Approximation of functions,which are analytic in a simply connected domain and can be represented by a Cauchy type integral,by sequences of rational fractions with poles prescribed by an array

Let G and {_kj} be the domain and the array mentioned in the title (the boundary of the domain is assumed to be rectifiable). One describes a general scheme for the approximation of fonctionsf in the domain G, representable in the form f(z)=(2i)^–1g()(–z)^–1d, where g L_z (G), by a sequence of rational fractions. The characteristic feature of this scheme is the fact that the poles _k of the fraction lie in the k-th row of the array {_kj}. There is given a condition on {_kj}, necessary and sufficient in order that each functionf, of the kind described above, should admit a uniform approximation inside G with the aid of the indicated scheme. In the case when this condition is not satisfied and \G.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 254–273, 1989.