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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
DOI: https://doi.org/10.1090/S0002-9939-96-03293-5
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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  • 1. G.A. Baker, Essentials of Padé Approximants, Academic Press, New York, 1975. MR 56:12710
  • 2. G.A. Baker and P.R. Graves--Morris, Padé Approximants, Part 1: Basic Theory, Encyclopaedia of Mathematics and its Applications, Vol. 13, Addison--Wesley, Reading, MA, 1981. MR 83a:41009a
  • 3. E. Bedford and B.A. Taylor,, The Complex Equilibrium Measure of a Symmetric Convex Set in $\mathbb {R}^{n}$, Trans. Amer. Math. Soc. 294 (1986), 705--717. MR 87f:32039
  • 4. P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Book (to appear).
  • 5. A. Cuyt, Multivariate Padé Approximants Revisited, BIT 26 (1986), 71--79. MR 87f:41031
  • 6. U. Cegrell, Capacities in Complex Analysis, Aspects of Mathematics, Vol. 14, Vieweg, Braunschweig, 1988. MR 89m:32001
  • 7. G.M. Goluzin, Geometric Theory of Functions of a Complex Variable, Translations of Math. Monographs, Vol. 26, American Mathematical Society, Providence, 1969. MR 40:308
  • 8. A.A. Goncar, Local Condition for the Single--Valuedness of Analytic Functions of Several Variables, Math. USSR. Sbornik 22 (1974), 305--322. MR 58:28632
  • 9. W.K. Hayman and P.B. Kennedy, Subharmonic Functions, Vol. 1,, Academic Press, London, 1976. MR 57:665
  • 10. E. Hille, Analytic Function Theory, Vol. 2, Chelsea, New York, 1987. MR 34:1490 (earlier ed.)
  • 11. J. Karlsson and H. Wallin, Rational Approximation by an Interpolation Procedure in Several Variables, (in) Padé and Rational Approximation: Theory and Applications (eds. E.B. Saff and R.S. Varga), Academic Press, New York, 1977, pp. 83--100. MR 58:1877
  • 12. N.S. Landkof, Foundations of Modern Potential Theory, Grundlehren der Mathematischen Wissenchaften, Vol. 180, Springer, Berlin, 1972. MR 50:2520
  • 13. J. Nuttall, The Convergence of Padé Approximants of Meromorphic Functions, J. Math. Anal. Applns. 31 (1970), 147--153. MR 44:5477
  • 14. B. Paneah, On a Lower Bound for the Absolute Value of a Polynomial of Several Variables, J. Approx. Theory 78 (1994), 402--409. MR 95j:32020
  • 15. C. Pommerenke, Padé Approximants and Convergence in Capacity, J. Math. Anal. Applns. 41 (1973), 775--780. MR 48:6432
  • 16. C.A. Rogers, Hausdorff Measures, Cambridge University Press, Cambridge, 1970. MR 43:7576
  • 17. A. Sadullaev, Plurisubharmonic Measures and Capacities on Complex Manifolds, Russian Math. Surveys 36 (1981), 61--119. MR 83c:32026 (Russian original)
  • 18. M. Schiffer and J. Siciak, Transfinite Diameter and Analytic Continuation of Functions of Two Complex Variables, (in) Studies in Math. Anal. and Related Topics, Stanford University Press, Stanford, 1962, pp. 341--358. MR 27:342
  • 19. G.W. Stewart, and J--G. Sun, Matrix Perturbation Theory, Academic Press, London, 1990. MR 92a:65017
  • 20. V.P. Zaharjuta, Transfinite Diameter, Chebyshev Constants, and Capacity for Compacta in $\mathbb {C}^{n}$, Math. USSR. Sbornik 25 (1975), 350--364. MR 58:6342

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Additional Information

A. Cuyt
Affiliation: Department of Mathematics, UIA, University of Antwerp, Universiteitsplein 1, B2610 Wilrijk, Belgium
Email: CUYT@WINS.UIA.AC.BE

K. Driver
Affiliation: Department of Mathematics, Witwatersrand University, P.O. Wits 2050, South Africa
Email: 036KAD@COSMOS.WITS.AC.ZA

D. S. Lubinsky
Affiliation: Department of Mathematics, Witwatersrand University, P.O. Wits 2050, South Africa
Email: 036DSL@COSMOS.WITS.AC.ZA

DOI: https://doi.org/10.1090/S0002-9939-96-03293-5
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

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