A CONVERSE APPROXIMATION THEOREM ON SUBSETS OF ELLIPTIC CURVES


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Abstract

Functions defined on closed subsets of elliptic curves G ⊂ E = {(ζ, w) ∈ ℂ 2:w2 = 4ζ3 − g2ζ − g3} are considered. The following converse theorem of approximation is established. Consider a function f:G → ℂ. Assume that there is a sequence of polynomials Pn(ζ, w) in two variables, deg Pn ≥ n, such that the following inequalities are valid: |f(ζ,w) − Pn(ζ,w)| = ≤ c(f,G)δαn (ζ,w), (ζ,w) ∈ δG, where 0 < α < 1. Then the function f necessarily belongs to the class Hα(G). The direct approximation theorem was proved in the previous paper by the authors. Thus, a constructive description of the class Hα(G) is obtained. Bibliography: 6 titles.

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