Convergence of Fourier series at jump discontinuities

Question

Suppose $f$ is a bounded piecewise-continuous periodic real-valued function. We assume the function is nice enough so that it has a Fourier series representation that converges to $f$ everywhere. For example, at least assume $f$ is differentiable everywhere except at the jump discontinuities and that $|f'|$ is bounded. Let there be a jump discontinuity at $\alpha$. Also suppose that $f$ is correctly defined at $\alpha$ so that $f(\alpha)=\tfrac{1}{2}\big(\lim_{x\to\alpha^{+}}f(x)+\lim_{x\to\alpha^{-}}f(x)\big)$. As I have done numerical examples, it seems that the partial sums of the Fourier series (either with or without a kernel like the Fejer kernel) best approximate the function exactly at $\alpha$. Slightly away from $\alpha$ the approximation is bad due to the jump discontinuity and the Gibb's effect. However, $\alpha$ and its translates seem to be more special than all other points. Is there a theorem about this?