History and Evolution of the Density Theorem for Gabor Frames

Abstract

The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the one-dimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.