An inverse approximation theorem on subsets of elliptic curves. (Russian) Zbl 1094.30043

Khaustov, A. V.; Shirokov, N. A.

Zap. Nauchn. Semin. POMI 314, 257-271, 291 (2004); translation in J. Math. Sci., New York 133, No. 6, 1756-1764 (2006).

Let \(\Psi\) be the conformal mapping of \(| z| >1\) onto the complement of a simply connected domain \(D\) to the extended complex plane. It is assumed that any arc of \(\partial D\) is of the order of the corresponding chord. Let \(\delta_n(z)=\text{dist}(z,\Psi\{| z| =1+1/n\})\) with \(n=1,2,...\); \(0<\alpha<1\). Let \(\mathcal{P}\mathbf{}(z)\) be the Weierstrass function. The authors prove that if for \(f:\overline{D}\to \mathbf{C}\) there exist a sequence of the polynomials \(p_n(\zeta,w)\) such that \(| f(z)-p_n(\mathcal{P}(z),\mathcal{P}^{\prime}(z))| \leq C(f,D) \delta^{\alpha}_n(z)\), then \(f\) belongs to the Hölder space \(H^{\alpha}(\overline{D})\).

Reviewer: Vladimir Mityushev (Kraków)

MSC:

30E10

Approximation in the complex plane

Keywords:

inverse approximation problem; Weierstrass function; polynomial approximation