In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Stieltjes moment problem, named after Thomas Joannes Stieltjes

Thomas Joannes Stieltjes

Thomas Joannes Stieltjes was a Dutch mathematician. He was born in Zwolle and died in Toulouse, France. He was a pioneer in the field of moment problems and contributed to the study of continued fractions....

, 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 (−∞, ∞).

Existence

Let


and


Then { m_n : n = 1, 2, 3, ... } is a moment sequence of some measure on 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 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

Carleman's condition

In mathematics, Carleman's condition is a sufficient condition for the determinacy of the moment problem.-Hamburger moment problem:For the Hamburger moment problem, the theorem, proved by Torsten Carleman, states the following:...

, which states that the solution is unique if