We study series expansions of signals with respect to Gabor wavelets and the equivalent problem of (irregular) sampling of the short time Fourier transform. Using Heisenberg group techniques rather than traditional Fourier analysis allows to design stable iterative algorithms for signal analysis and synthesis. These algorithms converge for a variety of norms and are compatible with the time-frequency localization of signals.

The paper provides a bridge between the engineering and signal processing terminology and group representation theory. Whereas the abstract terminology of representation theory talks of the Schroedinger representation of the (reduced) Heisenberg group (corresponding to the product of the LCA group Rd with its dual times the torus), or about a projective representation of the Abelian group  R^(2d) the more concrete view is to talk about time- and frequency shifts, which commute only up to phase factors (i.e. multiplicative factors from the unit circle = torus).

It is the main goal of this paper to show how the abstract theory of coorbit spaces as developed in a series of papers by Feichtinger-Gröchenig, as described in  Banach spaces related to integrable group representations and their atomic decompositions, I. and Banach spaces related to integrable group representations and their atomic decompositions, II applies in this specific situation, and that by relatively simple arguments the abstract statements (which concern the abstract wavelet transform associated with the Schroedinger representation, hence living on the reduced Heisenberg group) can be pushed down  to the time-frequency plane, which may be considered as a section of the Heisenberg group, by relatively simple arguments. From the point of view of frame theory the statements described in this paper imply that for large classes of function spaces - in particular for modulation spaces  irregular Gabor families (obtained by applying TF-shifts from an irregular subset of the TF-plane) form Banach frames  as long as the corresponding parameter set is sufficiently dense in the TF-plane. For the regular case such results can be deduced already from the earlier paper on  Atomic characterizations of modulation spaces through Gabor-type representations. 

The paper also contains the so-called piano-reconstruction theorem, stating that Weyl-Heisenberg families obtained by applying to a sufficiently `nice' Gabor atom a family of TF-shifts which is sufficiently well separated, forms a Riesz basis (actually a Riesz projection basis for a family of Banach spaces, cf. the article with G.Zimmermann on  A Banach space of test functions for Gabor analysis for more details on those bases). As that article the paper described here contains also a number of results concerning "Feichtinger's Segal algebra"  S_0(G), introduced in the paper On a new Segal algebra (published in 1980).  In particular a proof of the kernel theorem is given, which also plays a big role in Quantization of TF lattice-invariant operators on elementary LCA groups (Feichtinger-Kozek Chapter of the Gabor Book 1998). 

Keywords: Gabor analysis, irregular sampling, wavelets
Published: in: C.K. Chui, editor, Wavelets - A Tutorial in Theory and Applications, pages 359-397. Academic Press, Boston, 1992.

This paper is not available electronically. Contact the author fur any further information.