I'd love your help proving that if $f$ is an infinitely differentiable function, then $\lim_{n \to \infty} n^k \hat f = 0$, where $\hat f (n)$ is the Fourier coefficient for $n$. I wanted to use the Riemann-Lebesgue theorem that $\hat f (n)_{n \to \infty} \to 0$ and the fact that \hat f\,' (n)= in\hat f(n), and to use L’Hôpital's Theorem, but it didn't work.
Any help?
Thanks a lot!