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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 Reduced derivative ： ウィキペディア英語版 Reduced derivative In mathematics, the reduced derivative is a generalization of the notion of derivative that is well-suited to the study of functions of bounded variation. Although functions of bounded variation have derivatives in the sense of Radon measures, it is desirable to have a derivative that takes values in the same space as the functions themselves. Although the precise definition of the reduced derivative is quite involved, its key properties are quite easy to remember: * it is a multiple of the usual derivative wherever it exists; * at jump points, it is a multiple of the jump vector. The notion of reduced derivative appears to have been introduced by Alexander Mielke and Florian Theil in 2004. ==Definition== Let ''X'' be a separable, reflexive Banach space with norm || || and fix ''T'' > 0. Let BV−((); ''X'') denote the space of all left-continuous functions ''z'' : () → ''X'' with bounded variation on (). For any function of time ''f'', use subscripts +/− to denote the right/left continuous versions of ''f'', i.e. :$f\_\; (t)\; =\; \backslash lim\_\; f(s);$ :$f\_\; (t)\; =\; \backslash lim\_\; f(s).$ For any sub-interval () of (), let Var(''z'', ()) denote the variation of ''z'' over (), i.e., the supremum : The first step in the construction of the reduced derivative is the “stretch” time so that ''z'' can be linearly interpolated at its jump points. To this end, define :$\backslash hat\; \backslash colon\; (T)\; \backslash to\; ).$ The “stretched time” function ''τ̂'' is left-continuous (i.e. ''τ̂'' = ''τ̂''−); moreover, ''τ̂''− and ''τ̂''+ are strictly increasing and agree except at the (at most countable) jump points of ''z''. Setting ''T̂'' = ''τ̂''(''T''), this “stretch” can be inverted by :$\backslash hat\; \backslash colon\; (\backslash hat)\; \backslash to\; (T);$ :$\backslash hat(\backslash tau)\; =\; \backslash max\; \backslash .$ Using this, the stretched version of ''z'' is defined by :$\backslash hat\; \backslash in\; C^\; ((\backslash hat);\; X);$ :$\backslash hat(\backslash tau)\; =\; (1\; -\; \backslash theta)\; z\_(t)\; +\; \backslash theta\; z\_(t)$ where ''θ'' ∈ () and :$\backslash tau\; =\; (1\; -\; \backslash theta)\; \backslash hat\_\; (t)\; +\; \backslash theta\; \backslash hat\_\; (t).$ The effect of this definition is to create a new function ''ẑ'' which “stretches out” the jumps of ''z'' by linear interpolation. A quick calculation shows that ''ẑ'' is not just continuous, but also lies in a Sobolev space: :$\backslash hat\; \backslash in\; W^\; ((\backslash hat);\; X);$ :$\backslash left\backslash |\; \backslash frac\}\; \backslash right\backslash |\_\; );\; X)\}\; \backslash leq\; 1.$ The derivative of ''ẑ''(''τ'') with respect to ''τ'' is defined almost everywhere with respect to Lebesgue measure. The reduced derivative of ''z'' is the pull-back of this derivative by the stretching function ''τ̂'' : () → (). In other words, :$\backslash mathrm(z)\; \backslash colon\; (T)\; \backslash to\; \backslash ;$ :$\backslash mathrm(z)(t)\; =\; \backslash frac\}\; \backslash left(\; \backslash frac\; (t)\; +\; \backslash hat\_(t)\}\; \backslash right).$ Associated with this pull-back of the derivative is the pull-back of Lebesgue measure on (), which defines the differential measure ''μ''''z'': :$\backslash mu\_\; ([t\_,\; t\_))\; =\; \backslash lambda\; ([\backslash hat(t\_),\; \backslash hat(t\_))\; =\; \backslash hat\; (t\_)\; -\; \backslash hat(t\_)\; =\; t\_\; -\; t\_\; +\; \backslash int\_]\}\; \backslash |\; \backslash mathrm\; z\; \backslash |.$ 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「Reduced derivative」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.