a Fourier series question Suppose $f(x)$ is a function with the property that $f(x)$ and $f'(x)$ are both piecewise continuous on the interval $[a, b]$, then the Fourier series of $f(x)$ (either the sine, cosine or full Fourier series) converges pointwise to $f(x)$ on the interval $(a, b)$. 
I'm not sure how to construct a proof for this statement.  Can you please give me a hint or a sketch of the proof?  
 A: Here is a proof based on Paul Chernoff's Math 202A class notes from UC Berkeley (from a few decades ago):
By shifting & rescaling, we may assume that the interval in 
question is $[0,2\pi]$, this simplifies the following formulae.
If $f'$ is piecewise continuous then it is bounded and hence $f$ is
Lipschitz.
Pick $x^* \in  (0,2 \pi)$ (this avoids an issue with the fact that we may 
have $f(0) \neq f(2 \pi)$). Extend $f$ in a $2\pi$ periodic way to $\mathbb{R}$ to simplify the following formulae.
Let $g(x) = {f(x+x^*)-f(x^*) \over e^{ix}-1}$ and note that $g$ is
integrable (it is bounded near $0$ since $f$ is Lipschitz, this is the key element of this proof), hence it has a Fourier series.
Since $f(x+x^*) = f(x^*)+ (e^{ix}-1 ) g(x)$, we have
$e^{ikx^*}\hat{f}_k = f(x^*) \hat{1}_k  + \hat{g}_{k-1} - \hat{g}_k$.
With $m,n \ge 0$, we have
$\sum_{k=-m}^n e^{ikx^*}\hat{f}_k = f(x^*) + \hat{g}_{-m-1}- \hat{g}_n $, and using the Riemann Lebesgue lemma we have
$\lim_{m,n \to \infty} \sum_{k=-m}^n e^{ikx^*}\hat{f}_k = f(x^*)$.
