2017
Том 69
№ 9

All Issues

Approximation of Cauchy-Type Integrals

Savchuk V. V., Stepanets O. I.

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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.

English version (Springer): Ukrainian Mathematical Journal 54 (2002), no. 5, pp 869-911.

Citation Example: Savchuk V. V., Stepanets O. I. Approximation of Cauchy-Type Integrals // Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 706-740.

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