I am trying to prove the localization theorem which was mentioned in Rudin but not proved. It states that if $f$ and $g$ are continuous functions on $\mathbb R$.
Both of which have periods $2\pi$. Then if $f(x) = g(x)$ for $\lvert x - x_0 \rvert$ $\leq$ $\delta$. We have to show that both $S_{n,f}(x_0)$ and $S_{n,g}(x_0)$ either both converge to the same value or diverge. Here $S_{n,f}(x_0)$ is the n-th partial sum of the fourier series of f and $S_{n,g}(x_0)$ is the nth partial sum of the fourier series of g.
Now first we have to show that $S_{n,f}(x) =\frac{1}{\pi} \int_0^\delta$ $\frac{(f(x-u) + f(x+u)}{u} \sin((n+{1 \over 2})u)du$ $+$ $\epsilon_n(x)$. Where $\epsilon_n(x)$ is the error and goes to 0 as $n \rightarrow \infty$ How do we first show this and use it to prove the localization theorem?
A hint was given to me that we should use the Dirichlet kernel as $S_{n,f}(x)$ = $\frac{1}{\pi}$$\int_0^\pi$ ${(f(x-u) + f(x+u)}$$D_n(u)du$. Then manipulate this integral to get a sum of many integrals but I am not sure on how to do this.