I am trying to solve the following exercise:

Let $ f(x) = \sqrt{|x|}$ , $ x\in\mathbb{R}$ . 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 example of pointwise convergence 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.