Quadrature processes – development and new directions

G. V. Milovanovic

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

Full text of the article: (for faster download, first choose a mirror)

• Compressed DVI file (48 kilobytes)
• Compressed PostScript file (272 kilobytes)
• PDF file (292 kilobytes)