The duality condition for Weyl-Heisenberg frames. (English) Zbl 0890.42006

Feichtinger, Hans G. (ed.) et al., Gabor analysis and algorithms. Theory and applications. Boston, MA: Birkhäuser. Applied and Numerical Harmonic Analysis. 33-84, 453-488 (1998).
The first chapter of the book “Gabor analysis and algorithms” (H. G. Feichtinger and T. Strohmer, eds.) written by A. J. E. M. Janssen is concerned with the condition of duality of Gabor frames in the time domain, the frequency domain, the time-frequency domain, and, for rational time-frequency sampling factors, the Zak domain, both for the continuous-time and the discrete-time case. Many of the results are presented in the more general framework of shift-invariant systems or filter banks.
In all considered domains, the author derives formulas for the frame operator and for the frame bounds and characterizes and computes minimal dual systems.
For the entire collection see [Zbl 0890.42004].


42C15 General harmonic expansions, frames
42A65 Completeness of sets of functions in one variable harmonic analysis
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis