Duality and biorthogonality for Weyl-Heisenberg frames

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Abstract

Let (Formula presented.) and let (Formula presented.) In this paper we investigate the relation between the frame operator (Formula presented.) and the matrix (Formula presented.) whose entries (Formula presented.) are given by (Formula presented.) for (Formula presented.) Here (Formula presented.)(Formula presented.), for any (Formula presented.) We show that (Formula presented.) is bounded as a mapping of (Formula presented.) into (Formula presented.) if and only if (Formula presented.) is bounded as a mapping of(Formula presented.) into (Formula presented.) Also we show that (Formula presented.) if andonly if (Formula presented.) where (Formula presented.) denotes the identity operator of (Formula presented.) and (Formula presented.) respectively, and (Formula presented.)(Formula presented.) Next, when (Formula presented.) generates a frame, we have that (Formula presented.) has an upper frame bound, and the minimal dual function (Formula presented.) can be computed as (Formula presented.) The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a (Formula presented.) generating a frame are inherited by (Formula presented.) In particular, we show that (Formula presented.) when (Formula presented.) generates a frame (Formula presented.) Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula.

Original languageEnglish
Pages (from-to)403-436
Number of pages34
JournalJournal of Fourier Analysis and Applications
Volume1
Issue number4
DOIs
Publication statusPublished - 1 Jan 1994
Externally publishedYes

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