翻訳と辞書 Words near each other ・ Schuppanzigh Quartet ・ Schuppen ・ Schur ・ Schur algebra ・ Schur algorithm ・ Schur class ・ Schur complement ・ Schur complement method ・ Schur decomposition ・ Schur function ・ Schur functor ・ Schur multiplier ・ Schur orthogonality relations ・ Schur polynomial ・ Schur product theorem ・ Schur test ・ Schur's inequality ・ Schur's lemma ・ Schur's lemma (disambiguation) ・ Schur's lemma (from Riemannian geometry) ・ Schur's property ・ Schur's theorem ・ Schur-convex function ・ Schurman ・ Schurman Commission ・ Schurmann Family ・ Schurmsee ・ Schurr ・ Schurr High School ・ Schurre Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Schur test ： ウィキペディア英語版 Schur test In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the $L^2\backslash to\; L^2$ operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version.〔Paul Richard Halmos and Viakalathur Shankar Sunder, ''Bounded integral operators on $L^$ spaces'', Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 96., Springer-Verlag, Berlin, 1978. Theorem 5.2.〕 Let $X,\backslash ,Y$ be two measurable spaces (such as $\backslash mathbb^n$). Let $\backslash ,T$ be an integral operator with the non-negative Schwartz kernel $\backslash ,K(x,y)$, $x\backslash in\; X$, $y\backslash in\; Y$: :$T\; f(x)=\backslash int\_Y\; K(x,y)f(y)\backslash ,dy.$ If there exist functions $\backslash ,p(x)>0$ and $\backslash ,q(x)>0$ and numbers $\backslash ,\backslash alpha,\backslash beta>0$ such that :$(1)\backslash qquad\; \backslash int\_Y\; K(x,y)q(y)\backslash ,dy\backslash le\backslash alpha\; p(x)$ for almost all $\backslash ,x$ and :$(2)\backslash qquad\; \backslash int\_X\; p(x)K(x,y)\backslash ,dx\backslash le\backslash beta\; q(y)$ for almost all $\backslash ,y$, then $\backslash ,T$ extends to a continuous operator $T:L^2\backslash to\; L^2$ with the operator norm :$\backslash Vert\; T\backslash Vert\_\; \backslash le\backslash sqrt.$ Such functions $\backslash ,p(x)$, $\backslash ,q(x)$ are called the Schur test functions. In the original version, $\backslash ,T$ is a matrix and $\backslash ,\backslash alpha=\backslash beta=1$.〔I. Schur, ''Bemerkungen zur Theorie der Beschränkten Bilinearformen mit unendlich vielen Veränderlichen'', J. reine angew. Math. 140 (1911), 1–28.〕 ==Common usage and Young's inequality== A common usage of the Schur test is to take $\backslash ,p(x)=q(x)=1.$ Then we get: :$$ \Vert T\Vert^2_\le \sup_\int_Y|K(x,y)| \, dy \cdot \sup_\int_X|K(x,y)| \, dx. This inequality is valid no matter whether the Schwartz kernel $\backslash ,K(x,y)$ is non-negative or not. A similar statement about $L^p\backslash to\; L^q$ operator norms is known as Young's inequality:〔Theorem 0.3.1 in: C. D. Sogge, ''Fourier integral operators in classical analysis'', Cambridge University Press, 1993. ISBN 0-521-43464-5〕 if :$\backslash sup\_x\backslash Big(\backslash int\_Y|K(x,y)|^r\backslash ,dy\backslash Big)^\; +\; \backslash sup\_y\backslash Big(\backslash int\_X|K(x,y)|^r\backslash ,dx\backslash Big)^\backslash le\; C,$ where $r\backslash ,$ satisfies $\backslash frac\; 1\; r=1-\backslash Big(\backslash frac\; 1\; p-\backslash frac\; 1\; q\backslash Big)$, for some $1\backslash le\; p\backslash le\; q\backslash le\backslash infty$, then the operator $Tf(x)=\backslash int\_Y\; K(x,y)f(y)\backslash ,dy$ extends to a continuous operator $T:L^p(Y)\backslash to\; L^q(X)$, with $\backslash Vert\; T\backslash Vert\_\backslash le\; C.$ 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「Schur test」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.