TITLE: real analyticity, Fourier coefficients
QUESTION [5 upvotes]: Question. Suppose $f$ is periodic in $[0,2\pi]$. What conditions on the Fourier coefficients of $f$ would guarantee real analyticity of $f$? Please provide me with a reference.

REPLY [10 votes]: A function $f:\mathbb{R}\to\mathbb{C}$ periodic by $2\pi$ is real analytic if and only if it extends holomorphically to $\{z\in\mathbb{C}:\ |\Im z|<c\}$ for some $c>0$, because the interval $[0,2\pi]$ is compact. The latter condition is easily seen to be equivalent to the exponential decay of the Fourier coefficients: $a_n\ll e^{-c|n|}$ for come $c>0$. (Necessity of this condition follows from Cauchy's theorem about integrals of holomorphic functions. Sufficiency of the condition follows from Morera's theorem and Cauchy's theorem about the analyticity of holomorphic functions.)

REPLY [6 votes]: The condition is $|c_n|\le A e^{-B|n|}$ for some positive constants $A,B$ and all $n$. This is not difficult to prove directly (a reference might be exercise I.4.4 in Katznelson's Introduction to Harmonic Analysis, but it is given as an exercise...).