Approximation by Double Periodic Functions in the Classes \(C_{A}^{{r*}}\)

We consider approximations by polynomials of Weierstrass doubly periodic functions for functions that are analytic in a domain and continuous in its closure. This problem is closely related to the approximation by holomorphic bivariate polynomials of a function holomorphic in a domain on an elliptic curve. We assume that the length of the arc is commensurable with the length of the chord at the boundary of the domain on the plane. This condition can also be extended to the domain on an elliptic curve. The possibility of obtaining an approximation estimate that is consistent with the so-called inverse theorem, i.e., with restoring the smoothness of the function by the rate of approximation, was previously established for classes of functions analytic in the domain whose derivative of given order in the closure of the domain has the Hölder modulus of continuity with an order of less than one. The approximation method used earlier does not make it possible to study classes of analytic functions whose derivative of some order is bounded. In this paper, we use an alternative method of approximation by polynomials of Weierstrass doubly periodic functions for functions analytic in a domain for which the derivative of a given order is bounded in this domain.