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Universality and fine zero spacing on general sets. (English) Zbl 1180.42017
Summary: A recent approach of D. S. Lubinsky yields universality in random matrix theory and fine zero spacing of orthogonal polynomials under very mild hypothesis on the weight function, provided the support of the generating measure $\mu $ is $[-1,1]$. This paper provides a method with which analogous results can be proven on general compact subsets of $\bbfR$. Both universality and fine zero spacing involves the equilibrium measure of the support of $\mu $. The method is based on taking polynomial inverse images, by which results can be transferred from $[-1,1]$ to a system of intervals, and then to general sets.

42C05General theory of orthogonal functions and polynomials
Full Text: DOI
[1] Ancona, A., Démonstration d’une conjecture sur la capacité et l’effilement, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), 393--395. · Zbl 0544.31006
[2] Carleson, L., Selected Problems on Exceptional Sets, Van Nostrand Mathematical Studies 13, Van Nostrand, Princeton, NJ, 1967. · Zbl 0189.10903
[3] Deift, P. A., Orthogonal Polynomials and Random Matrices: a Riemann--Hilbert Approach, Courant Lecture Notes in Mathematics 3, Courant Institute of Mathematical Sciences, New York University, New York, 1999. · Zbl 0997.47033
[4] Deift, P. A., Kriecherbauer, T., McLaughlin, K. T. R., Venakides, S. and Zhou, X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335--1425. · Zbl 0944.42013 · doi:10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1
[5] DeVore, R. A. and Lorentz, G. G., Constructive Approximation, Grundlehren der Mathematischen Wissenschaften 303, Springer, Berlin--Heidelberg, 1993.
[6] Findley, M., Universality for locally Szego measures, to appear in J. Approx. Theory. · Zbl 1171.42015
[7] Geronimo, J. S. and Van Assche, W., Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc. 308 (1988), 559--581. · Zbl 0652.42009 · doi:10.1090/S0002-9947-1988-0951620-6
[8] Hoffman, K., Banach Spaces of Analytic Functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
[9] Kuijlaars, A. B. J. and Vanlessen, M., Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Int. Math. Res. Not. 2002:30 (2002), 1575--1600. · Zbl 1122.30303 · doi:10.1155/S1073792802203116
[10] Levin, A. L. and Lubinsky, D. S., Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials, J. Approx. Theory 150 (2008), 69--95. · Zbl 1138.33006 · doi:10.1016/j.jat.2007.05.003
[11] Levin, A. L. and Lubinsky, D. S., Universality limits for exponential weights, to appear in Constr. Approx. doi:10.107/s00365-008-9020-4
[12] Lubinsky, D. S., A new approach to universality limits involving orthogonal polynomials, to appear in Annals of Math. · Zbl 1176.42022
[13] Mehta, M. L., Random Matrices, Academic Press, Boston, MA, 1991.
[14] Rudin, W., Real and Complex Analysis, McGraw-Hill, New York, 1987. · Zbl 0925.00005
[15] Saff, E. B. and Totik, V., What parts of a measure’s support attract zeros of the corresponding orthogonal polynomials?, Proc. Amer. Math. Soc. 114 (1992), 185--190. · Zbl 0770.42015
[16] Saff, E. B. and Totik, V., Logarithmic Potentials with External Fields, Grundlehren der Mathematischen Wissenschaften 316, Springer, Berlin--Heidelberg, 1997. · Zbl 0881.31001
[17] Simon, B., Two extensions of Lubinsky’s universality theorem, to appear in J. Anal. Math. · Zbl 1168.42304
[18] Stahl, H. and Totik, V., General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications 43, Cambridge University Press, Cambridge, 1992. · Zbl 0791.33009
[19] Totik, V., Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math. 81 (2000), 283--303. · Zbl 0966.42017 · doi:10.1007/BF02788993
[20] Totik, V., Polynomial inverse images and polynomial inequalities, Acta Math. 187 (2001), 139--160. · Zbl 0997.41005 · doi:10.1007/BF02392833