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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Fourier duality for fractal measures with affine scales


Authors: Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal: Math. Comp. 81 (2012), 2253-2273
MSC (2010): Primary 47B32, 42B05, 28A35, 26A33, 62L20
Published electronically: May 22, 2012
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Abstract: For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in $ \mathbb{R}^d$, and they both have the same matrix scaling; but the two use different translation vectors, one by a subset $ B$ in $ \mathbb{R}^d$, and the other by a related subset $ L$. Among other things, we show that there is then a pair of infinite discrete sets $ \Gamma (L)$ and $ \Gamma (B)$ in $ \mathbb{R}^d$ such that the $ \Gamma (L)$-Fourier exponentials are orthogonal in $ L^2(\mu _B)$, and the $ \Gamma (B)$-Fourier exponentials are orthogonal in $ L^2(\mu _L)$. These sets of orthogonal ``frequencies'' are typically lacunary, and they will be obtained by scaling in the large. The nature of our duality is explored below both in higher dimensions and for examples on the real line.

Our duality pairs do not always yield orthonormal Fourier bases in the respective $ L^2(\mu )$-Hilbert spaces, but depending on the geometry of certain finite orbits, we show that they do in some cases. We further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these affine fractal measures.


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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
Email: ddutkay@mail.ucf.edu

Palle E. T. Jorgensen
Affiliation: University of Iowa, Department of Mathematics, 14 MacLean Hall, Iowa City, Iowa 52242-1419
Email: jorgen@math.uiowa.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02580-4
PII: S 0025-5718(2012)02580-4
Keywords: Affine fractal, Cantor set, Cantor measure, iterated function system, Hilbert space, Beurling density, Fourier bases.
Received by editor(s): November 5, 2009
Received by editor(s) in revised form: June 19, 2011
Published electronically: May 22, 2012
Additional Notes: This work was supported in part by the National Science Foundation.
Article copyright: © Copyright 2012 American Mathematical Society