Quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

In this paper we investigate the Szegő–Radau and Szegő–Lobatto quadrature formulas on the unit circle. These are ( n + m ) -point formulas for which m nodes are fixed in advance, with m = 1 and m = 2 respectively, and which have a maximal domain of validity in the space of Laurent polynomials. This means that the free parameters (free nodes and positive weights) are chosen such that the quadrature formula is exact for all powers z j , − p ≤ j ≤ p , with p = p ( n , m ) as large as possible.