History and evolution of the density theorem for Gabor frames. (English) Zbl 1133.42043

Heil, Christopher

J. Fourier Anal. Appl. 13, No. 2, 113-166 (2007).

The Density Theorem for Gabor frames is one of the fundamental results of time-frequency analysis. Centered around the Density Theorem for Gabor frames, for any reader who is interested in time-frequency analysis, this expository survey article provides an excellent source on the history, evolution, recent developments, and extended bibliography on this theorem. Written in a self-contained and easily-understood way, this article also surveys many related fundamental topics on time frequency analysis such as the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogenous Approximation Property for lattice and irregular Gabor systems.

Reviewer: Bin Han (Edmonton)

MSC:

42C15	General harmonic expansions, frames
42C30	Completeness of sets of functions in nontrigonometric harmonic analysis
46C99	Inner product spaces and their generalizations, Hilbert spaces

Keywords:

Gabor system; Balian-Low Theorem; duality principle; homogeneous approximation property; Janssen representation; Walnut representation; time-frequency analysis