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The support of the equilibrium measure in the presence of a monomial external field on $[-1,1]$

Authors: S. B. Damelin and A. B. J. Kuijlaars
Journal: Trans. Amer. Math. Soc. 351 (1999), 4561-4584
MSC (1991): Primary 31A15; Secondary 41A10, 45E05.
Published electronically: July 21, 1999
MathSciNet review: 1675178
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Abstract: The support of the equilibrium measure associated with an external field of the form $Q(x) = - cx^{2m+1}$, $x \in [-1,1]$, with $c > 0$ and $m$ a positive integer is investigated. It is shown that the support consists of at most two intervals. This resolves a question of Deift, Kriecherbauer and McLaughlin.

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  • 1. P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995. MR 97e:41001
  • 2. S. B. Damelin and D. S. Lubinsky, Jackson theorems for Erd\H{o}s weights in $L_p$ ($0<p\leq \infty$), J. Approx. Theory 94 (1998), 333-382. CMP 98:17
  • 3. P. Deift, T. Kriecherbauer and K. T-R McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), 388-475. CMP 99:05
  • 4. P. Deift and K. T-R McLaughlin, A continuum limit of the Toda lattice, Mem. Amer. Math. Soc. 624 (1998). MR 98h:58076
  • 5. Z. Ditzian and D. S. Lubinsky, Jackson and smoothness theorems for Freud weights in $L_p$ ($0<p\leq \infty$), Constr. Approx. 13 (1997), 99-152. MR 97k:41028
  • 6. F.D. Gakhov, Boundary Value Problems, Pergamon Press, Oxford, 1966. MR 33:6311
  • 7. A. A. Gonchar and E. A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sb. 125 (1984), 117-127. English transl: Math. USSR-Sb. 53 (1986), 119-130. MR 86f:41002
  • 8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Orlando, 1980. MR 81g:33001
  • 9. K. G. Ivanov and V. Totik, Fast decreasing polynomials, Constr. Approx. 6 (1990), 1-20. MR 90k:26023
  • 10. S. Karlin and W. J. Studden, Tchebycheff Systems with Applications in Analysis and Statistics, Wiley, New York, 1966. MR 34:4757
  • 11. A. B. J. Kuijlaars and P. D. Dragnev, Equilibrium problems associated with fast decreasing polynomials, Proc. Amer. Math. Soc. 127 (1999), 1065-1074. CMP 99:06
  • 12. A. B. J. Kuijlaars and W. Van Assche, A problem of Totik on fast decreasing polynomials, Constr. Approx. 14 (1998), 97-112. MR 99b:26026
  • 13. G. G. Lorentz, M. von Golitschek and Y. Makovoz, Constructive Approximation, Advanced Problems, Springer-Verlag, Berlin, 1996. MR 97k:41002
  • 14. D. S. Lubinsky and E. B. Saff, Strong Asymptotics for Extremal Polynomials associated with Weights on $\mathbb R$, Lect. Notes in Math. vol. 1305, Springer-Verlag, Berlin, 1988. MR 89m:41013
  • 15. H. N. Mhaskar, Introduction to the Theory of Weighted Polynomial Approximation, World Scientific, Singapore, 1996. MR 98i:41014
  • 16. H. N. Mhaskar and E. B. Saff, Where does the sup norm of a weighted polynomial live?, Constr. Approx. 1 (1985), 71-91. MR 86a:41004
  • 17. E. A. Rakhmanov, On asymptotic properties of polynomials orthogonal on the real axis, Mat. Sb. 119 (1982), 163-203. English transl.: Math. USSR-Sb. 47 (1984), 155-193. MR 84e:42025
  • 18. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, New York, 1997. CMP 98:05
  • 19. H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992. MR 93d:42029
  • 20. V. Totik, Weighted Approximation with Varying Weight, Lect. Notes in Math. vol. 1569, Springer-Verlag, Berlin, 1994. MR 96f:41002
  • 21. V. Totik, Fast decreasing polynomials via potentials, J. Anal. Math. 62 (1994), 131-154. MR 96e:41010

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Additional Information

S. B. Damelin
Affiliation: Department of Mathematics, University of the Witwatersrand, PO Wits 2050, South Africa

A. B. J. Kuijlaars
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Leuven, Belgium

Keywords: Balayage, equilibrium measure, external field, potential theory, weighted polynomials
Received by editor(s): June 22, 1997
Published electronically: July 21, 1999
Additional Notes: The research of the first author was begun while visiting the Mathematics Department at the Katholieke Universiteit Leuven, whose invitation to visit and hospitality are kindly acknowledged. This research was partly financed by FWO research project G.0278.97.
The second author was 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
Article copyright: © Copyright 1999 American Mathematical Society

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