Decay of Fourier coefficients of infinitely differentiable functions For a $C^n[-\pi,\pi]$ function $f$ we have that $|\hat{f}(k)|\in O(1/k^n)$. This implies that if $f$ is $C^\infty[-\pi,\pi]$ then its $k-th$ Fourier coefficient decays faster than any $1/k^n$, $n\geq0.$ My questions:
$\bullet$ There is a name for this order of convergence?(something like "Schwarz order" if there exists such a denomination)
$\bullet$ Can't we say more than this? Y mean, there are a stronger result that states that these coefficients decay at an exponential rate (for example)? 

Well first, what you say about $C^\infty([-\pi,\pi])$ is not true. For example, $f\in C^\infty([-\pi,\pi])$ if $f(t)=t$, but the coefficients of that function are not $O(1/k^2)$.
Let's talk about $2\pi$-periodic functions - that's what you meant, or what you should have meant. I'd just say the coefficients were rapidly decreasing, and specify what that meant if it mattered.
No, there's no more that can be said. It's easy to see that if $\hat f(k)=O(1/k^{n+2})$ then $f\in C^n$ (somewhat stronger results in this direction exist that are not quite as easy). Hence if $\hat f(k)=O(1/k^n)$ for all $n$ then $f\in C^\infty$.
