Yesterday I posted a question regarding the computation of complex Fourier coefficients for the functions
$$f(t) = \sin(2 \pi t)$$ $$f(t) = |\sin(2 \pi t)|$$
where $0 \leq t \leq 1$.
The first function can easily be represented as a finite sum with Fourier coefficients $c_{-1} = -1/2i$ and $c_1 = 1/2i$. However, the second function is not smooth at $t = 1/2$ and I was then informed that because of this, the Fourier series will be infinite.
I thus wanted to post the general question:
Do there exist any discontinuous functions that can be represented by a finite Fourier series?
My intuition says no, but does anyone have a link to a proof for this?
Thanks in advance!
A finite sum of trig functions is not only continuous but $C^\infty$: infinitely smoothly differentiable. So any kind of discontinuity in any derivative results in an infinite sum. Even if 9th derivative is discontinuous!