I think the following problem is a well known result, but I have not find some references about it on the Internet.
Let $f\in L^1(\mathbb{T})$, $\sigma_N f$ is the arithmetic means of the partial sums of $f$’s Fourier coefficients, and $x_0$ is a Lebesgue point of $f$. Prove that $$\lim_{N\to \infty}\sigma_N f(x_0)=f(x_0).$$
We know that $$\begin{aligned}\sigma_N f(x)&=\frac{1}{N+1}\sum_{k=0}^N S_k f(x)\\&=\int_0^1f(t)\frac{1}{N+1}\sum_{k=0}^N D_k(x-t)\,dt\\&=\int_0^1 f(t)F_N(x-t)\,dt,\end{aligned} $$ where $F_N(t)$ is the Fejer kernel, $$F_N(t)=\frac{1}{N+1}\sum_{k=0}^N D_k(t)=\frac{1}{N+1}\left(\frac{\sin(\pi(N+1)t)}{\sin(\pi t)}\right)^2\,.$$
Looking forward to your answer, thank you!