Stieltjes-type polynomials on the unit circle

Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the Gauss-Kronrod rule, provided that the integrand is analytic in a neighborhood of T.