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Problem of moments)
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 dμ
where M runs over a set of monomials. In the classical setting, μ is a measure on the real line, and M is in the sequence
- xn
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 what extent a probability measure may have specified its mean, variance and so on.
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 finite interval, which without loss of generality may be taken as [0,1]. For example, the uniqueness of μ in the Hausdorff moment problem follows because polynomials are dense in the uniform norm on [0,1]. It is the question of existence that matters. It was realised that this is closely connected to Hilbert spaces and spectral theory. In more concrete terms, the condition on a positive measure μ that
- ∫|P|2dμ > 0
for any complex-valued polynomial P(t) gives rise to matrix conditions, necessary on any sequence of moments, namely that some Hankel matrices are positive definite.