A~converse approximation theorem on subsets of elliptic curves

A. V. Khaustov ; N. A. Shirokov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 257-271

Résumé

Functions defined on closed subsets of elliptic curves $G\subset E=\{(\zeta,w)\in\mathbb C^2:w^2=4\zeta^3-g_2\zeta-g_3\}$ are considered. The following converse theorem of approximation is established. Consider a function $f\colon G\to\mathbb C$. Assume that there is a sequence of polynomials $P_n(\zeta, w)$, in two variables, $\deg{P_n}\leqslant n$, such that the following inequalities are valid: $$ |f(\zeta,w)-P_n(\zeta,w)|\leqslant c(f,G)\delta^\alpha_{1/n}(\zeta,w)\quad\text{при}\quad(\zeta,w)\in\partial G, $$ where $0\alpha1$. Then the function $f$ necessarily belongs to the class $H^\alpha(G)$. The direct approximation theorem was proved in the previous paper by the authors. Thus, a constructive description of the class $H^\alpha(G)$ is obtained.