This M.O question reminded me of a problem I've seen a while ago and was not able to solve. Any reference about this kind of problem would be nice.

The problem is the following :

Does there exist a *signed* measure $\mu$ on $[0,1]$ such that
$$\int_{0}^{1} x^n d\mu = e^{-n^2}$$
for all $n \geq 1$?

Note that this is related to the so-called Hausdorff moment problem, which however seems to deal only with positive measures.

It is quite easy to see that there is no *positive* measure satisfying the above, but does there exist a signed one?

Thank you, Malik