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

Damelin, S.; Kuijlaars, A.

doi:10.1090/S0002-9947-99-02509-X

The support of the equilibrium measure in the presence of a monomial external field on $[-1,1]$
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by S. B. Damelin and A. B. J. Kuijlaars PDF

Trans. Amer. Math. Soc. 351 (1999), 4561-4584 Request permission

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.

References

Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
S. B. Damelin and D. S. Lubinsky, Jackson theorems for Erdős weights in $L_p$ ($0<p\leq \infty$), J. Approx. Theory 94 (1998), 333–382.
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.
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, DOI 10.1090/memo/0624
Z. Ditzian and D. S. Lubinsky, Jackson and smoothness theorems for Freud weights in $L_p\ (0<p\leq \infty )$, Constr. Approx. 13 (1997), no. 1, 99–152. MR 1424365, DOI 10.1007/s003659900034
F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. Translation edited by I. N. Sneddon. MR 0198152
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
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
K. G. Ivanov and V. Totik, Fast decreasing polynomials, Constr. Approx. 6 (1990), no. 1, 1–20. MR 1027506, DOI 10.1007/BF01891406
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
A. B. J. Kuijlaars and P. D. Dragnev, Equilibrium problems associated with fast decreasing polynomials, Proc. Amer. Math. Soc. 127 (1999), 1065–1074.
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, DOI 10.1007/s003659900065
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, DOI 10.1007/978-3-642-60932-9
D. S. Lubinsky and E. B. Saff, Strong asymptotics for extremal polynomials associated with weights on $\textbf {R}$, Lecture Notes in Mathematics, vol. 1305, Springer-Verlag, Berlin, 1988. MR 937257, DOI 10.1007/BFb0082413
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
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, DOI 10.1007/BF01890023
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
E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, New York, 1997.
Herbert Stahl and Vilmos Totik, General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. MR 1163828, DOI 10.1017/CBO9780511759420
Vilmos Totik, Weighted approximation with varying weight, Lecture Notes in Mathematics, vol. 1569, Springer-Verlag, Berlin, 1994. MR 1290789, DOI 10.1007/BFb0076133
Vilmos Totik, Fast decreasing polynomials via potentials, J. Anal. Math. 62 (1994), 131–154. MR 1269202, DOI 10.1007/BF02835951