On bi-orthogonal systems of trigonometric functions and quadrature formulas for periodic integrands

In this paper, quadrature formulas with an arbitrary number of nodes and exactly integrating trigonometric polynomials up to degree as high as possible are constructed in order to approximate 2π-periodic weighted integrals. For this purpose, certain bi-orthogonal systems of trigonometric functions are introduced and their most relevant properties studied. Some illustrative numerical examples are also given. The paper completes the results previously given by Szeg? in Magy Tud Akad Mat Kut Intez Közl 8:255–273, 1963 and by some of the authors in Annales Mathematicae et Informaticae 32:5–44, 2005.