On uniform convergence 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 $\phi :\mathbb{T}\to \mathbb{T}$ is a continuous piecewise linear but not linear map, then $\Vert e^{in\phi }{\Vert }_{U(\mathbb{T})}\simeq \log n$.

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• Uniformly convergent Fourier series and multiplication of functions