An Error Expansion for some Gauss–Turán Quadratures and L1-Estimates of the Remainder Term

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• 作者：G. V. Milovanovi? and M. M. Spalevi?
• 关键词：Gauss–Tur&aacute ; n quadrature ; s ; orthogonal polynomials ; zeros ; multiple nodes ; weight ; remainder term for analytic functions ; contour integral representation ; error expansion ; error estimate
• 刊名：BIT Numerical Mathematics
• 出版年：2005
• 出版时间：March 2005
• 年：2005
• 卷：45
• 期：1
• 页码：117-136
• 全文大小：787 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Numeric Computing
  Mathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9125

文摘

Our aim in this paper is to obtain error expansions in the Gauss–Tur&aacute;n quadrature formula ∫<sub>?1sub>1f(t)w(t)?dt=∑<sub>ν=1sub>n∑<sub>i=0sub>2sA<sub>i,νsub>f(i)(τ<sub>νsub>)+R<sub>n,ssub>(f), in the case when f is an analytic function in some region of the complex plane containing the interval [?1,1] in its interior. Using a representation of the remainder term R<sub>n,ssub>(f) in the form of contour integral over confocal ellipses, we obtain R<sub>n,1sub>(f) for the four Chebyshev weights and R<sub>n,2sub>(f) for the Chebyshev weight of the first kind. Also, we get a few new L1-estimates of the remainder term, which are stronger than the previous ones. Some numerical results, illustrations and comparisons are also given.