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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 Chebyshev polynomial of the first kind ： ウィキペディア英語版 Chebyshev polynomials In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev,〔Chebyshev polynomials were first presented in: P. L. Chebyshev (1854) "Théorie des mécanismes connus sous le nom de parallélogrammes," ''Mémoires des Savants étrangers présentés à l’Académie de Saint-Pétersbourg'', vol. 7, pages 539–586.〕 are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted and Chebyshev polynomials of the second kind which are denoted . The letter T is used because of the alternative transliterations of the name ''Chebyshev'' as ''Tchebycheff'', ''Tchebyshev'' (French) or ''Tschebyschow'' (German). The Chebyshev polynomials or are polynomials of degree and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence. Chebyshev polynomials are polynomials with the largest possible leading coefficient, but subject to the condition that their absolute value on the interval is bounded by 1. They are also the extremal polynomials for many other properties.〔Rivlin, Theodore J. The Chebyshev polynomials. Pure and Applied Mathematics. ''Wiley-Interscience (Wiley & Sons ),'' New York-London-Sydney,1974. Chapter 2, "Extremal Properties", pp. 56--123.〕 Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw–Curtis quadrature. In the study of differential equations they arise as the solution to the Chebyshev differential equations :$(1-x^2)\backslash ,y\text{'}\text{'}\; -\; x\backslash ,y\text{'}\; +\; n^2\backslash ,y\; =\; 0\; \backslash ,\backslash !$ and :$(1-x^2)\backslash ,y\text{'}\text{'}\; -\; 3x\backslash ,y\text{'}\; +\; n(n+2)\backslash ,y\; =\; 0\; \backslash ,\backslash !$ for the polynomials of the first and second kind, respectively. These equations are special cases of the Sturm–Liouville differential equation. ==Definition== The Chebyshev polynomials of the first kind are defined by the recurrence relation :$$ \begin T_0(x) & = 1 \\ T_1(x) & = x \\ T_(x) & = 2xT_n(x) - T_(x). \end The ordinary generating function for ''T''''n'' is :$\backslash sum\_^T\_n(x)\; t^n\; =\; \backslash frac;\; \backslash ,\backslash !$ the exponential generating function is :$\backslash sum\_^T\_n(x)\; \backslash frac\; =\; \backslash tfrac\backslash left(\; e^+e^\backslash right)$ = e^ \cosh(t \sqrt). \,\! The generating function relevant for 2-dimensional potential theory and multipole expansion is :$\backslash sum\backslash limits\_^T\_\backslash left(\; x\backslash right)\; \backslash frac=\backslash ln\; \backslash frac.$ The Chebyshev polynomials of the second kind are defined by the recurrence relation :$$ \begin U_0(x) & = 1 \\ U_1(x) & = 2x \\ U_(x) & = 2xU_n(x) - U_(x). \end The ordinary generating function for ''U''''n'' is :$\backslash sum\_^U\_n(x)\; t^n\; =\; \backslash frac;\; \backslash ,\backslash !$ the exponential generating function is :$\backslash sum\_^U\_n(x)\; \backslash frac\; =\; e^$ \left( \cosh(t \sqrt) + \frac) \right). \,\! 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「Chebyshev polynomials」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.