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Harmonic Analysis of Fractal Processes via C * ‐ Algebras
Author(s) -
Jorgensen Palle E. T.
Publication year - 1999
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19992000105
Subject(s) - endomorphism , mathematics , wavelet , pure mathematics , hilbert space , commutative property , harmonic , lattice (music) , fourier transform , algebra over a field , mathematical analysis , quantum mechanics , physics , artificial intelligence , computer science , acoustics
Abstract We construct a harmonic analysis of iteration systems which include those which arise from wavelet algorithms based on multiresolutions. While traditional discretizations lead to asymptotic formulas, we argue here for a direct Fourier duality; but it is based on a non ‐ commutative harmonic analysis, specifically on representations of the Cuntz C * ‐algebras. With this approach the waling from the wavelet takes the form of an endomorphism of B(H), H a Hilbert space derived from the lattice of translations. We use this to describe, and to calculate, new invariants for the wavelets. those iteration systems which arise from wavelets and from Julia sets, we show that the associated endomorphisms are in fact Powers shifts.