Moment problem

In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure &mu; to the sequences of moments


 * $$\int_{-\infty}^\infty M_n(x)\,d\mu(x)\,$$

where Mn(x) is the nth in a list of monomials, for n = 0, 1, 2, 3, ... .

Introduction
In the classical setting, &mu; is a measure on the real line, and M is in the sequence { xn : n = 0, 1, 2, ... }, giving moments mn for n = 0, 1, 2, 3, ... . It is in this form that the question would appear in probability theory, of asking to whether there is a  probability measure having specified mean, variance and so on.

There are three named classical moment problems: the Hamburger moment problem in which the support of &mu; is allowed to be the whole real line; the Stieltjes moment problem, for [0, +&infin;) ; and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as [0, 1].

Existence
It was realized that this is closely connected to Hilbert spaces and spectral theory. In more concrete terms, there is a condition on a positive measure &mu;, namely that


 * $$\int \left|P(x)\right|^2 \, d\mu(x) > 0\,$$

for every complex-valued polynomial P(x), unless P vanishes on the support of &mu;. This gives rise to matrix conditions, necessary on any sequence of moments, namely that certain Hankel matrices are positive semi-definite.

Uniqueness (or determinacy)
The uniqueness of &mu; in the Hausdorff moment problem follows because polynomials are dense in the uniform norm on [0, 1]. For the problem on an infinite interval, uniqueness is a more delicate question; see Carleman's condition, Krein's condition and Ref. 2.

Variations
An important variation is the truncated moment problem, which studies the properties of measures with fixed first k moments (for a finite k). Results on the truncated moment problem have numerous applications to extremal problems, optimisation and limit theorems in probability theory. See also: Chebyshev-Markov-Stieltjes inequalities and Ref. 3.