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 (−∞, ∞).

Existence

Let be a Hankel matrix, and Then { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on [0,\infty) with infinite support if and only if for all n, both { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on [0,\infty) with finite support of size m if and only if for all n \leq m, both and for all larger n

Uniqueness

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