Divergence of the mock and scrambled Fourier series on fractal measures

By Dorin Ervin Dutkay, Deguang Han, Qiyu Sun. 2013

84 citations recorded by the source corpus.

We study divergence properties of the Fourier series on Cantortype fractal measures, also called the mock Fourier series. We show that in some cases the L1-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 L1norm 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66a3f12dce5b71bd0945291cab51f7ff6739435c

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In Applied Mathematics

1. J. Miao
2. Wenxia Li
3. Zhiqiang Wang
4. Li-Xiang An
5. Xiaoye Fu
6. Chun-Kit Lai
7. Xinggang He
8. Xin-Rong Dai

In Mathematical Physics

1. J. Miao
2. Wenxia Li
3. Zhiqiang Wang
4. Li-Xiang An
5. Xiaoye Fu
6. Chun-Kit Lai
7. Xinggang He
8. Xin-Rong Dai

Citations in the corpus

• On spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem (2019, background): “measures [9, 10, 11], as well as the convergence properties of the associated Fourier series [38, 39, 13] and the rescaling properties of a given spectrum [19, 16, 21].”
• Weak convergence and spectrality of infinite convolutions (2022, background): “A surprising phenomenon is that the convergence of the mock Fourier series may be very different for distinct spectra of a singular spectral measure [16].”