Approximation by double periodic functions in the class C^r_A

Abstract

In this work we will consider approximation by polynomials in doubly periodic Weierstrass functions for functions that are analytic in a domain and continuous in its closure. This problem is closely related to the approximation by holomorphic polynomials in two variables of a function that is holomorphic in a domain on an elliptic curve. We assume that at the boundary of the region on the plane the length of the are is commensurable with length of the chord. This condition also applies to the region on the elliptic curve. The possibility of obtaining an approximation estimate that is consistent with the so-called the inverse theorem. i. e., with the restoration of the smoothness of the function by the speed of approximation, it was previously obtained for classed of function analytic in a domain for which in the closure of the domain the derivative of a given order has a modulus of continuity of the Holder type, with an order less than unity. The approximation method used earlier did not make it possible to study classes of analytic functions whose derivative of some order is limited. In this paper, we use a different approximation method to approximate by polynomials in doubly periodic Weierstrass functions the functions that are analytic in a domain and whose derivative of a given order is bounded in the domain.

Keywords:

doubly periodic Weierstrass functions, polynomials, analytic functions, smooth in the closure of a domain