Given a finite set of moments $(m_n)$ for $n\in V \subset \mathbb Z^+$, is there a way to determine that there exists a probability distribution $d\mu(x)$ such that $m_n = \int x^n d\mu(x)$ for $n\in V$? (Assuming we always have $m_0 = 1$.)
In particular, can this question be answered without constructing all possible $(\tilde m_n)_{n\in \mathbb Z^+}$ such that $\tilde m_n = m_n$ for $n\in V$? If I have all moments, then a positive semidefinite matrix given by the elements $H_{i,j} = \tilde m_{i+j}$ implies I can construct such a probability distribution, but is there a way to see that this matrix can be made positive semidefinite without testing all ways to construct $\tilde m_n$?