Better dual functions for Gabor time-frequency lattices. (English) Zbl 0927.42017

Chui, C. K. (ed.) et al., Approximation theory VIII. Vol. 2. Wavelets and multilevel approximation. Papers from the 8th Texas international conference, College Station, TX, USA, January 8–12, 1995. Singapore: World Scientific. Ser. Approx. Decompos. 6, 113-116 (1995).
Summary: Gabor time-frequency lattices are sets of functions of the form \(g_{m\alpha,n\beta}(t)= e^{-2\pi i\alpha mt}g(t- n\beta)\) generated from a given function \(g(t)\) by discrete translations in time and frequency. It was recently observed by Wexler and Raz that the behavior of a lattice \((m\alpha, n\beta)\) can be connected to that of a dual lattice \(\left({m\over\beta}, {n\over \alpha}\right)\). We establish this interesting relationship rigorously and study its properties. We also exploit the connection between the two lattices to construct expansions having improved convergence and localization properties.
For the entire collection see [Zbl 0902.00035].


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems