DIVERGENCE OF THE MOCK AND SCRAMBLED FOURIER SERIES ON FRACTAL MEASURES

    Authors

    D. E. Dutkay; D. G. Han;Q. Y. Sun

    Comments

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    Abbreviated Journal Title

    Trans. Am. Math. Soc.

    Keywords

    Fourier series; Dirichlet kernel; Hilbert space; fractal; selfsimilar; iterated function system; Hadamard matrix; ITERATED FUNCTION SYSTEMS; CANTOR MEASURES; CONVERGENCE; Mathematics

    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.

    Journal Title

    Transactions of the American Mathematical Society

    Volume

    366

    Issue/Number

    4

    Publication Date

    1-1-2014

    Document Type

    Article

    Language

    English

    First Page

    2191

    Last Page

    2208

    WOS Identifier

    WOS:000331471800019

    ISSN

    0002-9947

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