Quadrature formulae connected to σ-orthogonal polynomials

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:140

Article; Proceedings Paper

Milovanovic, GV ; Spalevic, MM

关键词: nonnegative measure;    moments;    weights;    quadratures with multiple nodes;    nodes;    degree of precision;    error term;    influence function;    convergence;    orthogonal polynomials;    s- and sigma-orthogonal polynomials;   

DOI  :  10.1016/S0377-0427(01)00476-9

来源: Elsevier

【 摘 要 】

Let dlambda(t) be a given nonnegative measure on the real line R, with compact or infinite support, for which all moments mu(k) = integral(R) t(k) dlambda.(t), k = 0,1,..., exist and are finite, and mu(0) > 0. Quadrature formulas of Chakalov-Popoviciu type with multiple nodes integralRf(t)dlambda(t)=Sigma(v=1)(n)Sigma(t=0)(2sv) A(t,v)f((i))(tauv) + R(f), where sigma = sigma(n) =(s(1), s(2),..., s(n)) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness d(max) = 2 Sigma(v=1)(n) s(v) + 2n - 1 if and only if integral(R) v(=1)(n) (t -tau(v))(2sv+1)t(k) dlambda(t)=0, k=0,1...,n-1. The proof of the uniqueness of the extremal nodes tau(1),tau(2),...,tau(n) was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1-15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error terrn R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes tau(v), v = 1, 2,..., n, which are the zeros of the corresponding sigma-orthogonal polynomial, is presented, Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included. (C) 2002 Elsevier Science B.V. All rights reserved.

【 授权许可】

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