Quadrature rules with multiple nodes

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 A. Ghizzeti and A. Ossicini [Quadrature Formulae, Academie–Verlag, 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 so called s− and σ−orthogonal trigonometric polynomials. Numerical method for the construction of such quadrature rules are given, as well as numerical the example which illustrate obtained theoretical results.