Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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. Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960
  • 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. 131 (1998), no. 624, x+216. MR 1407901, 10.1090/memo/0624
  • 5. Z. Ditzian and D. S. Lubinsky, Jackson and smoothness theorems for Freud weights in 𝐿_{𝑝}(0<𝑝≤∞), Constr. Approx. 13 (1997), no. 1, 99–152. MR 1424365, 10.1007/s003659900034
  • 6. F. D. Gakhov, Boundary value problems, Translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. MR 0198152
  • 7. A. A. Gonchar and E. A. Rakhmanov, The equilibrium measure and distribution of zeros of extremal polynomials, Mat. Sb. (N.S.) 125(167) (1984), no. 1(9), 117–127 (Russian). MR 760416
  • 8. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
  • 9. K. G. Ivanov and V. Totik, Fast decreasing polynomials, Constr. Approx. 6 (1990), no. 1, 1–20. MR 1027506, 10.1007/BF01891406
  • 10. Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922
  • 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), no. 1, 97–112. MR 1486392, 10.1007/s003659900065
  • 13. George G. Lorentz, Manfred v. Golitschek, and Yuly Makovoz, Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 304, Springer-Verlag, Berlin, 1996. Advanced problems. MR 1393437
  • 14. D. S. Lubinsky and E. B. Saff, Strong asymptotics for extremal polynomials associated with weights on 𝑅, Lecture Notes in Mathematics, vol. 1305, Springer-Verlag, Berlin, 1988. MR 937257
  • 15. H. N. Mhaskar, Introduction to the theory of weighted polynomial approximation, Series in Approximations and Decompositions, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1469222
  • 16. H. N. Mhaskar and E. B. Saff, Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials), Constr. Approx. 1 (1985), no. 1, 71–91. MR 766096, 10.1007/BF01890023
  • 17. E. A. Rakhmanov, Asymptotic properties of orthogonal polynomials on the real axis, Mat. Sb. (N.S.) 119(161) (1982), no. 2, 163–203, 303 (Russian). MR 675192
  • 18. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, New York, 1997. CMP 98:05
  • 19. Herbert Stahl and Vilmos Totik, General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. MR 1163828
  • 20. Vilmos Totik, Weighted approximation with varying weight, Lecture Notes in Mathematics, vol. 1569, Springer-Verlag, Berlin, 1994. MR 1290789
  • 21. Vilmos Totik, Fast decreasing polynomials via potentials, J. Anal. Math. 62 (1994), 131–154. MR 1269202, 10.1007/BF02835951

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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