EMIS ELibM Electronic Journals Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques naturelles / sciences mathematiques
Vol. CXXXVII, No. 33, pp. 11–41 ()

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Quadrature processes – development and new directions

G. V. Milovanovic

Department Mathematics, Faculty of Electronic Engineering, University of Nis, P. O. Box 73, 18000 Nis, Serbia

Abstract: We present a survey on quadrature processes, beginning with Newton's idea of approximate integration and Gauss' discovery of his famous quadrature method, as well as significant contributions of Jacobi and Christoffel. Beside the stable construction of Gauss-Christoffel quadratures for classical and non-classical weights we give some recent applications in the summation of slowly convergent series and moment-preserving spline approximation. Also, we consider quadratures of the maximal degree of precision with multiple nodes, as well as a more general concept of orthogonality with respect to a given linear moment functional and corresponding quadratures of Gaussian type. A short account of non-standard quadratures of Gaussian type is also included. Finally, we mention the Gaussian integration which is exact on the space of Müntz polynomials.

Keywords: Quadrature process; Newton-Cotes formula, Gauss-Christoffel quadrature formula; Orthogonal polynomials; Moments; Moment functional; Three-term recurrence relation; Weight; Node; Multiple nodes; Summation of series; Moment-preserving spline approximation; Non-standard quadratures; Müntz polynomials

Classification (MSC2000): 41A55, 33C45, 33C47, 42C05, 65D30, 65D32

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Electronic fulltext finalized on: 7 Sep 2008. This page was last modified: 20 Jun 2011.

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