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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 Bernstein's theorem on monotone functions ： ウィキペディア英語版 Bernstein's theorem on monotone functions In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average, or expected value. Total monotonicity (sometimes also ''complete monotonicity'') of a function ''f'' means that ''f'' is continuous on [0, ∞), infinitely differentiable on (0, ∞), and satisfies :$(-1)^n\; f(t)\; \backslash geq\; 0$ for all nonnegative integers ''n'' and for all ''t'' > 0. Another convention puts the opposite inequality in the above definition. The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on [0, ∞), with cumulative distribution function ''g'', such that :$f(t)\; =\; \backslash int\_0^\backslash infty\; e^\; \backslash ,dg(x),$ the integral being a Riemann–Stieltjes integral. Nonnegative functions whose derivative is completely monotone are called ''Bernstein functions''. Every Bernstein function has the Lévy-Khintchine representation: :$f(t)\; =\; a\; +\; b\; t\; +\backslash int\_0^\backslash infty\; (1-e^)\backslash mu(dx)$ where $a,b\; \backslash geq\; 0$ and $\backslash mu$ is a measure on the positive real half-line such that : In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on [0,∞). In this form it is known as the Bernstein–Widder theorem, or Hausdorff–Bernstein–Widder theorem. Felix Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem. ==References== * * * 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「Bernstein's theorem on monotone functions」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.