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.
Multi-dimensional moment problemMulti-dimensional moment problem
