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| In [[mathematics]], the '''Stieltjes [[moment problem]]''', named after [[Thomas Joannes Stieltjes]], seeks necessary and sufficient conditions for a sequence { ''m''<sub>''n''</sub>, : ''n'' = 0, 1, 2, ... } to be of the form
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| :<math>m_n=\int_0^\infty x^n\,d\mu(x)\,</math>
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| for some measure ''μ''. If such a function ''μ'' exists, one asks whether it is unique.
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| The essential difference between this and other well-known [[moment problem]]s is that this is on a half-line <nowiki>[</nowiki>0, ∞<nowiki>)</nowiki>, whereas in the [[Hausdorff moment problem]] one considers a bounded interval [0, 1], and in the [[Hamburger moment problem]] one considers the whole line (−∞, ∞).
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| ==Existence==
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| Let
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| :<math>\Delta_n=\left[\begin{matrix}
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| m_0 & m_1 & m_2 & \cdots & m_{n} \\
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| m_1 & m_2 & m_3 & \cdots & m_{n+1} \\
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| m_2& m_3 & m_4 & \cdots & m_{n+2} \\
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| \vdots & \vdots & \vdots & \ddots & \vdots \\
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| m_{n} & m_{n+1} & m_{n+2} & \cdots & m_{2n}
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| \end{matrix}\right]</math>
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| and
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| :<math>\Delta_n^{(1)}=\left[\begin{matrix}
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| m_1 & m_2 & m_3 & \cdots & m_{n+1} \\
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| m_2 & m_3 & m_4 & \cdots & m_{n+2} \\
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| m_3 & m_4 & m_5 & \cdots & m_{n+3} \\
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| \vdots & \vdots & \vdots & \ddots & \vdots \\
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| m_{n+1} & m_{n+2} & m_{n+3} & \cdots & m_{2n+1}
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| \end{matrix}\right].</math>
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| Then { ''m''<sub>''n''</sub> : ''n'' = 1, 2, 3, ... } is a moment sequence of some measure on <math>[0,\infty)</math> with infinite support if and only if for all ''n'', both
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| :<math>\det(\Delta_n) > 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) > 0.</math> | |
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| { ''m''<sub>''n''</sub> : ''n'' = 1, 2, 3, ... } is a moment sequence of some measure on <math>[0,\infty)</math> with finite support of size ''m'' if and only if for all <math>n \leq m</math>, both
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| :<math>\det(\Delta_n) > 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) > 0</math>
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| and for all larger <math>n</math>
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| :<math>\det(\Delta_n) = 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) = 0.</math>
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| ==Uniqueness==
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| There are several sufficient conditions for uniqueness, for example, [[Carleman's condition]], which states that the solution is unique if
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| :<math> \sum_{n \geq 1} m_n^{-1/(2n)} = \infty~.</math>
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| == References ==
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| *{{citation|first=Michael|last=Reed|first2=Barry|last2=Simon|title=Fourier Analysis, Self-Adjointness|year=1975|ISBN=0-12-585002-6|series=Methods of modern mathematical physics|volume=2|publisher=Academic Press|page= 341 (exercise 25)}}
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| [[Category:Probability theory]]
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| [[Category:Mathematical analysis]]
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| [[Category:Theory of probability distributions]]
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| [[Category:Mathematical problems]]
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Oscar is what my spouse enjoys to call me and I completely dig that title. Managing people is his occupation. My family members life in Minnesota and my family enjoys it. What I love doing is performing ceramics but I haven't produced a dime with it.
Also visit my web site over the counter std test (their website)