Journal article
An Operator-Fractal
Numerical Functional Analysis and Optimization: Operator Algebras and Representation Theory: Frames, Wavelets, and Fractals, Vol.33(7-9), pp.1070-1094
07/01/2012
DOI: 10.1080/01630563.2012.682127
Abstract
Certain Bernoulli convolution measures μ are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be spectral, the ONB need not be unique; indeed, there are often families of such spectral bases. Let for a natural number n and consider the Bernoulli measure with scale factor λ. It is known that L 2 (μ λ ) has a Fourier basis. We first show that there are Cuntz operators acting on this Hilbert space which create an orthogonal decomposition, thereby offering powerful algorithms for computations for Fourier expansions. When L 2 (μ λ ) has more than one Fourier basis, there are natural unitary operators U, indexed by a subset of odd scaling factors p; each U is defined by mapping one ONB to another. We show that the unitary operator U can also be orthogonally decomposed according to the Cuntz relations. Moreover, this operator-fractal U exhibits its own self-similarity.
Details
- Title: Subtitle
- An Operator-Fractal
- Creators
- Palle E. T Jorgensen - Department of Mathematics , University of IowaKeri A Kornelson - Department of Mathematics , University of OklahomaKaren L Shuman - Department of Mathematics and Statistics , Grinnell College
- Resource Type
- Journal article
- Publication Details
- Numerical Functional Analysis and Optimization: Operator Algebras and Representation Theory: Frames, Wavelets, and Fractals, Vol.33(7-9), pp.1070-1094
- DOI
- 10.1080/01630563.2012.682127
- ISSN
- 0163-0563
- eISSN
- 1532-2467
- Publisher
- Taylor & Francis Group
- Language
- English
- Date published
- 07/01/2012
- Academic Unit
- Mathematics
- Record Identifier
- 9983985941602771
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