TITLE: Recovering a measure from its moments
QUESTION [6 upvotes]: Suppose we are given moments of a measure on the interval [0,1]. Is there some practical way to recover the measure itself? I am particularly interested in the case where the measure density is given by some smooth positive function, and I would like to determine the behavior of this function near x=0.

REPLY [3 votes]: An answer on your question is given in the following book: 
[ST] J. A. Shohat and J. D. Tamarkin "The problem of moments". 
See [ST], page 90 and what follows. In particular, the case of an absolutely continuous measure is considered on page 95.