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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 Delta operator ： ウィキペディア英語版 Delta operator In mathematics, a delta operator is a shift-equivariant linear operator ''$Q\backslash colon\backslash mathbb()\; \backslash longrightarrow\; \backslash mathbb()$'' on the vector space of polynomials in a variable $x$ over a field $\backslash mathbb$ that reduces degrees by one. To say that $Q$ is shift-equivariant means that if $g(x)\; =\; f(x\; +\; a)$, then :$.\backslash ,$ In other words, if ''$f$'' is a "shift" of ''$g$'', then ''$Qf$'' is also a shift of ''$Qg$'', and has the same "shifting vector" ''$a$''. To say that ''an operator reduces degree by one'' means that if ''$f$'' is a polynomial of degree ''$n$'', then ''$Qf$'' is either a polynomial of degree $n-1$, or, in case $n\; =\; 0$, ''$Qf$'' is 0. Sometimes a ''delta operator'' is defined to be a shift-equivariant linear transformation on polynomials in ''$x$'' that maps ''$x$'' to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition, since shift-equivariance is a fairly strong condition. ==Examples== * The forward difference operator :: $(\backslash Delta\; f)(x)\; =\; f(x\; +\; 1)\; -\; f(x)\backslash ,$ :is a delta operator. * Differentiation with respect to ''x'', written as ''D'', is also a delta operator. * Any operator of the form ::$\backslash sum\_^\backslash infty\; c\_k\; D^k$ : (where ''D''''n''(ƒ) = ƒ(''n'') is the ''n''th derivative) with $c\_1\backslash neq0$ is a delta operator. It can be shown that all delta operators can be written in this form. For example, the difference operator given above can be expanded as ::$\backslash Delta=e^D-1=\backslash sum\_^\backslash infty\; \backslash frac.$ * The generalized derivative of time scale calculus which unifies the forward difference operator with the derivative of standard calculus is a delta operator. * In computer science and cybernetics, the term "discrete-time delta operator" (δ) is generally taken to mean a difference operator :: $\{(\backslash delta\; f)(x)\; =\; ,$ : the Euler approximation of the usual derivative with a discrete sample time $\backslash Delta\; t$. The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling. 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「Delta 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.