Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation (Q2762768)

Based on the Hermite interpolation formula and the associated Cotes numbers of higher order, the Gaussian quadrature formula has been extended to multiple nodes [cf. \textit{L. Chakalov}, C. R. Acad. Bulg. Sci. 1, No. 2-3, 9-12 (1948; MR 10-743)]. In addition, the Gauss-Legendre quadrature formula has been extended to higher orders [cf. \textit{P. Turán}, Acta Sci. Math. Szeged 12, 30-37 (1950; Zbl 0045.33606)]. In both cases, the class of polynomials for which the numerical quadrature formulae are exact have been explicitly determined. The generalized Gauss-Turán quadrature with respect to a non-negative measure \(\lambda\) of compact support on the real line \textbf{R} and the moments associated with \(\lambda\) lead to the concept of power orthogonal polynomials [cf. \textit{A. Ossicini}, Ann. Mat. Pura Appl. 72, 213-237 (1966; Zbl 0143.38603); \textit{A. Ossicini} and \textit{F. Rosati} [Boll. Un. Mat. Ital. 11, 224-237 (1975; Zbl 0311.41024)]. NEWLINENEWLINENEWLINEThe paper under review provides an analysis of the case of weights [cf. \textit{R. D. Riess}, omputing 15, 173-179 (1975; Zbl 0311.65016)], and presents a less detailed survey of the application of numerical quadrature formulae to moment--preserving approximation by defective splines.NEWLINENEWLINENEWLINEThis article is a reprint from J. Comput. Appl. Math. 127, No. 1-2, 267-286 (2001; Zbl 0970.65023).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00018].