If a function $f$ is smooth and compactly supported in $[a, b]$, can we assume anything on the exponential growth of $\hat f$?
For the context, I'm trying to understand how the author of this paper infers "$\hat G(x) D(...)$ is of exponential type in x ..."
(page 14 in the paper; D is a Dirichlet polynomial)
I know that if $f$ is in the Schwartz space, so is its Fourier transform, but this results looks stronger.
Thanks in advance!
For $f \in C^\infty_c(\Bbb{R})$ with $[a,b]$ the convex closure of its support, $c = \max(|a|,|b|)$
Then $$\hat{f}(z) = \int_{-\infty}^\infty f(t) e^{-2i \pi z t}dt, \qquad z \in \Bbb{C}$$ is entire, it is Schwartz on every hozirontal line/strip,
Moreover $|\hat{f}(z)| \le \|f\|_\infty e^{2 \pi c |z|}$ and $\hat{f}(z) \ne O(e^{2 \pi (c-\epsilon) |z|})$
Thus it is an entire function of exponential type $2\pi c$.
Any entire function $G$ of exponential type $2\pi c$ which is Schwartz on every horizontal strip (ie. the decay at $\infty$ is locally uniform) is of this form : letting $y \to sign(t)\infty$ in $g(t) = \int_{-\infty}^\infty G(x+iy) e^{2i \pi (x+iy)t}dx$ shows that $g$ is supported on $[-c,c]$.