I am trying to solve the following exercise:
Let $f(x) = \sqrt{|x|}$, $x\in[-\pi,\pi]$. Show that the Fourier series $s_n(0)$ converges to $f(0)$.
The hint is that one should consider the convolution with the Dirichlet Kernel and the Riemann-Lebesgue lemma. This approach yields $$s_n(0) = \int_{-\pi}^\pi f(t)\frac{\sin[(N+1/2)t]}{\sin t/2}dt = \int_{-\pi}^\pi f(t)\sin nt\cot(t/2)dt + \int_{-\pi}^\pi f(t)\cos nt dt,$$ and while the integral on the right tends to zero, by R-L, I could not estimate the integral on the left. This seems to come primarily from the fact that $\cot t/2$ behaves quite poorly around $t = 0$, with $\lim_{x\to 0} f(x)\cot x = \infty$.
I have searched a few elementary texts on PDE's, including Folland's, Evans' and Strauss', and I could not find any examples of pointwise convergence questions with a function of unbounded derivative. Moreover, the only related question that I found on MSE was this one one, but in this case the function is odd and the integrals vanish trivially. Any help would be appreciated.
The integrand is even so this reduces to an integral over $[0,\pi]$. Consider the contributions from $[0,\delta]$ and $[\delta, \pi]$ separately.
The contribution over $[\delta,\pi]$ converges to $0$ directly by the Riemann-Lebesgue lemma since $f(t)/\sin(t/2)$ is integrable on that interval.
For the contribution over $[0,\delta]$ use the fact that $f$ is nondecreasing and continuous. By the second mean value theorem, there exists $\xi \in (0, \delta)$ such that
$$\int_0^\delta f(t) D_N(t) \, dt = f(\delta)\int_\xi^\delta D_N(t) \, dt$$
By continuity, for any $\epsilon > 0$ there exists a sufficiently small $\delta$ such that for $0 \leqslant x \leqslant \delta$, we have $f(x) < \epsilon$.
Again using the Riemann-Lebesgue lemma, we have as $N \to \infty$,
$$\left|f(\delta)\int_\xi^\delta D_N(t) \, dt \right| \leqslant \epsilon\left|\int_\xi^\delta D_N(t) \, dt \right| \to 0 $$
Therefore, $\lim_{N \to \infty} S_N(0) = 0 = f(0).$