Let $C_0^\infty(\mathbb{R}^N)$ denote the space of infinite differentiable functions with compact support. My question is: Is there any characterization of $C_0^\infty(\mathbb{R}^N)$ in terms of Fourier tranform, i.e. is there a statemant like this:
$f\in C_0^\infty(\mathbb{R}^N)$ if and only $\hat{f}$ satisfies ... ?
Thank you
This is about the Paley-Wiener theorems, as enhanced by Schwartz and others: the Fourier transforms of the test functions are exactly the functions that extend to holomorphic functions on $\mathbb C^n$ bounded by (writing in the single-variable case for simplicity) $|\hat{f}(x+iy)| \cdot (1+x^2)^M \ll e^{Ny}$ for all $M$ and for some $N$.
Proofs are not too hard, and are written many places, e.g., http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/paley-wiener.pdf
