We let $f\in$$PC_{2\pi}$ and $f$ be $f(x)=e^{-|x|}$, $x\in[-\pi,\pi$]. I have to found out if the Fourier serie for the function is pointwise convergent on $x\in(-\pi,\pi$) or not?
I have this theorem I think I can use: If $f\in$$PC_{2\pi}$ and let $x_0$$\in R$ be an point where $f$ is continuous and differentiable from both sides. Then is the Fourier serie for $f$ pointwise convergent in $x_0$ with the sum $f(x_0)$.
I think I have to check if: $\frac{f(x_0+t)-f(x_0)}{t}$ have a limiting value for $t->$$0$. Is that correct. But how can I formally check that?
$|e^{-|x|}-e^{-|y|}| \leq ||x|-|y||e^{\pi}$ by MVT applied to the function $e^{x}$. This gives $|e^{|x|}-e^{|y|}| \leq |x-y|e^{\pi}$. Hence $f$ is of bounded varation. For any continuous function of bounded variation the Fourier series converges uniformly.