# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\left[-1,1\right]$. This paper provides a method with which analogous results can be proven on general compact subsets of $ℝ$. 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 $\left[-1,1\right]$ to a system of intervals, and then to general sets.
##### MSC:
 42C05 General theory of orthogonal functions and polynomials
##### References:
 [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. [2] Carleson, L., Selected Problems on Exceptional Sets, Van Nostrand Mathematical Studies 13, Van Nostrand, Princeton, NJ, 1967. [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. [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. · 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. [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. · 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. [13] Mehta, M. L., Random Matrices, Academic Press, Boston, MA, 1991. [14] Rudin, W., Real and Complex Analysis, McGraw-Hill, New York, 1987. [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. [16] Saff, E. B. and Totik, V., Logarithmic Potentials with External Fields, Grundlehren der Mathematischen Wissenschaften 316, Springer, Berlin–Heidelberg, 1997. [17] Simon, B., Two extensions of Lubinsky’s universality theorem, to appear in J. Anal. Math. [18] Stahl, H. and Totik, V., General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications 43, Cambridge University Press, Cambridge, 1992. [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