Positivity of Hankel matrices of moments

Remarks

For real $c = (c_0, \ldots, c_n)$, let $p(x) = \sum_{j=0}^{n} c_j x^j$. Then $c^{T} H_n c = \sum_{j,k} c_j c_k \int x^{j+k} \, d\mu = \int p(x)^2 \, d\mu(x) \ge 0$, which is clause 1. If $c \ne 0$, then $p$ is a nonzero polynomial with at most $n$ real roots. If the support is infinite, then $p^2 > 0$ on a set of positive measure, so the integral is positive, which is clause 2. Example: for Lebesgue measure on $[0, 1]$, $m_k = 1/(k+1)$, and $H_n$ is the Hilbert matrix $(1/(j + k + 1))$, which is positive definite. For $\mu = \delta_0$, the support is finite, $H_n$ has $m_0 = 1$ in one corner and zeros elsewhere, and it is only semidefinite for $n \ge 1$. The converse of clause 1 is Hamburger's theorem, part of the Hamburger moment problem. It says that a real sequence with all $H_n$ positive semidefinite is the moment sequence of a positive measure on $\R$.

Going deeper

For a student

This is the necessary condition in the Hamburger moment problem, and Hamburger showed that it is also sufficient. For the Stieltjes problem on $[0, \infty)$ one adds the shifted Hankel matrices $(m_{j+k+1})$, whose positivity reflects $x \ge 0$. The positive definite case gives an inner product $\ip{p}{q} = \int p q \, d\mu$ on polynomials, and Gram–Schmidt then yields the orthogonal polynomials of $\mu$.