The Classical Moment Problem And Some Related Questions In Analysis -

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:

$$ m_n = \int_\mathbbR x^n , d\mu(x) $$

$$ x P_n(x) = P_n+1(x) + a_n P_n(x) + b_n P_n-1(x) $$ At first glance, this seems like a straightforward

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 $\mu$: You ask: "What is the average squared position

Imagine you are given a mysterious black box. You cannot see inside it, but you are allowed to ask for specific "moments." You ask: "What is the average position?" The box replies: $m_1 = 0$. You ask: "What is the average squared position?" It replies: $m_2 = 1$. You continue: $m_3 = 0$, $m_4 = 3$, and so on. $m_4 = 3$

$$ \sum_i,j=0^N a_i a_j m_i+j \ge 0 $$