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Divergence of mock Fourier series for spectral measures

Part of: Harmonic analysis in one variable Harmonic analysis in several variables Classical measure theory

Published online by Cambridge University Press:  17 October 2022

Wu-yi Pan  and
Wen-hui Ai
Wu-yi Pan
Affiliation:
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China (pwyyyds@163.com, awhxyz123@163.com)
Wen-hui Ai
Affiliation:
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China (pwyyyds@163.com, awhxyz123@163.com)
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Abstract

In this paper, we study divergence properties of the Fourier series on Cantor-type fractal measure, also called the mock Fourier series. We give a sufficient condition under which the mock Fourier series for doubling spectral measure is divergent on a set of strictly positive measure. In particular, there exists an example of the quarter Cantor measure whose mock Fourier sums are not almost everywhere convergent.

Keywords

Mock Fourier seriesspectral measuredivergencequarter Cantor measure

MSC classification

Primary: 28A80: Fractals
Secondary: 42A20: Convergence and absolute convergence of Fourier and trigonometric series 42B05: Fourier series and coefficients
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 6 , December 2023 , pp. 1818 - 1832
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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