Orbits of bounded bijective operators and Gabor frames

Abstract

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of L2(R) , which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over Z, which are orbits of bounded operators on L2(R). Two classes of overcomplete Gabor frames which cannot be ordered over Z and represented by orbits of operators in GL(L2(R)) are given. Some results about operator representation are stated in a general context for arbitrary frames, covering also certain wavelet frames.