I have some difficulties with the last part of an old exam exercise.
For the 1-periodic function $f$ defined on $[0, 1[$ by $f(x)=e^{2 \pi i \alpha x}$ with $0<\alpha <1$. I have found that its Fourier coefficients are $$c_k=\frac{1}{2 \pi i (\alpha - k)}\left(e^{2\pi i(\alpha - k)}-1\right)$$
Then I showed that $$\vert c_k\vert ^2=\frac{\sin^2(\pi\alpha)}{\pi^2(\alpha -k)^2} \quad \quad (1)$$ and $$\frac{\pi^2}{\sin^2(\pi\alpha)}=\sum_{k=-\infty}^{+\infty}\frac{1}{(\alpha-k)^2} \quad \quad \quad(2)$$
My problem is with the following point : How do I find where the serie $\sum_{k=-\infty}^{+\infty} c_k e^{2\pi i k x}$ converge to, for $x \in \mathbb{R}$. Is it simply to $f$ ? How do I prove it ?
Is the convergence uniform or/and does it converge in the norm of $L^2$ (I don't know if it has a specific name in english but in french it's convergence en moyenne quadratique : $\lim_{n\rightarrow \infty} \Vert f_n-f\Vert_2=0$, i.e. $\int_0^1\vert f_n(x)-f(x)\vert^2 dx \underset{n\rightarrow \infty}{\longrightarrow}0$).
I suppose I have to use $(1)$ and $(2)$ but I could not figure out how. Any indication would be greatly appreciated.
The Fourier series of a smooth (in fact one only needs $C^\alpha$ for some $\alpha>0$) periodic function converges everywhere to the function. This is a famous theorem by Dirichlet.
Convergence in $L^2$ also holds by the Fischer-Riesz theorem for any square-integrable function.
Convergence in $L^2$ can also be referred to as "square mean convergence" in English.