Divergence of mock and scrambled Fourier series

TL;DR

This paper investigates divergence phenomena of Fourier series on fractal measures and scrambled series, revealing exponential growth of Dirichlet kernels and non-convergence, linked to Mahler measures and spectral properties.

Contribution

It demonstrates exponential divergence of Fourier series on fractal measures and introduces scrambled Fourier series with worse divergence behavior than classical bounds.

Findings

Dirichlet kernel $L^1$-norm grows exponentially

Fourier series are not pointwise convergent in some cases

Scrambled Fourier series exhibit worse divergence than classical bounds

Abstract

We study divergence properties of Fourier series on Cantor-type fractal measures, also called 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, which we call 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.