Approximation by holomorphic polynomials in two variables of functions holomorphic in domains on elliptic curves in {C^2} has been studied since the early 2000s. Technically, this question reduces to approximating a function analytic in a domain lying strictly inside the fundamental parallelogram of some doubly periodic Weierstrass function using polynomials in this Weierstrass function and its derivative. Constructive descriptions of functional classes of analytic functions defined in the uniform metric have been obtained. The classes under consideration are absolutely continuous functions defined on several intervals lying in the fundamental parallelogram of the doubly periodic Weierstrass function with a norm expressed through an integral over intervals, while it turned out that to prove inverse approximation theorems, it is important to have an estimate for the growth of approximating polynomials on the entire plane. This paper proves that the estimates used in the inverse theorem are valid for polynomials that are used in the proof of the direct theorem of approximation by polynomials of the doubly periodic Weierstrass function and its derivative function, defined on several intervals.
Shagay et al. (Sun,) studied this question.