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

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

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

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

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 { m_n : 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

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

{ m_n : 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

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

References

• Reed, Michael; Simon, Barry (1975), Fourier Analysis, Self-Adjointness, Methods of modern mathematical physics 2, Academic Press, p. 341 (exercise 25), ISBN 0-12-585002-6