If a function is locally Lipschitz, does its Fourier series converge uniformly?

Question

Let $f$ be a $2\pi$-periodic function that is Riemann integrable on $[-\pi,\pi]$. Suppose $f$ is locally Lipschitz. That is, for every $x \in \mathbb{R}$, there exists a $\delta > 0$ and $M < \infty$ such that $|f(x+t)-f(x)| \leq M|t|$ for all $t \in (-\delta,\delta)$. Then does the Fourier series of $f$ converge to $f$ uniformly?

Thank you.

Answer 1

Locally Lipschitz implies Lipschitz by compactness of $[-\pi, \pi]$. Hence the Fourier series converges uniformly.