Some properties of dual quadrature formulae

摘要

We investigate dual formulae Q̄nƒ which result from a variation of an interpolatory technique used by Markoff to generate Gaussian quadrature formulae Qnƒ. Dual formulae Q̄nƒ with n nodes prove to have degree of precision greater than or equal to 2n − 2 and thus include Gaussian quadrature. Discussing the position of the nodes of dual formulae we give a complete characterisation of a system of polynomials which are orthogonal with respect to a sign-changing weight function. Further we demonstrate some interesting properties of Gaussian quadrature in comparison with dual formulae.