Rational approximants associated with measures supported on the unit circle and the real line

As a continuation of the well known connection between the theory of orthogonal polynomials on the unit circle and the interval [−1,1][−1,1], in this paper properties concerning error and convergence of certain rational approximants associated with the measures dμ(t) and dσ(θ)=|dμ(cosθ)| supported on [−1,1][−1,1] and the unit circle respectively are deduced. Numerical illustrations are also given.

► We analyze a connection between certain rational approximants to Markov functions and to Herglotz–Riesz transform.
► We compute rational approximats from Szegő quadrature formulas.
► Error estimates and convergence are analyzed.
► An application to the numerical calculation of certain special functions is carried out.