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?
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 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. The central question of the is: Can you
$$ x P_n(x) = P_n+1(x) + a_n P_n(x) + b_n P_n-1(x) $$
$$ S(z) = \int_\mathbbR \fracd\mu(x)x - z, \quad z \in \mathbbC\setminus\mathbbR $$ If yes, is that measure unique
At first glance, this seems like a straightforward problem of "matching moments." But as we will see, it opens a Pandora's box of deep analysis, touching functional analysis, orthogonal polynomials, complex analysis, and even quantum mechanics. In probability and analysis, a moment is a generalization of the idea of "average power." For a real random variable $X$ with distribution $\mu$ (a positive measure on $\mathbbR$), the $n$-th moment is:
For the Stieltjes problem (support on $[0,\infty)$), we need an extra condition: both the Hankel matrix of $(m_n)$ and the shifted Hankel matrix of $(m_n+1)$ must be positive semidefinite. In probability and analysis, a moment is a
$$ m_n = \int_\mathbbR x^n , d\mu(x) $$
$$ \sum_i,j=0^N a_i a_j m_i+j \ge 0 $$
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 $\mu$: