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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Divergence of the mock and scrambled Fourier series on fractal measures


Authors: Dorin Ervin Dutkay, Deguang Han and Qiyu Sun
Journal: Trans. Amer. Math. Soc. 366 (2014), 2191-2208
MSC (2010): Primary 28A80, 28A78, 42B05
Published electronically: September 4, 2013
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Abstract: We study divergence properties of the Fourier series on Cantor-type fractal measures, also called the mock Fourier series. We show that in some cases the $ L^1$-norm of the corresponding Dirichlet kernel grows exponentially fast, and therefore the Fourier series are not even pointwise convergent. We apply these results to the Lebesgue measure to show that a certain rearrangement of the exponential functions, with affine structure, which we call a scrambled Fourier series, have a corresponding Dirichlet kernel whose $ L^1$-norm grows exponentially fast, which is much worse than the known logarithmic bound. The divergence properties are related to the Mahler measure of certain polynomials and to spectral properties of Ruelle operators.


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Additional Information

Dorin Ervin Dutkay
Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, P.O. Box 161364, Orlando, Florida 32816-1364
Email: Dorin.Dutkay@ucf.edu

Deguang Han
Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, P.O. Box 161364, Orlando, Florida 32816-1364
Email: Deguang.Han@ucf.edu

Qiyu Sun
Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, P.O. Box 161364, Orlando, Florida 32816-1364
Email: qiyu.sun@ucf.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-06021-7
PII: S 0002-9947(2013)06021-7
Keywords: Fourier series, Dirichlet kernel, Hilbert space, fractal, selfsimilar, iterated function system, Hadamard matrix
Received by editor(s): March 22, 2011
Received by editor(s) in revised form: August 27, 2012
Published electronically: September 4, 2013
Additional Notes: This research was partially supported in part by NSF grants (DMS-1106934 and DMS-1109063) and by a grant from the Simons Foundation #228539.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.