History and evolution of the Density Theorem for Gabor Frames

作     者：Heil, Christopher 

作者机构：Georgia Inst Technol Sch Math Atlanta GA 30332 USA 

出 版 物：《JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS》 (J. Fourier Anal. Appl.)

年 卷 期：2007年第13卷第2期

页      面：113-166页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：National Science Foundation, NSF, (0139261) National Science Foundation, NSF 

主　　题：Balian-Low Theorem complete sequence density theorem duality principle frames Gabor system homogeneous approximation property Janssen representation localized frames modulation spaces Riesz bases Schauder bases time-frequency analysis von Neumann algebras Walnut representation Wexler-Raz biorthogonality relations Weyl-Heisenberg system 

摘      要：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 con text 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.