Let $f$ be a `very good' function on the real line; say, infinitely differentiable and compactly supported. We are given its Fourier transform on the imaginary axis: $$g(x)=\int_{\mathbb R}f(t)e^{xt}\,dt, \qquad x\in\mathbb R.$$ How can the values of $f$ be restored?
If we know an algorithm of finding $f(0)$, then the value $f(a)$ for any real $a$ can be found by the same algorithm applied to $e^{-iat}g$ in place of $g$. So it suffices to find an answer for a single point $a=0$.
The `bad' answer is, for instance, as follows: find all derivatives of $g$ at the origin, construct the corresponding entire function, take its values on the real axis and apply the inverse Fourier transform. I am interested in more direct procedures; although, unfortunately, I cannot formalize my criteria to consider an answer satisfactory.
As the Fourier transform of an infinitely differentiable and compactly supported function is analytic, you certainly can.