Do all analytic and $2\pi$ periodic functions have a finite Fourier series?

Question

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

Answer 1

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.

Answer 2

[This is false - do not believe it:]

Any uniformly convergent sum of analytic functions is again analytic. So you can construct as many counterexamples to your question as you want by taking sequences $\{ \dots, a_{-1}, a_0, a_1, \dots \}$ whose sum $\sum_{n=-\infty}^\infty a_n$ absolutely converges; the corresponding Fourier series $\sum_{n=-\infty}^\infty a_n e^{inx}$ will then be an analytic function.