Rational Approximation on Closed Curves

Abstract

In this paper, we study a problem of approximation for the classes of functions determined only on the boundary of domain in weighted integral spaces by means of the rational functions of the form (1) where b is a point lying strictly inside the considered curve. Notice that the approximation estimations, generally speaking, coincide with the esti- mations of polynomial approximation for p E classes (Smirnov's class). Approximation problem for the classes of functions de- termined only on the boundary of domain is of great impor- tance alongside with the study of approximation of functions by means of polynomials analytic in the domain G and with some conditions on the boundary Γ . Obviously, it is im- possible in general to approximate such classes of functions by means of polynomials(12). Therefore, various kinds of rational functions or so called generalized polynomials are mostly used in this case as an approximation tool(12). J. I. Mamedkhanov, D. M. Israfilov and I. M. Botchaev investi- gated the approximation problems of functions determined only on the boundary of domain by means of rational func- tions of the form ( ) ( ) ,1 nn R z P z z = for certain classes of curves in terms of uniform metric(1-4). In this paper, we study the approximation problems of a function from the class ( ) , p L ϑ Γ by means of a rational function of the form ( ) ( ) n k nk kn R z a z b − = = −