On uniform convergence of Fourier series

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 $φ: \mathbb T\rightarrow\mathbb T$ is a continuous piecewise linear but not linear map, then $\|e^{inφ}\|_{U(\mathbb T)}\simeq\log n$.

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