On uniform convergence of Fourier series

TL;DR

This paper investigates the uniform convergence properties of Fourier series for continuous functions on the circle, revealing that certain nonlinear transformations induce logarithmic growth in the Fourier partial sums.

Contribution

It demonstrates that for continuous piecewise linear but nonlinear maps, the Fourier series norm grows logarithmically, highlighting specific convergence behaviors.

Findings

Fourier series norms grow as log n for certain nonlinear maps

Piecewise linear but nonlinear maps cause specific convergence rates

Provides insight into uniform convergence behavior of Fourier series

Abstract

We consider the space $U(\mathbb T)$ of all continuous functions on the circle $\mathbb T$ with uniformly convergent Fourier series. We show that if $\varphi: \mathbb T\rightarrow\mathbb T$ is a continuous piecewise linear but not linear map, then $\|e^{in\varphi}\|_{U(\mathbb T)}\simeq\log n$.