Continuity and discontinuity in fourier series?

Question

Can somebody please explain continuity and discontinuity in fourier series?

Answer 1

1. If you have a removable discontinuity at a point, the Fourier series will converge to the limit of the function at the point.

2. If you have a jump discontinuity at some point, then the Fourier series will converge to the average of the values of left and right limits. But the higher harmonics are significant, resulting in the "Gibbs phenomenon".

3. For more complicated discontinuities, we really cannot say anything. You can have a look at the wikipedia page for more details.

The best one can say about the convergence of Fourier series is that it always converges almost everywhere, according to Carleson's theorem. But this kind of theorems involve more sophisticated levels of mathematical analysis.