# Paley–Wiener theorem for functions with exponential decay

I feel like this should be well-known, but haven't been able to find any reference so far. Consider the set of all smooth functions on $\mathbb{R}$ such that $$\sup_{x\in \mathbb{R}} |e^{\alpha x} f^{(n)}(x)|<\infty \qquad \mbox{for all n\in \mathbb{N}, \alpha \in \mathbb{C}. }$$ Then the two-sided laplace transform of such a functions is well defined, and gives rise to a holomorphic function on $\mathbb{C}$. Is there a Paley-Wiener kind of theorem that gives a characterization of the space of functions that can be generated in this way? (That is, as Laplace transforms of smooth functions with exponential decay for all its derivatives.)

Let me write $F$ for the FT of $f$. Then I claim that $f$ satisfies your conditions if and only if $F$ is an entire function such that for any $n\ge 0$, $L\ge 1$, we have that $(x+iy)^nF(x+iy)$ is bounded on $-L\le y\le L$ and $$\sup_{-L\le y\le L} \int_{-\infty}^{\infty} |x+iy|^n |F(x+iy)|\, dx < \infty .$$
It is obvious that $F(z)=\int f(t) e^{itz}\, dt$ has these properties if $f$ is as above (note that the smoothness of $f$ gives $F$ enough decay to make it integrable). Conversely, if $F$ satisfies these conditions, then we obtain the desired estimates on $f^{(n)}(x)=\int (-iz)^n F(z) e^{-itz}\, dz$ by moving the path of integration into the complex plane, to pick up the required exponentially small factors.