Equilibrium problems associated with fast decreasing polynomials
Authors:
A. B. J. Kuijlaars and P. D. Dragnev
Journal:
Proc. Amer. Math. Soc. 127 (1999), 10651074
MSC (1991):
Primary 41A10, 31A15
MathSciNet review:
1469419
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Abstract: The determination of the support of the equilibrium measure in the presence of an external field is important in the theory of weighted polynomials on the real line. Here we present a general condition guaranteeing that the support consists of at most two intervals. Applying this to the external fields associated with fast decreasing polynomials, we extend previous results of Totik and KuijlaarsVan Assche. In the proof we use the iterated balayage algorithm which was first studied by Dragnev.
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 [1]
 P.D. Dragnev, Constrained energy problems for logarithmic potentials, Ph.D. Thesis, University of South Florida, Tampa, FL, 1997.
 [2]
 P.D. Dragnev and E.B. Saff, Constrained energy problems with applications to orthogonal polynomials of a discrete variable, J. Anal. Math. 72 (1997), 223259. CMP 98:04
 [3]
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 A.A. Gonchar and E.A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sb. 125 (1984), 117127 (Russian). English transl. in Math. USSR Sb. 53 (1986), 119130. MR 86f:41002
 [5]
 I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Orlando, 1980. MR 81g:33001
 [6]
 K.G. Ivanov and V. Totik, Fast decreasing polynomials, Constr. Approx. 6 (1990), 120. MR 90k:26023
 [7]
 A.B.J. Kuijlaars and W. Van Assche, A problem of Totik on fast decreasing polynomials, Constr. Approx. 14 (1998), 97112. CMP 98:05
 [8]
 , A contact problem in elasticity related to weighted polynomials on the real line, Rend. Circ. Mat. Palermo Suppl. 52 (1998), 575587.
 [9]
 N.S. Landkof, Foundations of Modern Potential Theory, SpringerVerlag, New York, 1972. MR 50:2520
 [10]
 D.S. Lubinsky and E.B. Saff, Strong Asymptotics for Extremal Polynomials Associated with Weights on , Lecture Notes Math., vol. 1305, SpringerVerlag, New York, 1988. MR 89m:41013
 [11]
 D.S. Lubinsky and V. Totik, How to discretize a logarithmic potential?, Acta Sci. Math. (Szeged) 57 (1993), 419428.
 [12]
 H.N. Mhaskar and E.B. Saff, Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc. 285 (1984), 203234. MR 86b:41024
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 N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953.MR 15:434e
 [14]
 E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, SpringerVerlag, Berlin, 1997.
 [15]
 H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992.
 [16]
 V. Totik, Fast decreasing polynomials via potentials, J. Anal. Math. 62 (1994), 131154. MR 96e:41010
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 , Weighted Approximation with Varying Weight, Lecture Notes Math., vol. 1569, SpringerVerlag, Berlin, 1994. MR 96f:41002
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 , Problem 13.5, in: Linear and Complex Analysis Problem Book 3, Part II (V.P. Havin and N.K. Nikolski, eds.), Lecture Notes Math., vol. 1574, SpringerVerlag, Berlin, 1994.
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Additional Information
A. B. J. Kuijlaars
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B3001 Leuven, Belgium
Email:
arno@wis.kuleuven.ac.be
P. D. Dragnev
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620
Address at time of publication:
Department of Mathematical Sciences, Indiana University Purdue University Fort Wayne, Fort Wayne, Indiana 46805
Email:
dragnevp@ipfw.edu
DOI:
http://dx.doi.org/10.1090/S0002993999045906
PII:
S 00029939(99)045906
Keywords:
Equilibrium measures,
extremal support,
balayage,
fast decreasing polynomials
Received by editor(s):
December 12, 1996
Received by editor(s) in revised form:
July 16, 1997
Additional Notes:
The first author is supported by a postdoctoral fellowship of the Belgian National Fund for Scientific Research, Scientific Research Network nr WO.011.96N: Fundamental Methods and Techniques in Mathematics. The research of the second author is in partial fulfillment of the Ph.D. requirements at the University of South Florida.
Communicated by:
J. Marshall Ash
Article copyright:
© Copyright 1999
American Mathematical Society
