# Pointwise convergence of Fourier series of function $\sqrt{|x|}$

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.

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