Paley–Wiener theorem for functions with exponential decay

Question

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.)

Answer 1

We can certainly characterize these functions in terms of their Fourier transforms, but what I'm about to write down just rephrases the well known connection between holomorphic (on a strip) Fourier transforms and exponential decay, and it's not very exciting.

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.