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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 Hausdorff moment problem ： ウィキペディア英語版 Hausdorff moment problem In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence be the sequence of moments :$m\_n\; =\; \backslash int\_0^1\; x^n\backslash ,d\backslash mu(x)\backslash ,$ of some Borel measure ''μ'' supported on the closed unit interval (). In the case ''m''0 = 1, this is equivalent to the existence of a random variable ''X'' supported on (), such that E ''X''n = ''m''''n''. The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the Stieltjes moment problem one considers a half-line [0, ∞), and in the Hamburger moment problem one considers the whole line (−∞, ∞). The Stieltjes moment problems and the Hamburger moment problems, if they are solvable, may have infinitely many solutions (indeterminate moment problem) whereas a Hausdorff moment problem always has a unique solution if it is solvable (determinate moment problem). In the indeterminate moment problem case, there are infinitely measures correspond to the same prescribed moments and they consist of a convex set. The set of polynomials may or may not be dense in the associated Hilbert spaces if the moment problem is indeterminate, and it depends on whether measure is extremal or not. But in the determinate moment problem case, the set of polynomials are dense in the associated Hilbert space. In 1921, Hausdorff showed that is such a moment sequence if and only if the sequence is completely monotonic, i.e., its difference sequences satisfy the equation :$(-1)^k(\backslash Delta^k\; m)\_n\; \backslash geq\; 0$ for all ''n'',''k'' ≥ 0. Here, Δ is the difference operator given by :$(\backslash Delta\; m)\_n\; =\; m\_\; -\; m\_n.$ The necessity of this condition is easily seen by the identity :$(-1)^k(\backslash Delta^k\; m)\_n\; =\; \backslash int\_0^1\; x^n\; (1-x)^k\; d\backslash mu(x),$ which is ''≥ 0'', being the integral of an almost sure non-negative function. For example, it is necessary to have :$\backslash Delta^4\; m\_6\; =\; m\_6\; -\; 4m\_7\; +\; 6m\_8\; -\; 4m\_9\; +\; m\_\; =\; \backslash int\; x^6\; (1-x)^4\; d\backslash mu(x)\; \backslash geq\; 0.$ ==See also== * Total monotonicity 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「Hausdorff 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.