Let $\overline{f}$ be a function on the whole real line, such that $\overline{f}$ is continuous and differentiable everywhere, and its derivative $\overline{f}'$ is also continuous everywhere. Now, restrict $\overline{f}$ to a function $f$ defined only on the interval $(0, \pi)$.
Does Fourier sine series of $f$ always converges to $f$ pointwise on $(0, \pi)$? I know that it does not converge uniformly on $(0, \pi)$.
What about Fourier cosine series? Does Fourier cosine series of $f$ always converge to $f$ uniformly on $(0, \pi)$?
My professor hasn't covered much in convergence, so I want to know more about the convergence of functions.