
Accession Number : ADA304529
Title : WeylHeisenberg Frames and Riesz Bases in L2(Rd).
Descriptive Note : Technical summary rept.
Corporate Author : WISCONSIN UNIVMADISON CENTER FOR MATHEMATICAL SCIENCES
Personal Author(s) : Ron, Amos ; Shen, Zuowei
PDF Url : ADA304529
Report Date : OCT 1994
Pagination or Media Count : 42
Abstract : We study WeylHeisenberg (= 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 fiberizationdecomposition techniques of shiftinvariant 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.
Descriptors : *SET THEORY, *MAPPING(TRANSFORMATIONS), STABILITY, MULTIVARIATE ANALYSIS, MATRICES(MATHEMATICS), ESTIMATES, APPROXIMATION(MATHEMATICS), SYSTEMS ANALYSIS, OPERATORS(MATHEMATICS), ORTHOGONALITY, SEQUENCES(MATHEMATICS), INTEGRAL TRANSFORMS, BESSEL FUNCTIONS.
Subject Categories : Theoretical Mathematics
Distribution Statement : APPROVED FOR PUBLIC RELEASE