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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
:\int f(x) \; d\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
:\langle f, g \rangle = \int f(x) g(x) \; d\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
: \deg P_n = n~, \quad \langle P_m, \, P_n \rangle = 0 \quad \text \quad m \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,
: \langle P_n, P_n \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)
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