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.
$$ x P_n(x) = P_n+1(x) + a_n P_n(x) + b_n P_n-1(x) $$
$$ m_n = \int_\mathbbR x^n , d\mu(x) $$
For the Stieltjes problem (support on $[0,\infty)$), we need an extra condition: both the Hankel matrix of $(m_n)$ and the shifted Hankel matrix of $(m_n+1)$ must be positive semidefinite.
For the Hamburger problem, this condition is also sufficient (a theorem of Hamburger, 1920): A sequence $(m_n)$ is a Hamburger moment sequence if and only if the Hankel matrix is positive semidefinite.
$$ S(z) = \int_\mathbbR \fracd\mu(x)x - z, \quad z \in \mathbbC\setminus\mathbbR $$
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:
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.
$$ x P_n(x) = P_n+1(x) + a_n P_n(x) + b_n P_n-1(x) $$ Imagine you are given a mysterious black box
$$ m_n = \int_\mathbbR x^n , d\mu(x) $$
For the Stieltjes problem (support on $[0,\infty)$), we need an extra condition: both the Hankel matrix of $(m_n)$ and the shifted Hankel matrix of $(m_n+1)$ must be positive semidefinite. You continue: $m_3 = 0$, $m_4 = 3$, and so on
For the Hamburger problem, this condition is also sufficient (a theorem of Hamburger, 1920): A sequence $(m_n)$ is a Hamburger moment sequence if and only if the Hankel matrix is positive semidefinite. $$ S(z) = \int_\mathbbR \fracd\mu(x)x - z, \quad
$$ S(z) = \int_\mathbbR \fracd\mu(x)x - z, \quad z \in \mathbbC\setminus\mathbbR $$
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: