Applications of the work of Stone and von Neumann to wavelets

Details Applications of the work of Stone and von Neumann to wavelets Applications of the work of Stone and von Neumann to wavelets

2004-07-02

arxiv.org/abs/math/0407037v1

Mathematics

Functional Analysis

27 pages, will appear in Contemporary Mathematics Volume edited by R. Doran and R. Kadison, minor corrections added

Scientific paper

This survey paper examines the work of J. von Neumann and M.H. Stone as it relates to the abstract theory of wavelets. In particular, we discuss the direct integral theory of von Neumann and how it can be applied to representations of certain discrete groups to study the existence of normalized tight frames in the setting of Gabor systems and wavelets, via the use of group representations and von Neumann algebras. Then the extension of Stone's theorem due to M. Naimark, W. Ambrose and R. Godement is reviewed, and its relationship to the multiresolution analyses of S. Mallat and Y. Meyer and the generalized multiresolution analyses of L. Baggett, H. Medina, and K. Merrill. Finally, the paper ends by discussing some recent work due to the author, Baggett, P. Jorgensen and Merrill, and its relationship to operator theory.

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