Moment problem

In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ to the sequence of moments More generally, one may consider for an arbitrary sequence of functions Mn. In the classical setting, μ is a measure on the real line, and M is the sequence { xn : n = 0, 1, 2, ... }. In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique. There are three named classical moment problems: the Hamburger moment problem in which the support of μ is allowed to be the whole real line; the Stieltjes moment problem, for [0, +∞); and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as [0, 1]. A sequence of numbers mn is the sequence of moments of a measure μ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices Hn, should be positive semi-definite. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional such that and (non-negative for sum of squares of polynomials). Assume can be extended to . In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional is positive for all the non-negative polynomials in the univariate case. By Haviland's theorem, the linear functional has a measure form, that is . A condition of similar form is necessary and sufficient for the existence of a measure supported on a given interval [a, b]. One way to prove these results is to consider the linear functional that sends a polynomial to If mkn are the moments of some measure μ supported on [a, b], then evidently Vice versa, if () holds, one can apply the M. Riesz extension theorem and extend to a functional on the space of continuous functions with compact support C0([a, b]), so that By the Riesz representation theorem, () holds iff there exists a measure μ supported on [a, b], such that for every ƒ ∈ C0([a, b]).