Stieltjes moment problem

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


 * $$m_n = \int_0^\infty x^n\,d\mu(x)$$

for some measure &mu;. If such a function &mu; 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, &infin; ), whereas in the Hausdorff moment problem one considers a bounded interval [0, 1], and in the Hamburger moment problem one considers the whole line (−&infin;, &infin;).

Existence
Let


 * $$\Delta_n=\left[\begin{matrix}

m_0 & m_1 & m_2 & \cdots & m_{n}   \\ m_1 & m_2 & m_3 & \cdots & m_{n+1} \\ m_2& m_3 & m_4 & \cdots & m_{n+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ m_{n} & m_{n+1} & m_{n+2} & \cdots & m_{2n} \end{matrix}\right]$$

be a Hankel matrix, and


 * $$\Delta_n^{(1)}=\left[\begin{matrix}

m_1 & m_2 & m_3 & \cdots & m_{n+1}   \\ m_2 & m_3 & m_4 & \cdots & m_{n+2} \\ m_3 & m_4 & m_5 & \cdots & m_{n+3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ m_{n+1} & m_{n+2} & m_{n+3} & \cdots & m_{2n+1} \end{matrix}\right].$$

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


 * $$\det(\Delta_n) > 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) > 0.$$

{ 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


 * $$\det(\Delta_n) > 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) > 0$$

and for all larger $$n$$


 * $$\det(\Delta_n) = 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) = 0.$$

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


 * $$ \sum_{n \geq 1} m_n^{-1/(2n)} = \infty~.$$