Approximation of Cauchy-Type Integrals

Abstract

We investigate approximations of analytic functions determined by Cauchy-type integrals in Jordan domains of the complex plane. We develop, modify, and complete (in a certain sense) our earlier results. Special attention is given to the investigation of approximation of functions analytic in a disk by Taylor sums. In particular, we obtain asymptotic equalities for upper bounds of the deviations of Taylor sums on the classes of ψ-integrals of functions analytic in the unit disk and continuous in its closure. These equalities are a generalization of the known Stechkin's results on the approximation of functions analytic in the unit disk and having bounded rth derivatives (here, r is a natural number).

On the basis of the results obtained for a disk, we establish pointwise estimates for the deviations of partial Faber sums on the classes of ψ-integrals of functions analytic in domains with rectifiable Jordan boundaries. We show that, for a closed domain, these estimates are exact in order and exact in the sense of constants with leading terms if and only if this domain is a Faber domain.