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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 Mercer's theorem ： ウィキペディア英語版 Mercer's theorem In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in , is one of the most notable results of the work of James Mercer. It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite kernel.〔http://www.cs.berkeley.edu/~bartlett/courses/281b-sp08/7.pdf〕 == Introduction == To explain Mercer's theorem, we first consider an important special case; see below for a more general formulation. A ''kernel'', in this context, is a symmetric continuous function that maps :$K:\; ()\; \backslash times\; ()\; \backslash rightarrow\; \backslash mathbb$ where symmetric means that ''K''(''x'', ''s'') = ''K''(''s'', ''x''). ''K'' is said to be ''non-negative definite'' (or positive semidefinite) if and only if :$\backslash sum\_^n\backslash sum\_^n\; K(x\_i,\; x\_j)\; c\_i\; c\_j\; \backslash geq\; 0$ for all finite sequences of points ''x''1, ..., ''x''''n'' of () and all choices of real numbers ''c''1, ..., ''c''''n'' (cf. positive definite kernel). Associated to ''K'' is a linear operator (more specifically a Hilbert–Schmidt integral operator) on functions defined by the integral :$(\backslash varphi)(x)\; =\backslash int\_a^b\; K(x,s)\; \backslash varphi(s)\backslash ,\; ds.$ For technical considerations we assume φ can range through the space ''L''2() (see Lp space) of square-integrable real-valued functions. Since ''T'' is a linear operator, we can talk about eigenvalues and eigenfunctions of ''T''. Theorem. Suppose ''K'' is a continuous symmetric non-negative definite kernel. Then there is an orthonormal basis i of ''L''2() consisting of eigenfunctions of ''T''''K'' such that the corresponding sequence of eigenvalues ''i'' is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on () and ''K'' has the representation :$K(s,t)\; =\; \backslash sum\_^\backslash infty\; \backslash lambda\_j\; \backslash ,\; e\_j(s)\; \backslash ,\; e\_j(t)$ where the convergence is absolute and uniform. 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「Mercer's theorem」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.