Let $f \in C^\infty(\mathbb R) \cap L^1(\mathbb R)$. I want to learn necessary and sufficient conditions for the Fourier transform $$\hat f (y) := \int_{\mathbb R} f(x) \exp(-2 \pi i xy) dx$$ to have rapid decay. Meaning that we have $$\lim_{|y| \to \infty} y^k \hat f(y) = 0$$ for all $k \in \mathbb N_0$.
Using the Riemann–Lebesgue lemma and the fact $$\hat{f^{(k)}}(y) = (2 \pi i y)^k \hat f(y)$$ in case $f^{(k)} \in L^1$ we get $\hat f$ has rapid decay if all derivatives of $f$ are in $L^1$. So this condition is sufficient. Is it also necessary?
Which other sufficient and necessary conditions are there?