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?
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This might be true if the variation in the function away from $\alpha$ is small relative to the jump. Otherwise, you can make the function experience a huge, continuous rise at another place, relative to the jump discontinuity. – Alex R. Jan 21 '20 at 21:22
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There are continuous functions whose Fourier series diverge at a point, or even at a dense set of points. Adding a jump discontinuity is unlikely to improve matters. – Robert Israel Jan 21 '20 at 21:25
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Robert, you are correct about that. A classic example is explained in Volume 1 of the Princeton Lecture Series in Analysis by Stein and Shakarchi. I'm not interested in pathological behavior. So, I'll assume whatever is necessary to eliminate weird cases. – Alan C. Jan 21 '20 at 23:39