Extremal polynomials on the unit circle with preassigned zeros and Two-point Partial Padé approximation.

It is well known that for a general class of measures the sequence of paraorthogonal polynomials { B n } n ∈ N satisfies lim n → ∞ | B n (z) | 1 n = max (| z | , 1) , uniformly on compact subsets of ℂ ˆ ∖ T. This is an essential property to obtain the exact rate of convergence for two-point Padé approximants to the Herglotz–Riesz transform of the measure when these paraorthogonal polynomials are used as denominators. In this paper we obtain a similar result when we appropriately preselect some of the zeros of these paraorthogonal polynomials. As an application, we obtain the corresponding exact rate of convergence of the approximants to the Herglotz–Riesz transform with rational perturbation when these polynomials are used as denominators. [ABSTRACT FROM AUTHOR]