Equilibrium problems associated with fast decreasing polynomials

Kuijlaars, A.; Dragnev, P.

doi:10.1090/S0002-9939-99-04590-6

Equilibrium problems associated with fast decreasing polynomials
HTML articles powered by AMS MathViewer

by A. B. J. Kuijlaars and P. D. Dragnev PDF

Proc. Amer. Math. Soc. 127 (1999), 1065-1074 Request permission

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 Kuijlaars-Van Assche. In the proof we use the iterated balayage algorithm which was first studied by Dragnev.

References

P.D. Dragnev, Constrained energy problems for logarithmic potentials, Ph.D. Thesis, University of South Florida, Tampa, FL, 1997.
P.D. Dragnev and E.B. Saff, Constrained energy problems with applications to orthogonal polynomials of a discrete variable, J. Anal. Math. 72 (1997), 223–259.
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
A.B.J. Kuijlaars and W. Van Assche, A problem of Totik on fast decreasing polynomials, Constr. Approx. 14 (1998), 97–112.
—, A contact problem in elasticity related to weighted polynomials on the real line, Rend. Circ. Mat. Palermo Suppl. 52 (1998), 575–587.
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027, DOI 10.1007/978-3-642-65183-0
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
D.S. Lubinsky and V. Totik, How to discretize a logarithmic potential?, Acta Sci. Math. (Szeged) 57 (1993), 419–428.
H. N. Mhaskar and E. B. Saff, Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc. 285 (1984), no. 1, 203–234. MR 748838, DOI 10.1090/S0002-9947-1984-0748838-0
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, 1997.
H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992.
Vilmos Totik, Fast decreasing polynomials via potentials, J. Anal. Math. 62 (1994), 131–154. MR 1269202, DOI 10.1007/BF02835951
Vilmos Totik, Weighted approximation with varying weight, Lecture Notes in Mathematics, vol. 1569, Springer-Verlag, Berlin, 1994. MR 1290789, DOI 10.1007/BFb0076133
—, 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, Springer-Verlag, Berlin, 1994.