Gaussian-type Quadrature Rules for Müntz Systems

Abstract

A method for constructing the generalized Gaussian quadrature rules for Muntz polynomials on $(0,1)$ is given. Such quadratures possess several properties of the classical Gaussian formulae (for polynomial systems), such as positivity of the weights, rapid convergence, etc. They can be applied to the wide class of functions, including smooth functions, as well as functions with end-point singularities, such as those in boundary-contact value problems, integral equations with singular kernels, complex analysis, potential theory, etc. The constructive method is based on an application of orthogonal Muntz polynomials introduced by Badalyan [{\it Akad. Nauk Armyan. SSR. Izv. Fiz.-Mat. Estest. Tehn. Nauk}, 8 (1955), pp. 1--28; 9 (1956), pp. 3--22 (Russian; Armenian summary)] and studied intensively by Borwein, Erdelyi, and Zhang [{\it Trans. Amer. Math. Soc}., 342 (1994), pp. 523--542], as well as by Milovanovic on a numerical procedure for evaluation of such polynomials with high precision [{\it Muntz orthogonal polynomials and their numerical evaluation}, in Applications and Computation of Orthogonal Polynomials, Internat. Ser. Numer. Math. 131, W. Gautschi, G. H. Golub, and G. Opfer, eds., Birkhauser, Basel, 1999, pp. 179--194]. The method is numerically stable. Some numerical examples are included.