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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 Pincherle derivative ： ウィキペディア英語版 Pincherle derivative In mathematics, the Pincherle derivative ''T’'' of a linear operator ''T'':K() → K() on the vector space of polynomials in the variable ''x'' over a field K is the commutator of ''T'' with the multiplication by ''x'' in the algebra of endomorphisms End(K()). That is, ''T’'' is another linear operator ''T’'':K() → K() :$T\text{'}\; :=\; ()\; =\; Tx-xT\; =\; -\backslash operatorname(x)T,\backslash ,$ so that :$T\text{'}\backslash =T\backslash -xT\backslash \backslash qquad\backslash forall\; p(x)\backslash in\; \backslash mathbb().$ This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936). == Properties == The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators $\backslash scriptstyle\; S$ and $\backslash scriptstyle\; T$ belonging to $\backslash scriptstyle\; \backslash operatorname\; \backslash left(\; \backslash mathbb\; K()\; \backslash right)$ #$\backslash scriptstyle$ ; #$\backslash scriptstyle$ where $\backslash scriptstyle$ is the composition of operators ; One also has $\backslash scriptstyle$ where $\backslash scriptstyle$ is the usual Lie bracket, which follows from the Jacobi identity. The usual derivative, ''D'' = ''d''/''dx'', is an operator on polynomials. By straightforward computation, its Pincherle derivative is : $D\text{'}=\; \backslash left(\_\; =\; 1.$ This formula generalizes to : $(D^n)\text{'}=\; \backslash left(\backslash right)\text{'}\; =\; nD^,$ by induction. It proves that the Pincherle derivative of a differential operator : $\backslash partial\; =\; \backslash sum\; a\_n\; \backslash over\; \}\; =\; \backslash sum\; a\_n\; D^n$ is also a differential operator, so that the Pincherle derivative is a derivation of $\backslash scriptstyle\; \backslash operatorname(\backslash mathbb\; K\; ())$. The shift operator : $S\_h(f)(x)\; =\; f(x+h)\; \backslash ,$ can be written as : $S\_h\; =\; \backslash sum\_\; \backslash over\; \}D^n$ by the Taylor formula. Its Pincherle derivative is then : $S\_h\text{'}\; =\; \backslash sum\_\; \backslash over\; \}D^\; =\; h\; \backslash cdot\; S\_h.$ In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars $\backslash scriptstyle$. If ''T'' is shift-equivariant, that is, if ''T'' commutes with ''S''''h'' or $\backslash scriptstyle$, then we also have $\backslash scriptstyle$, so that $\backslash scriptstyle\; T\text{'}$ is also shift-equivariant and for the same shift $\backslash scriptstyle\; h$. The "discrete-time delta operator" : $(\backslash delta\; f)(x)\; =\; \backslash over\; h\; \}$ is the operator : $\backslash delta\; =\; (S\_h\; -\; 1),$ whose Pincherle derivative is the shift operator $\backslash scriptstyle$. 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「Pincherle 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.