What is the convergence rate of Fourier series?

[zheng89120]

Okay, so I didn't really understand the professor when he talked about the speed of convergence of Fourier series. The question is what kind of functions converge faster than what kind of other functions using Fourier series representation. My guess from what I have absorbed is that functions with the least continuity and resembles a cosine function the most converges the fastest. Obviously this is not entirely right, and I would like more explanation on this matter.

Thank you.

P.S. How does Gibb's phenomenon tie into the convergence of Fourier series. My guess from what I know is that functions that converge slower will have a higher Gibb's "overshoot".

 

[fresh_42]

You should look up "convergence rate of Fourier series", e.g.
http://math.uchicago.edu/~may/REU2014/REUPapers/Zou.pdf
http://www.math.utk.edu/~freire/m435f07/m435f07fourierconvergence.pdf
And yes, the Gibb's phenomenon has to do with it, see e.g.
http://www.ee.ic.ac.uk/hp/staff/dmb/courses/E1Fourier/00500_GibbsPhenomenon_p.pdf