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In mathematics, a delta operator is a shift-equivariant linear operator '''' on the vector space of polynomials in a variable over a field that reduces degrees by one. To say that is shift-equivariant means that if , then : In other words, if '''' is a "shift" of '''', then '''' is also a shift of '''', and has the same "shifting vector" ''''. To say that ''an operator reduces degree by one'' means that if '''' is a polynomial of degree '''', then '''' is either a polynomial of degree , or, in case , '''' is 0. Sometimes a ''delta operator'' is defined to be a shift-equivariant linear transformation on polynomials in '''' that maps '''' 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 :: :is a delta operator. * Differentiation with respect to ''x'', written as ''D'', is also a delta operator. * Any operator of the form :: : (where ''D''''n''(ƒ) = ƒ(''n'') is the ''n''th derivative) with 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 :: * 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 :: : the Euler approximation of the usual derivative with a discrete sample time . 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
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