翻訳と辞書 Words near each other ・ Orthogonal array testing ・ Orthogonal basis ・ Orthogonal collocation ・ Orthogonal complement ・ Orthogonal convex hull ・ Orthogonal coordinates ・ Orthogonal Defect Classification ・ Orthogonal diagonalization ・ Orthogonal frequency-division multiple access ・ Orthogonal frequency-division multiplexing ・ Orthogonal functions ・ Orthogonal group ・ Orthogonal instruction set ・ Orthogonal matrix ・ Orthogonal polarization spectral imaging ・ Orthogonal polynomials ・ Orthogonal polynomials on the unit circle ・ Orthogonal Procrustes problem ・ Orthogonal symmetric Lie algebra ・ Orthogonal trajectory ・ Orthogonal transformation ・ Orthogonal wavelet ・ Orthogonality ・ Orthogonality (programming) ・ Orthogonality (term rewriting) ・ Orthogonality principle ・ Orthogonalization ・ Orthogonia ・ Orthogonia grisea ・ Orthogonia plana Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Orthogonal polynomials ： ウィキペディア英語版 Orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the Gegenbauer polynomials, the Chebyshev polynomials, and the Legendre polynomials. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A. A. Markov and T. J. Stieltjes. Some of the mathematicians who have worked on orthogonal polynomials include Gábor Szegő, Sergei Bernstein, Naum Akhiezer, Arthur Erdélyi, Yakov Geronimus, Wolfgang Hahn, Theodore Seio Chihara, Mourad Ismail, Waleed Al-Salam, and Richard Askey. == Definition for 1-variable case for a real measure == Given any non-decreasing function ''α'' on the real numbers, we can define the Lebesgue–Stieltjes integral :$\backslash int\; f(x)\; \backslash ;\; d\backslash alpha(x)$ of a function ''f''. If this integral is finite for all polynomials ''f'', we can define an inner product on pairs of polynomials ''f'' and ''g'' by :$\backslash langle\; f,\; g\; \backslash rangle\; =\; \backslash int\; f(x)\; g(x)\; \backslash ;\; d\backslash alpha(x).$ This operation is a positive semidefinite inner product on the vector space of all polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero. Then the sequence (''P''''n'')''n''=0∞ of orthogonal polynomials is defined by the relations :$\backslash deg\; P\_n\; =\; n~,\; \backslash quad\; \backslash langle\; P\_m,\; \backslash ,\; P\_n\; \backslash rangle\; =\; 0\; \backslash quad\; \backslash text\; \backslash quad\; m\; \backslash neq\; n~.$ In other words, the sequence is obtained from the sequence of monomials 1, ''x'', ''x''2, ... by the Gram–Schmidt process with respect to this inner product. Usually the sequence is required to be orthonormal, namely, :$\backslash langle\; P\_n,\; P\_n\; \backslash rangle\; =\; 1~,$ however, other normalisations are sometimes used. 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「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.