On the Rate of Approximation of Closed Jordan Curves by Lemniscates

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 $H_n(\Gamma)=O(q^n)$, $0\le q=q(\Gamma)<1$, $n\to\infty$.