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 { mn, : 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[edit]

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 { 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[edit]

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[edit]

  • 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