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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 Lorentz space ： ウィキペディア英語版 Lorentz space In mathematical analysis, Lorentz spaces, introduced by George Lorentz in the 1950s,〔G. Lorentz, "Some new function spaces", ''Annals of Mathematics'' 51 (1950), pp. 37-55.〕〔G. Lorentz, "On the theory of spaces Λ", ''Pacific Journal of Mathematics'' 1 (1951), pp. 411-429.〕 are generalisations of the more familiar $L^$ spaces. The Lorentz spaces are denoted by $L^$. Like the $L^$ spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the $L^$ norm does. The two basic qualitative notions of "size" of a function are: how tall is graph of the function, and how spread out is it. The Lorentz norms provide tighter control over both qualities than the $L^$ norms, by exponentially rescaling the measure in both the range ($p$) and the domain ($q$). The Lorentz norms, like the $L^$ norms, are invariant under arbitrary rearrangements of the values of a function. ==Definition== The Lorentz space on a measure space $(X,\; \backslash mu)$ is the space of complex-valued measurable functions $f$ on ''X'' such that the following quasinorm is finite :$\backslash |f\backslash |\_\; =\; p^\}\; \backslash left\; \backslash |t\backslash mu\backslash ^\}\; \backslash right\; \backslash |\_\; \backslash right)\}$ where and . Thus, when , :$\backslash |f\backslash |\_=p^\}\backslash left(\backslash int\_0^\backslash infty\; t^q\; \backslash mu\backslash left\backslash ^\}\backslash ,\backslash frac\backslash right)^\}.$ and, when $q\; =\; \backslash infty$, :$\backslash |f\backslash |\_^p\; =\; \backslash sup\_\backslash left(t^p\backslash mu\backslash left\backslash \backslash right).$ It is also conventional to set $L^(X,\; \backslash mu)\; =\; L^(X,\; \backslash mu)$. 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「Lorentz space」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.