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From Wikipedia, the free encyclopedia
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 (−∞, ∞).
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
There are several sufficient conditions for uniqueness, for example, Carleman's condition, which states that the solution is unique if
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