# I really don't understand convergence of a Fourier Series...

I'm studying Fourier Series right now so I'm pretty new to most the concepts that come with it. I've practiced finding a function's Fourier Series so I think I can do that pretty well now but I'm still quite lost when it comes to proving convergence of a Fourier Series and especially questions that ask me to find the point where a Fourier Series converges.

I've looked online and in my notes but I just can't really understand it. What I've gathered so far is that for a Fourier series to be convergent we need to check for Piecewise Continuity? But again, I'm not quite sure what this means and how I can apply this.

This is more of a general question so I'm not sure if it's suitable to post here but I thought it was worth a shot since everywhere else doesn't really help much.

Any help would be much appreciated

• Two handy things to keep in mind: the Fourier series of an $L^2$ function $f$ converges in the sense of $L^2$ to $f$. The Fourier series of a bounded variation function $f$ converges pointwise to the local average of $f$ everywhere.
– Ian
May 3, 2021 at 15:41
• @nicomezi Your statement of the Fejer theorem is definitely wrong: you need to Cesaro-sum the Fourier series rather than summing it in the usual way.
– Ian
May 3, 2021 at 15:41
• Which statement ? :o) I tried to make some "global view" and by correcting me multiple times it ended up being completely false. @Ian May 3, 2021 at 16:22
• Have a look at Dirichlet theorem. May 3, 2021 at 16:23
• @nicomezi From your original comment the only problematic one was your claim that the Fejer theorem says the Fourier series of a continuous periodic function converges uniformly to it. The Fejer theorem says that when you Cesaro sum the series rather than summing it the normal way. Otherwise the comment was good.
– Ian
May 3, 2021 at 16:25