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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 Hilbert–Schmidt integral operator ： ウィキペディア英語版 Hilbert–Schmidt integral operator In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in ''n''-dimensional Euclidean space R''n'', a Hilbert–Schmidt kernel is a function ''k'' : Ω × Ω → C with : (that is, the ''L''2(Ω×Ω; C) norm of ''k'' is finite), and the associated Hilbert–Schmidt integral operator is the operator ''K'' : ''L''2(Ω; C) → ''L''2(Ω; C) given by :$(K\; u)\; (x)\; =\; \backslash int\_\; k(x,\; y)\; u(y)\; \backslash ,\; dy.$ Then ''K'' is a Hilbert–Schmidt operator with Hilbert–Schmidt norm :$\backslash Vert\; K\; \backslash Vert\_\backslash mathrm\; =\; \backslash Vert\; k\; \backslash Vert\_.$ Hilbert–Schmidt integral operators are both continuous (and hence bounded) and compact (as with all Hilbert–Schmidt operators). The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. Specifically, let ''X'' be a locally compact Hausdorff space equipped with a positive Borel measure. Suppose further that ''L''2(''X'') is a separable Hilbert space. The above condition on the kernel ''k'' on Rn can be interpreted as demanding ''k'' belong to ''L''2(''X × X''). Then the operator :$(Kf)(x)\; =\; \backslash int\_\; k(x,y)f(y)\backslash ,dy$ is compact. If :$k(x,y)\; =\; \backslash overline$ then ''K'' is also self-adjoint and so the spectral theorem applies. This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces. See Chapter 2 of the book by Bump in the references for examples. == See also == * Hilbert–Schmidt operator 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「Hilbert–Schmidt integral operator」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.