Title

Divergence Of The Mock And Scrambled Fourier Series On Fractal Measures

Keywords

Dirichlet kernel; Fourier series; Fractal; Hadamard matrix; Hilbert space; Iterated function system; Selfsimilar

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 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 L1-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. © 2013 American Mathematical Society Reverts to public domain 28 years from publication.

Publication Date

1-30-2014

Publication Title

Transactions of the American Mathematical Society

Volume

366

Issue

4

Number of Pages

2191-2208

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1090/S0002-9947-2013-06021-7

Socpus ID

84893020948 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84893020948

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