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On the size of lemniscates of polynomials in one and several variables

Authors: A. Cuyt, K. Driver and D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 124 (1996), 2123-2136
MSC (1991): Primary 30C10, 32A30, 41A10, 41A21
MathSciNet review: 1322919
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Abstract: In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set $E:=\big \{z\,:\, |z|\le r$ and $|P(z)|\le \epsilon ^{n}\big \}$, for a polynomial $P$ of degree $\le n$. Usually, $P$ is taken to be monic, and either Cartan's Lemma or potential theory is used to estimate the size of $E$, in terms of Hausdorff contents, planar Lebesgue measure $m_{2}$, or logarithmic capacity cap. Here we normalize $\|P\|_{L_{\infty }\bigl (|z|\le r\bigr )}=1$ and show that cap$(E)\le 2r\epsilon $ and $m_{2} (E)\le \pi (2r\epsilon )^{2}$ are the sharp estimates for the size of $E$. Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on $\mathbb {C}^{n}$ or product capacity and Favarov's capacity. Several of our estimates are sharp with respect to order in $r$ and $\epsilon $.

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Additional Information

A. Cuyt
Affiliation: Department of Mathematics, UIA, University of Antwerp, Universiteitsplein 1, B2610 Wilrijk, Belgium

K. Driver
Affiliation: Department of Mathematics, Witwatersrand University, P.O. Wits 2050, South Africa

D. S. Lubinsky
Affiliation: Department of Mathematics, Witwatersrand University, P.O. Wits 2050, South Africa

Keywords: Polynomials, several complex variables, logarithmic capacity, product capacities, lemniscates, potential theory, Favarov's capacity
Received by editor(s): September 19, 1994
Received by editor(s) in revised form: January 30, 1995
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society