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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 Laplace–Stieltjes transform ： ウィキペディア英語版 Laplace–Stieltjes transform The Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For real-valued functions, it is the Laplace transform of a Stieltjes measure, however it is often defined for functions with values in a Banach space. It is useful in a number of areas of mathematics, including functional analysis, and certain areas of theoretical and applied probability. ==Real-valued functions== The Laplace–Stieltjes transform of a real-valued function ''g'' is given by a Lebesgue–Stieltjes integral of the form :$\backslash int\backslash mathrm^\backslash ,dg(x)$ for ''s'' a complex number. As with the usual Laplace transform, one gets a slightly different transform depending on the domain of integration, and for the integral to be defined, one also needs to require that ''g'' be of bounded variation on the region of integration. The most common are: * The bilateral (or two-sided) Laplace–Stieltjes transform is given by ::$\backslash (s)\; =\; \backslash int\_^\; \backslash mathrm^\backslash ,dg(x).$ * The unilateral (one-sided) Laplace–Stieltjes transform is given by ::$\backslash (s)\; =\; \backslash int\_^\; \backslash mathrm^\backslash ,dg(x).$ :where the lower limit 0− means ::$\backslash lim\_\backslash int\_^\backslash infty.$ :This is necessary to ensure that the transform captures a possible jump in ''g''(''x'') at ''x'' = 0, as is needed to make sense of the Laplace transform of the Dirac delta function. * More general transforms can be considered by integrating over a contour in the complex plane; see . The Laplace–Stieltjes transform in the case of a scalar-valued function is thus seen to be a special case of the Laplace transform of a Stieltjes measure. To wit, :$\backslash mathcal^$ *g = \mathcal(dg). In particular, it shares many properties with the usual Laplace transform. For instance, the convolution theorem holds: :$\backslash (s)\; =\; \backslash (s)\backslash (s).$ Often only real values of the variable ''s'' are considered, although if the integral exists as a proper Lebesgue integral for a given real value ''s'' = σ, then it also exists for all complex ''s'' with re(''s'') ≥ σ. The Laplace–Stieltjes transform appears naturally in the following context. If X is a random variable with cumulative distribution function ''F'', then the Laplace–Stieltjes transform is given by the expectation: :$\backslash (s)\; =\; \backslash mathrm\backslash left().$ 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「Laplace–Stieltjes transform」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.