2004/07/02 by Judith Packer, Judith A. Packer, Packer, Judith A.
Computer Science · Mathematics · #22D20 #22D30 #42C40 #47C05 #47L30 #Digital Filter Design and Implementation #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA) #Primary 46L99 #Secondary 42C15 #math.FA #math.OA #msc:22D20 #msc:22D30 #msc:42C15 #msc:42C40 #msc:46L99 #msc:47C05 #msc:47L30
paper · pdf · doi:10.48550/arxiv.math/0407037
27 pages, will appear in Contemporary Mathematics Volume edited by R. Doran and R. Kadison, minor corrections added
arxiv created 2004/07/02 · openalex publication_date 2004/07/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.