Moment problem

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

More generally, one may consider

for an arbitrary sequence of functions M_n.

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 m_n is the sequence of moments of a measure μ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices H_n,

should be positive semi-definite. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional ${\displaystyle \Lambda }$ such that ${\displaystyle \Lambda (x^{n})=m_{n}}$ and ${\displaystyle \Lambda (f^{2})\geq 0}$ (non-negative for sum of squares of polynmials). Assume ${\displaystyle \Lambda }$ can be extended to ${\displaystyle \mathbb {R} [x]^{*}}$. In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional ${\displaystyle \Lambda }$ is positive for all the non-negative polynomials in the univariate case. By Haviland theorem, the linear functional has a measure form, that is ${\displaystyle \Lambda (x^{n})=\int _{-\infty }^{\infty }x^{n}d\mu }$. A condition of similar form is necessary and sufficient for the existence of a measure ${\displaystyle \mu }$ supported on a given interval [a, b].

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