Weyl-Heisenberg Frames and Riesz Bases in L2(Rd).

Abstract
We study Weyl-Heisenberg (= Gabor) expansions for either L2(Rd) or a subspace of it. These are expansions in terms of the spanning set, involving K and L are some discrete lattices in Rd, P, in L2(Rd), is finite, E is the translation operator, and M is a modulation operator. Such sets X are known as WH systems. The analysis of the 'basis' properties of WH systems (e.g. being a frame or a Riesz basis) is our central topic, with the fiberization-decomposition techniques of shift-invariant systems, developed in a previous paper of us, being the main tool. Of particular interest is the notion of the adjoint of a WH set, and the duality principle which characterizes a WH (tight) frame in term of the stability (orthonormality) of its adjoint. The actions of passing to the adjoint and passing to the dual system commute, hence the dual WH frame can be computed via the dual basis of the adjoint. Estimates for the underlying frame/basis bounds are obtained by two different methods. The Gramian analysis applies to all WH systems, albeit provides estimates that might be quite crude. This approach is invoked to show how, under only mild conditions on X, a frame can be obtained by oversampling a Bessel sequence. Finally, finer estimates of the frame bounds, based on the Zak transform, are obtained for a large collection of WH systems.

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