On the Rate of Approximation of Closed Jordan Curves by Lemniscates

O. N. Kosukhin

Matematičeskie zametki, Tome 77 (2005) no. 6, pp. 861-876

Résumé

As proved by Hilbert, it is, in principle, possible to construct an arbitrarily close approximation in the Hausdorff metric to an arbitrary closed Jordan curve $\Gamma$ in the complex plane $\{z\}$ by lemniscates generated by polynomials $P(z)$. In the present paper, we obtain quantitative upper bounds for the least deviations $H_n(\Gamma)$ (in this metric) from the curve $\Gamma$ of the lemniscates generated by polynomials of a given degree $n$ in terms of the moduli of continuity of the conformal mapping of the exterior of $\Gamma$ onto the exterior of the unit circle, of the mapping inverse to it, and of the Green function with a pole at infinity for the exterior of $\Gamma$. For the case in which the curve $\Gamma$ is analytic, we prove that $H_n(\Gamma)=O(q^n)$, $0\le q=q(\Gamma)1$, $n\to\infty$.