Divergence of the mock and scrambled Fourier series on fractal measures
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- by Dorin Ervin Dutkay, Deguang Han and Qiyu Sun PDF
- Trans. Amer. Math. Soc. 366 (2014), 2191-2208 Request permission
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.References
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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
- MR Author ID: 608228
- 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
- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 2191-2208
- MSC (2010): Primary 28A80, 28A78, 42B05
- DOI: https://doi.org/10.1090/S0002-9947-2013-06021-7
- MathSciNet review: 3152727