Consider a function $f:\mathbb{R}\to\mathbb{R}$ which is periodic with period $2\pi$. Let us impose the condition that $f$ is analytic. Now does that imply that $f$ has a finite Fourier series?
PS : Although this question seems to be related to this, I couldn't find anything that I can understand there
Best Answer
Fourier series represents an analytic function if and only if its coefficients decrease at least as a geometric progression: $$\limsup_{n\to\infty}\,(|a_n|+|b_n|)^{1/n}=q<1.$$ This fact can be found in books on Forier series.