Approximation of dual Gabor frames, window decay, and wireless communications

We consider two problems involving Gabor frames that have recently received much attention. The first problem concerns the approximation of dual Gabor frames in L-2(R) by finite-dimensional methods. Utilizing the duality rela tions for Gabor frames we derive a method to approximate the dual Gabor fra me, that is much simpler than previously proposed techniques. Furthermore i t enables us to give estimates for the approximation rate when the dimensio n of the finite model approaches infinity. The second problem concerns the relation between the decay of the window function g and its canonical dual window gamma = S-1 g as well as its canonical tight window psi = S-1/2 g. B ased on results on commutative Banach algebras and Laurent operators we der ive a general condition under which y and h inherit the decay properties of g. These derivations are of relevance in the context of wireless communica tions. More precisely, our results provide a theoretical foundation for a r ecently proposed method for the design of time-frequency well-localized pul se shapes for orthogonal frequency division multiplexing systems. (C) 2001 Academic Press.