Stieltjes moment problem


In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence 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 , and in the Hamburger moment problem one considers the whole line.

Existence

Let
be a Hankel matrix, and
Then is a moment sequence of some measure on with infinite support if and only if for all n, both
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.
Carleman's condition: The solution is unique if
Hardy's criterion: If is a probability distribution supported on, such that, then all its moments are finite, and is the unique distribution with these moments.