive studies on non-uniform Gabor bases and frames in recent years. But interestingly there have not been a single example of a compactly supported orthonormal Gabor basis in which either the frequency set or the translation set is non-uniform. Nor has there been an example in which the modulus of the generating function is not a characteristic function of a set. In this paper, we prove that in the one dimension and if we assume that the generating function g(x) of an orthonormal Gabor basis is supported on an interval, then both the frequency and the translation sets of the Gabor basis must be lattices. In fact, the Gabor basis must be the “trivial” one in the sense that |g(x)| = cχΩ(x) for some fundamental interval of the translation set. We also give examples showing that compactly supported non-uniform orthonormal Gabor bases exist in higher dimensions.