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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 Classical orthogonal polynomials ： ウィキペディア英語版 Classical orthogonal polynomials In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials), Chebyshev polynomials, and Legendre polynomials.〔See 〕 They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others. Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials. For given polynomials $Q,\; L:\; \backslash R\; \backslash to\; \backslash R$ and $\backslash forall\backslash ,n\; \backslash in\; \backslash N\_0$ the classical orthogonal polynomials $f\_n:\backslash R\; \backslash to\; \backslash R$ are characterized by being solutions of the differential equation :$Q(x)\; \backslash ,\; f\_n^\; +\; L(x)\backslash ,f\_n^\; +\; \backslash lambda\_n\; f\_n\; =\; 0$ with to be determined constants $\backslash lambda\_n\; \backslash in\; \backslash R$. There are several more general definitions of orthogonal classical polynomials; for example, use the term for all polynomials in the Askey scheme. == Definition == In general, the orthogonal polynomials $P\_n$ with respect to a weight $W:\backslash mathbb\; R\; \backslash rightarrow\; \backslash mathbb\; R^+$ :$\backslash begin$ &\deg P_n = n~, \quad n = 0,1,2,\ldots\\ &\int P_m(x) \, P_n(x) \, W(x)\,dx = 0~, \quad m \neq n~. \end The relations above define $P\_n$ up to multiplication by a number. Various normalisations are used to fix the constant, e.g. :$\backslash int\; P\_n(x)^2\; W(x)\backslash ,dx\; =\; 1~.$ The classical orthogonal polynomials correspond to the three families of weights: :$\backslash begin$ \text\quad &W(x) = \begin (1 - x)^\alpha (1+x)^\beta~, & -1 \leq x \leq 1 \\ 0~, &\text \end \\ \text\quad &W(x) = \exp(- x^2) \\ \text\quad &W(x) = \begin x^\alpha \exp(- x)~, &\quad x \geq 0 \\ 0~, &\text \end \end The standard normalisation (also called ''standardization'') is detailed below. 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「Classical orthogonal 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.