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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 Secondary measure ： ウィキペディア英語版 Secondary measure In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system. ==Introduction== Under certain assumptions that we will specify further, it is possible to obtain the existence of a secondary measure and even to express it. For example, if one works in the Hilbert space ''L''2((1 ), R, ρ) : $\backslash forall\; x\; \backslash in\; (),\; \backslash qquad\; \backslash mu(x)=\backslash frac\; +\; \backslash pi^2\backslash rho^2(x)\}$ with : $\backslash varphi(x)\; =\; \backslash lim\_\; 2\backslash int\_0^1\backslash frac\; \backslash ,\; dt$ in the general case, or: : $\backslash varphi(x)\; =\; 2\backslash rho(x)\backslash text\backslash left(\backslash frac\backslash right)\; -\; 2\; \backslash int\_0^1\backslash frac\; \backslash ,\; dt$ when ρ satisfies a Lipschitz condition. This application φ is called the reducer of ρ. More generally, μ et ρ are linked by their Stieltjes transformation with the following formula: : $S\_(z)=z-c\_1-\backslash frac$ in which ''c''1 is the moment of order 1 of the measure ρ. These secondary measures, and the theory around them, lead to some surprising results, and make it possible to find in an elegant way quite a few traditional formulas of analysis, mainly around the Euler Gamma function, Riemann Zeta function, and Euler's constant. They also allowed the clarification of integrals and series with a tremendous effectiveness, though it is a priori difficult. Finally they make it possible to solve integral equations of the form : $f(x)=\backslash int\_0^1\backslash frac\backslash rho(t)\backslash ,dt$ where ''g'' is the unknown function, and lead to theorems of convergence towards the Chebyshev and Dirac measures. 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「Secondary measure」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.