Title

DIVERGENCE OF THE MOCK AND SCRAMBLED FOURIER SERIES ON FRACTAL MEASURES

Authors

Authors

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

Comments

Authors: contact us about adding a copy of your work at STARS@ucf.edu

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

Share

COinS