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 { mn, : n = 0, 1, 2, ... } 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 (−∞, ∞).

Existence

Let

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

Uniqueness

There are several sufficient conditions for uniqueness, for example, Carleman's condition, which states that the solution is unique if

References

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