The duality condition for Weyl-Heisenberg frames.

*(English)*Zbl 0890.42006Feichtinger, 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.

Reviewer: G.Steidl (Mannheim)

##### MSC:

42C15 | General harmonic expansions, frames |

42A65 | Completeness of sets of functions |

42C30 | Completeness of sets of functions of non-trigonometric Fourier analysis |