Some Related Questions In Analysis: The Classical Moment Problem And

$$ x P_n(x) = P_n+1(x) + a_n P_n(x) + b_n P_n-1(x) $$

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

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

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 $\mu$: $$ x P_n(x) = P_n+1(x) + a_n P_n(x)