Let $f:\mathbb R \to \mathbb C$ be a smooth (with that I mean $C^\infty$) $1$-periodic function. So there are $a_n \in \mathbb C$ with $$f(x) = \sum_{n \in \mathbb Z} a_n \exp(2 \pi i nx).$$ Question 1: What can we say about the decay of the $a_n$?
On the other hand: Let's have $b_n \in \mathbb C$ such that $$g(x) := \sum_{n \in \mathbb Z} b_n \exp(2 \pi i nx)$$ converges absolutely for all $x$, which is equivalent to $$\sum_{n \in \mathbb Z} |b_n| < \infty.$$ Question 2: Can we infer that for very fast decay of the $b_n$ (if yes, how fast?) the function $g$ is smooth?
The best result would be an equivalence. So that there is one magical bound sich that the function is smooth if and only if the Fourier coefficients decay faster than this magical bound.