I know that for a continously differentiable $2 \pi$ periodic function $f \in L^2(-\pi,\pi)$ its Fourier series converges uniformly and that if I only suppose that the function is continous everywhere and continously differentiable except that for a finite number of points, where $f$ admits right and left derivative , there is pointwise convergence.
I was asked to prove that in this second situation, we have uniform convergence on the compact subsets of $(-\pi,\pi)$ that do not contain any of the points where the derivative isn't defined.
I tried to write the Fourier serie as convolution of $f$ with $D_n$, the Dirichlet kernel and refining the standard estime about the pointwise convergence, but i wasn't able to find a uniform condition because that estime is made by Riemann-Lebesgue lemma, which does not seem uniform in any way to me.
Can someone help me?
On intervals where $f'$ is continuous on $(a,b)$ with a left derivative at $a$ and a right derivative at $b$, integration by parts gives a uniform Riemann-Lebesgue estimate: $$ \int_{a}^{b}f(t)e^{-int}dt= \left.f(t)\frac{e^{-int}}{-in}\right|_{a}^{b}+\frac{1}{in}\int_{a}^{b}f'(t)e^{-int}dt. $$