Quadrature rules with multiple nodes

Abstract

© Springer International Publishing Switzerland 2016. In this paper a brief historical survey of the development of quadrature rules with multiple nodes and the maximal algebraic degree of exactness is given. The natural generalization of such rules are quadrature rules with multiple nodes and the maximal degree of exactness in some functional spaces that are different from the space of algebraic polynomial. For that purpose we present a generalized quadrature rules considered by Ghizzeti and Ossicini (Quadrature Formulae, Academie, Berlin, 1970) and apply their ideas in order to obtain quadrature rules with multiple nodes and the maximal trigonometric degree of exactness. Such quadrature rules are characterized by the so-called s-and σ-orthogonal trigonometric polynomials. Numerical method for constructing such quadrature rules is given, as well as a numerical example to illustrate the obtained theoretical results.