What is the analog of the Paley-Weiner theorem for Fourier series or what is the applicable reference. I would think it is that for the Fourier series expansion $f(x) = \sum_n c_n e^{inx}$, the coefficients $c_n$ decay as $\exp\{-|b|n\}$ as $n\rightarrow \infty$ if $f(z)$ is analytic in a strip of width $b$ in the complex plane. Could you point out if this is wrong, or point me to a reference which states the theorem and proof in its entirety.
Sorry if this is a repeat, I tried to search but only found posts about the Fourier Transform.
We assume the width of the extension strip on both sides of the real axis is $\frac{b}{2}$.
Consider the analytic function $\sum_{n \ge 0}{c_nw^n}, w=e^{a+ix}$. By Riemann Lebesgue this is well defined in the unit disc $|w| \le 1$ or $-\infty \le a \le 0$ as $c_n \to 0$ hence are bounded and your condition implies it extends to the disc of radius $e^{\frac{b}{2}}$ so by the usual theorems on power series coefficients $|c_n| =O_{\epsilon} (\exp{(-n(\frac{b}{2}-\epsilon)}), n \ge 0$.
Similarly, considering the analytic function $\sum_{n \ge 0}{c_{-n}w^n}, w=e^{-a-ix}$ we get the same estimates for $|c_{-n}| , n \ge 0$
I do not think you can say more since you can just pick an arbitrary analytic function with convergence radius $e^{\frac{b}{2}}$ and "unpack" it in a Fourier series for $c_n, n \ge 0$ and then take another (or same) and unpack it for $c_n, n \le 0$ and then obviously you can extend in the strip of width $\frac{b}{2}$ on each side of the real axis but no more.