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Moment problem ：ウィキペディア英語版

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
:

m_n = \int_^\infty x^n \,d\mu(x)\,.\,\!

More generally, one may consider
:

m_n = \int_^\infty M_n(x) \,d\mu(x)\,.\,\!

for an arbitrary sequence of functions ''M''_''n''.
== Introduction ==

In the classical setting, μ is a measure on the real line, and ''M'' is the sequence . 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].

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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