Error Estimates for the Carathéodory-Fejér Method in Polynomial Approximation

Abstract

In the Caratheodory-Fejer method one computes — starting from a complex power series absolutely convergent on the unit disk — a polynomial which is (hopefully) a better approximation to the function given by the power series than the truncated power series (with respect to the supremum norm). In this article we show that — under fairly restrictive conditions on the coefficients of the power series — the Caratheodory-Fejer method gives an asymptotically optimal approximation and in some cases is really a better approximation than the truncated power series.