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Stieltjes moment problem
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
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
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
When k = n or n−1 we saw in the previous section that Vk(Fn) is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group. These are listed in the table at the right.
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