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