The support of the equilibrium measure in the presence of a monomial external field on
Authors:
S. B. Damelin and A. B. J. Kuijlaars
Journal:
Trans. Amer. Math. Soc. 351 (1999), 45614584
MSC (1991):
Primary 31A15; Secondary 41A10, 45E05.
Published electronically:
July 21, 1999
MathSciNet review:
1675178
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The support of the equilibrium measure associated with an external field of the form , , with and 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.
 1.
Peter
Borwein and Tamás
Erdélyi, Polynomials and polynomial inequalities,
Graduate Texts in Mathematics, vol. 161, SpringerVerlag, New York,
1995. MR
1367960 (97e:41001)
 2.
S. B. Damelin and D. S. Lubinsky, Jackson theorems for Erd\H{o}s weights in (), J. Approx. Theory 94 (1998), 333382. CMP 98:17
 3.
P. Deift, T. Kriecherbauer and K. TR McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), 388475. CMP 99:05
 4.
P.
Deift and K.
TR McLaughlin, A continuum limit of the Toda lattice, Mem.
Amer. Math. Soc. 131 (1998), no. 624, x+216. MR 1407901
(98h:58076), http://dx.doi.org/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
(97k:41028), http://dx.doi.org/10.1007/s003659900034
 6.
F.
D. Gakhov, Boundary value problems, Translation edited by I.
N. Sneddon, Pergamon Press, OxfordNew YorkParis; AddisonWesley
Publishing Co., Inc., Reading, Mass.London, 1966. MR 0198152
(33 #6311)
 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
(86f:41002)
 8.
I.
S. Gradshteyn and I.
M. Ryzhik, Table of integrals, series, and products, Academic
Press [Harcourt Brace Jovanovich, Publishers], New YorkLondonToronto,
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 (81g:33001)
 9.
K.
G. Ivanov and V.
Totik, Fast decreasing polynomials, Constr. Approx.
6 (1990), no. 1, 1–20. MR 1027506
(90k:26023), http://dx.doi.org/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 YorkLondonSydney, 1966. MR 0204922
(34 #4757)
 11.
A. B. J. Kuijlaars and P. D. Dragnev, Equilibrium problems associated with fast decreasing polynomials, Proc. Amer. Math. Soc. 127 (1999), 10651074. 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
(99b:26026), http://dx.doi.org/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, SpringerVerlag, Berlin, 1996. Advanced problems.
MR
1393437 (97k:41002)
 14.
D.
S. Lubinsky and E.
B. Saff, Strong asymptotics for extremal polynomials associated
with weights on 𝑅, Lecture Notes in Mathematics,
vol. 1305, SpringerVerlag, Berlin, 1988. MR 937257
(89m:41013)
 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 (98i:41014)
 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
(86a:41004), http://dx.doi.org/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
(84e:42025)
 18.
E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, SpringerVerlag, 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
(93d:42029)
 20.
Vilmos
Totik, Weighted approximation with varying weight, Lecture
Notes in Mathematics, vol. 1569, SpringerVerlag, Berlin, 1994. MR 1290789
(96f:41002)
 21.
Vilmos
Totik, Fast decreasing polynomials via potentials, J. Anal.
Math. 62 (1994), 131–154. MR 1269202
(96e:41010), http://dx.doi.org/10.1007/BF02835951
 1.
 P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, SpringerVerlag, New York, 1995. MR 97e:41001
 2.
 S. B. Damelin and D. S. Lubinsky, Jackson theorems for Erd\H{o}s weights in (), J. Approx. Theory 94 (1998), 333382. CMP 98:17
 3.
 P. Deift, T. Kriecherbauer and K. TR McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), 388475. CMP 99:05
 4.
 P. Deift and K. TR 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 (), Constr. Approx. 13 (1997), 99152. 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), 117127. English transl: Math. USSRSb. 53 (1986), 119130. 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), 120. 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), 10651074. CMP 99:06
 12.
 A. B. J. Kuijlaars and W. Van Assche, A problem of Totik on fast decreasing polynomials, Constr. Approx. 14 (1998), 97112. MR 99b:26026
 13.
 G. G. Lorentz, M. von Golitschek and Y. Makovoz, Constructive Approximation, Advanced Problems, SpringerVerlag, Berlin, 1996. MR 97k:41002
 14.
 D. S. Lubinsky and E. B. Saff, Strong Asymptotics for Extremal Polynomials associated with Weights on , Lect. Notes in Math. vol. 1305, SpringerVerlag, 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), 7191. MR 86a:41004
 17.
 E. A. Rakhmanov, On asymptotic properties of polynomials orthogonal on the real axis, Mat. Sb. 119 (1982), 163203. English transl.: Math. USSRSb. 47 (1984), 155193. MR 84e:42025
 18.
 E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, SpringerVerlag, 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, SpringerVerlag, Berlin, 1994. MR 96f:41002
 21.
 V. Totik, Fast decreasing polynomials via potentials, J. Anal. Math. 62 (1994), 131154. 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
Email:
036sbd@cosmos.wits.ac.za
A. B. J. Kuijlaars
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B3001 Leuven, Belgium
Email:
arno@wis.kuleuven.ac.be
DOI:
http://dx.doi.org/10.1090/S000299479902509X
PII:
S 00029947(99)02509X
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
