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Spectral measures and Cuntz algebras

Authors: Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal: Math. Comp. 81 (2012), 2275-2301
MSC (2010): Primary 28A80, 42B05, 46C05, 46L89
Published electronically: February 14, 2012
MathSciNet review: 2945156
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Abstract: We consider a family of measures $ \mu $ supported in $ \mathbb{R}^d$ and generated in the sense of Hutchinson by a finite family of affine transformations. It is known that interesting sub-families of these measures allow for an orthogonal basis in $ L^2(\mu )$ consisting of complex exponentials, i.e., a Fourier basis corresponding to a discrete subset $ \Gamma $ in $ \mathbb{R}^d$. Here we offer two computational devices for understanding the interplay between the possibilities for such sets $ \Gamma $ (spectrum) and the measures $ \mu $ themselves. Our computations combine the following three tools: duality, discrete harmonic analysis, and dynamical systems based on representations of the Cuntz $ C^*$-algebras $ \mathcal O_N$.

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Additional Information

Dorin Ervin Dutkay
Affiliation: University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, Florida 32816-1364

Palle E. T. Jorgensen
Affiliation: University of Iowa, Department of Mathematics, 14 MacLean Hall, Iowa City, Iowa 52242-1419

Keywords: Spectrum, Hilbert space, fractal, Fourier bases, selfsimilar, iterated function system, operator algebras.
Received by editor(s): January 25, 2010
Received by editor(s) in revised form: July 14, 2011
Published electronically: February 14, 2012
Additional Notes: With partial support by the National Science Foundation
Article copyright: © Copyright 2012 American Mathematical Society

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