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Stieltjes moment problem

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In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (m0, m1, m2, ...) to be of the form

for some measure μ. If such a function μ exists, one asks whether it is unique.

The essential difference between this and other well-known moment problems is that this is on a half-line [0, ), whereas in the Hausdorff moment problem one considers a bounded interval [0, 1], and in the Hamburger moment problem one considers the whole line (−, ).

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Existence

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Let

be a Hankel matrix, and

Then { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on with infinite support if and only if for all n, both

{ mn : n = 1, 2, 3, ... } is a moment sequence of some measure on with finite support of size m if and only if for all , both

and for all larger

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Uniqueness

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There are several sufficient conditions for uniqueness.

Carleman's condition: The solution is unique if

Hardy's criterion: If is a probability distribution supported on , such that , then all its moments are finite, and is the unique distribution with these moments.[1][2][3]

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References

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