Generalized weighted Birkhoff–Young quadratures with the maximal degree of exactness

Abstract

Several types of quadratures of Birkhoff–Young type, as well as a sequence of the weighted generalized quadrature rules and their connection with multiple orthogonal polynomials, are considered. Beside a short account on a recent result on the generalized (4n+1)-point Birkhoff–Young quadrature, general weighted quadrature formulas of Birkhoff–Young type with the maximal degree of exactness are given. It includes a characterization and uniqueness of such rules, as well as numerical construction of nodes and weight coefficients. An explicit form of the node polynomial of such kind of quadratures with respect to the generalized Gegenbauer weight function is obtained. Finally, a sequence of generalized quadrature formulas is studied and their node polynomials are interpreted in terms of multiple orthogonal polynomials.