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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hamburger moment problem ： ウィキペディア英語版 Hamburger moment problem In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence , does there exist a positive Borel measure ''μ'' on the real line such that :$m\_n\; =\; \backslash int\_^\backslash infty\; x^n\backslash ,d\; \backslash mu(x)\backslash \; ?$ In other words, an affirmative answer to the problem means that is the sequence of moments of some positive Borel measure ''μ''. The Stieltjes moment problem, Vorobyev moment problem, and the Hausdorff moment problem are similar but replace the real line by $[0,+\backslash infty)$ (Stieltjes and Vorobyev; but Vorobyev formulates the problem in the terms of matrix theory), or a bounded interval (Hausdorff). == Characterization == The Hamburger moment problem is solvable (that is, is a sequence of moments) if and only if the corresponding Hankel kernel on the nonnegative integers :$$ A = \left(\begin m_0 & m_1 & m_2 & \cdots \\ m_1 & m_2 & m_3 & \cdots \\ m_2 & m_3 & m_4 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ \end\right) is positive definite, i.e., :$\backslash sum\_m\_c\_j\backslash bar\; c\_k\backslash ge0$ for an arbitrary sequence ''j'' ≥ 0 of complex numbers with finite support (i.e. ''cj'' = 0 except for finitely many values of ''j''). For the "only if" part of the claims simply note that :$\backslash sum\_m\_c\_j\backslash bar\; c\_k=\; \backslash int\_^\backslash infty\; \backslash left|\backslash sum\_\; c\_j\; x^j\backslash right|^2\backslash ,d\; \backslash mu(x)$ which is non-negative if $\backslash mu$ is non-negative. We sketch an argument for the converse. Let Z+ be the nonnegative integers and ''F''0(Z+) denote the family of complex valued sequences with finite support. The positive Hankel kernel ''A'' induces a (possibly degenerate) sesquilinear product on the family of complex valued sequences with finite support. This in turn gives a Hilbert space :$(\backslash mathcal,\; \backslash langle,\; \backslash ;\; \backslash rangle)$ whose typical element is an equivalence class denoted by (). Let ''en'' be the element in ''F''0(Z+) defined by ''en''(''m'') = ''δnm''. One notices that :$\backslash langle\; (),\; ()\; \backslash rangle\; =\; A\_\; =\; m\_\; =\; \backslash langle\; (),\; ()\backslash rangle.$ Therefore the "shift" operator ''T'' on $\backslash mathcal$, with ''T''() = (), is symmetric. On the other hand, the desired expression :$m\_n\; =\; \backslash int\_^\backslash infty\; x^n\backslash ,d\; \backslash mu(x).$ suggests that ''μ'' is the spectral measure of a self-adjoint operator. If we can find a "function model" such that the symmetric operator ''T'' is multiplication by ''x'', then the spectral resolution of a self-adjoint extension of ''T'' proves the claim. A function model is given by the natural isomorphism from ''F''0(Z+) to the family of polynomials, in one single real variable and complex coefficients: for ''n'' ≥ 0, identify ''en'' with ''xn''. In the model, the operator ''T'' is multiplication by ''x'' and a densely defined symmetric operator. It can be shown that ''T'' always has self-adjoint extensions. Let :$\backslash bar\; \backslash ,$ be one of them and ''μ'' be its spectral measure. So :$\backslash langle\; \backslash bar^n\; (),\; ()\; \backslash rangle\; =\; \backslash int\; x^n\; d\; \backslash mu(x).$ On the other hand, :$\backslash langle\; \backslash bar^n\; (),\; ()\; \backslash rangle\; =\; \backslash langle\; T^n\; (),\; ()\; \backslash rangle\; =\; m\_n.\; \backslash ,$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hamburger moment problem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.