Quadrature processes - development and new directions

G. V. Milovanović

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 33 (2008), p. 11

Résumé

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 Muntz polynomials.

Détail

Classification : 41A55 33C45 33C47 42C05 65D30 65D32
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, Muntz polynomials