
> 
> **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.
> 
> 
> 



real analyticity, Fourier coefficients
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.)


