Anti-Szego quadrature rules

Szegő quadrature rules are discretization methods for approximating integrals of the form $\int _{-\pi }^{\pi } f(e^{it}) d\mu (t)$. This paper presents a new class of discretization methods, which we refer to as anti-Szegő quadrature rules. Anti-Szegő rules can be used to estimate the error in Szegő quadrature rules: under suitable conditions, pairs of associated Szegő and anti-Szegő quadrature rules provide upper and lower bounds for the value of the given integral. The construction of anti-Szegő quadrature rules is almost identical to that of Szegő quadrature rules in that pairs of associated Szegő and anti-Szegő rules differ only in the choice of a parameter of unit modulus. Several examples of Szegő and anti-Szegő quadrature rule pairs are presented.