Optimal extension of the Szego quadrature
All the possible extensions of the Szeg}o quadrature with the highest degree of exactness and self-reciprocal
nodal polynomial are constructed. For a large class of weight functions we prove that the nodal polynomial fullls strong asymptotic behavior uniformly 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 suciently large number of nodes, the extended quadratures have positive weights and
simple nodes on the unit circle. Finally, some interesting consequences related to Gauss-Kronrod quadrature
in the real line are discussed.