The Classical Moment Problem And Some Related Questions In Analysis _top_ Page

Given a measure $\mu$, we can orthogonalize the monomials $1, x, x^2, \dots$ in $L^2(\mu)$ to get orthogonal polynomials $P_n(x)$. The recurrence relation

: Always determinate. The moments uniquely define the measure due to the Weierstrass Approximation Theorem Hamburger/Stieltjes Problems : Can be indeterminate. For example, the log-normal distribution Given a measure $\mu$, we can orthogonalize the

In quantum mechanics, moments of position correspond to expectation values $\langle x^n \rangle$. The question "Is the Hamiltonian self-adjoint?" is intimately related to the moment problem. The classic example: the "Stieltjes moment problem" appears in the study of the anharmonic oscillator $H = p^2 + x^4$. The measure of the ground state is determinate, guaranteeing a unique quantum theory. Given a measure $\mu$