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$f(x)=|x|, x\in[-\pi,\pi[$ is a $2\pi$-periodic function. Does the Fourier series for $f(x)$ converge uniformly to $f(x),x\in\mathbb{R}$?

Answer in my book is yes, but how can it when $f'(x)=\frac{x}{|x|}$ is not defined at $x=0$ i.e. function is not piecewise differentiable (smooth)?

Mathias
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  • The uniform convergence of the Fourier series does not tell you anything about $f'$; the convergence rate does. What is the confusion? –  Nov 12 '21 at 17:07
  • @Zarrax you are correct, I was confusing this with the Holder condition. Thanks for the correction! – Josh B. Nov 12 '21 at 17:20
  • The fact that the function is Lipschitz is enough. See https://en.wikipedia.org/wiki/Convergence_of_Fourier_series#Uniform_convergence – Zarrax Nov 12 '21 at 17:21

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