Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T20:12:13.063Z Has data issue: false hasContentIssue false
  • English
  • Français

Universality Under Szegő’s Condition

Published online by Cambridge University Press:  20 November 2018

Vilmos Totik
Vilmos Totik*
Affiliation:
MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Szeged, Aradi v. tere 1, 6720, Hungary and Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave, CMCh z, Tampa, FL 33620-5700, USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper presents a theoremon universality on orthogonal polynomials/randommatrices under a weak local condition on the weight function $w$ . With a new inequality for polynomials and with the use of fast decreasing polynomials, it is shown that an approach of D. S. Lubinsky is applicable. The proof works at all points that are Lebesgue-points for both the weight function $w$ and $\log \,w$ .

Keywords

42C0560B2030C8531A15universalityrandom matricesChristoòel functionsasymptoticspotential theory
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 1 , 01 March 2016 , pp. 211 - 224
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Armitage, D. H. and Gardiner, S. J., Classical potential theory. Springer Mongraphs in Mathematics, Springer-Verlag, London, 2001.Google Scholar
[2] Avila, A., Last, Y.. and Simon, B.. Bulk universality and dock spacing of zeros for ergodic Jacobi matrices with absolutely continuous spectrum. Anal. PDE 3(2010), no. 1, 81–108. http://dx.doi.Org/10.2140/apde.20103.81 Google Scholar
[3] Deift, P., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.Google Scholar
[4] Findley, E. M., Universality for locally Szegô measures. J. Approx. Theory 155(2008), no. 2,136-54.Google Scholar
[5] Levin, A. L. and Lubinsky, D. S., Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials. J. Approx. Theory 150(2008), no. 1, 69–95. http://dx.doi.Org/10.1016/j.jat.2007.05.003 Google Scholar
[6] Lubinsky, D. S., A new approach to universality limits involving orthogonal polynomials. Annals of Math. 170(2009), no. 2, 915–939. http://dx.doi.Org/10.4007/annals.2009.170.915 Google Scholar
[7] Lubinsky, D. S., Universality in the bulk for arbitrary measures on compact sets. J. Anal. Math. 106(2008), 373–394. http://dx.doi.Org/10.1007/s11854-008-0053-1 Google Scholar
[8] Mehta, M. L., Random matrices. Second, éd., Academic Press, Boston, MA, 1991.Google Scholar
[9] Pastur, L. A., Spectral and probabilistic aspects of matrix models. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996, pp. 207–242.Google Scholar
[10] Ransford, T., Potential theory in the complex plane. London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995 Google Scholar
[11] Simon, B., Two extensions of Lubinsky's universality theorem. J. Anal. Math. 105(2008), 345–362. http://dx.doi.Org/10.1007/s11854-008-0039-z Google Scholar
[12] Stahl, H. and Totik, V.. General orthogonal polynomials. Encyclopedia of Mathematics and its Applications, 43, Cambridge University Press, Cambridge, 1992.Google Scholar
[13] Totik, V., Asymptotics for Christoffel functions for general measures on the real line. J. Anal. Math. 81(2000), 283–303. http://dx.doi.Org/10.1007/BF02788993 Google Scholar
[14] Totik, V., Universality and fine zero spacing on general sets. Arkiv for Math. 47(2009), no. 2, 361–391. http://dx.doi.Org/10.1007/s1512-008-0071-3 Google Scholar
[15] Totik, V., Christoffel functions on curves and domains. Trans. Amer. Math. Soc. 362(2010), no. 4, 2053–2087. http://dx.doi.Org/10.1090/S0002-9947-09-05059-4 Google Scholar
[16] Totik, V., Szegô's problem on curves. Amer. J. Math. 135(2013), no. 6,1507-1524. http://dx.doi.Org/10.1353/ajm.2013.0053 Google Scholar
[17] Tsuji, M., Potential theory in modern function theory. Maruzen, Tokyo, 1959.Google Scholar
[18] Walsh, J. L., Interpolation and approximation by rational functions in the complex domain. Third éd., American Mathematical Society Colloquium Publications, XX, American Mathematical Society, Providence, RI, 1960.Google Scholar