For the Hamburger problem, this condition is also sufficient (a theorem of Hamburger, 1920): A sequence $(m_n)$ is a Hamburger moment sequence if and only if the Hankel matrix is positive semidefinite.
$$ \sum_i,j=0^N a_i a_j m_i+j \ge 0 $$
We assume all moments exist (are finite). The classical moment problem asks: Given a sequence $(m_n)_n=0^\infty$, does there exist some measure $\mu$ that has these moments? If yes, is that measure unique? For the Hamburger problem, this condition is also
$$ S(z) = \int_\mathbbR \fracd\mu(x)x - z, \quad z \in \mathbbC\setminus\mathbbR $$ If yes, is that measure unique
encodes all the moments. The measure is determinate iff the associated (a tridiagonal matrix) is essentially self-adjoint in $\ell^2$. Indeterminacy corresponds to a deficiency of self-adjoint extensions—a concept from quantum mechanics. Complex Analysis and the Stieltjes Transform Define the Stieltjes transform of $\mu$: For the Hamburger problem
The central question of the is: Can you uniquely reconstruct the contents of the box—specifically, a measure or a probability distribution—from this infinite sequence of moments?