The Classical Moment Problem And Some Related Questions In Analysis 🎁 🎁

for all finite sequences $(a_0,\dots,a_N)$. This means the infinite $H = (m_i+j)_i,j=0^\infty$ must be positive semidefinite (all its finite leading principal minors are $\ge 0$).

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

$$ x P_n(x) = P_n+1(x) + a_n P_n(x) + b_n P_n-1(x) $$ for all finite sequences $(a_0,\dots,a_N)$

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?

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

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?

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

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 $\mu$: