A $C^{\infty}$ function on $\mathbb R$ is real analytic if for every $x\in \mathbb R$, $f$ is the sum of its Taylor series expansion bases at $x$ in some neighbourhood of $x$.
If $f$ is periodic we may regard $f$ as a function on $S=\{z\in \mathbb C: |z|=1\}$, this condition is equivalent to the condition that $f$ be the restriction on $S$ of a holomorphic function on some neighbourhood of $S.$ (Why?)
Also, how to show that $f\in C^{\infty}(\mathbb T)$ is real analytic iff $|\hat{f}(k)|\leq C e^{-\epsilon |k|}$ for some $C>0$ and $\epsilon>0.$