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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

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The case of a signed measure was studied by Hausdorff himself:

F. Hausdorff, Momentprobleme für ein endliches Intervall. Math. Ztschr. 16, 220-248 (1923).

http://www.digizeitschriften.de/dms/resolveppn/?PPN=GDZPPN002446111

The paper contains some conditions for this case, and you can try to check them.

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