Stieltjes moment problem

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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

[math]\displaystyle{ m_n = \int_0^\infty x^n\,d\mu(x) }[/math]

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

[math]\displaystyle{ \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] }[/math]

and

[math]\displaystyle{ \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]. }[/math]

Then { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on [math]\displaystyle{ [0,\infty) }[/math] with infinite support if and only if for all n, both

[math]\displaystyle{ \det(\Delta_n) \gt 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) \gt 0. }[/math]

mn : n = 1, 2, 3, ... } is a moment sequence of some measure on [math]\displaystyle{ [0,\infty) }[/math] with finite support of size m if and only if for all [math]\displaystyle{ n \leq m }[/math], both

[math]\displaystyle{ \det(\Delta_n) \gt 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) \gt 0 }[/math]

and for all larger [math]\displaystyle{ n }[/math]

[math]\displaystyle{ \det(\Delta_n) = 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) = 0. }[/math]

Uniqueness

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

[math]\displaystyle{ \sum_{n \geq 1} m_n^{-1/(2n)} = \infty~. }[/math]

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