On the Rate of Approximation of Closed Jordan Curves by Lemniscates |
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Authors: | O. N. Kosukhin |
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Affiliation: | (1) M. V. Lomonosov Moscow State University, Moscow, Russia |
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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 Γ 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 Hn(Γ) (in this metric) from the curve Γ 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 Γ 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 Γ. For the case in which the curve Γ is analytic, we prove that Hn(Γ) = O(qn), 0 ≤ q = q(Γ) < 1, n → ∞.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 861–876.Original Russian Text Copyright ©2005 by O. N. Kosukhin. |
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Keywords: | closed Jordan curve lemniscate Hausdorff distance Faber polynomial Laplace operator harmonic capacity |
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