Optimal extension of the Szeg quadrature

All the possible extensions of the Szegő quadrature with the highest degree of exactness and self-inversive nodal polynomial are constructed and algebraically characterized. Using ideas of F. Peherstorfer we prove that, for a large class of weight functions, the nodal polynomial fulfils strong asymptotic behaviour on the unit circle. This asymptotic representation is analogous to that of para-orthogonal polynomials. We additionally prove that, for the class of weight functions considered and a sufficiently large number of nodes, the extended quadratures have positive weights and simple nodes on the unit circle.