There are several approaches to practical resolution of the problem. The moment data can typically be interpreted as linear constraints on the density function that is recovered by minimizing a functional. Depending on the application, the functional might be energy, Shannon entropy, or some other application-specific quantity.

Take a look at my answer to an earlier question. Despite the fact that the earlier question dealt with multivariate measures, the references in the answer should give you a good head start.

https://mathoverflow.net/questions/86856/multi-dimensional-moment-problem/86858#86858