Stieltjes moment problem

In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence { m_n, : 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 (−∞, ∞).

Let

and

Then { m_n : n = 1, 2, 3, ... } is a moment sequence of some measure on ${\displaystyle [0,\infty )}$ with infinite support if and only if for all n, both

{ m_n : n = 1, 2, 3, ... } is a moment sequence of some measure on ${\displaystyle [0,\infty )}$ with finite support of size m if and only if for all ${\displaystyle n\leq m}$, both

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