Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T02:52:52.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials

Published online by Cambridge University Press:  20 November 2018

A. Kroó  and
D. S. Lubinsky
A. Kroó
Affiliation:
Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u.13-15, Budapest H-1053, Hungary, e-mail: [email protected]
D. S. Lubinsky
Affiliation:
Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u.13-15, Budapest H-1053, Hungary, 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.

We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain. In particular, this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.

Keywords

42C0542C9942B0560B20orthogonal polynomialsrandom matricesunitary ensemblescorrelation functionsChristoffel functions
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 3 , 01 June 2013 , pp. 600 - 620
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Bos, L., Asymptotics for Christoffel functions for Jacobi like weights on a ball in Rm. New Zealand J. Math. 23(1994), no. 2, 99–109.Google Scholar
[2] Bos, L., Della Vecchia, B., and Mastroianni, G., On the asymptotics of Christoffel functions for centrally symmetric weights functions on the ball in Rn. Rend. Circ. Mat. Palermo 52(1998), 277–290.Google Scholar
[3] Bloom, T. and Levenberg, N., Asymptotics for Christoffel functions of planar measures, J. Anal. Math. 106(2008), 353–371. http://dx.doi.org/10.1007/s11854-008-0052-2 Google Scholar
[4] Deift, P., Orthogonal polynomials and random matrices: A Riemann-Hilbert approach. Courant Institute Lecture Notes, 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.Google Scholar
[5] Forrester, P. J., Log-gases and random matrices. London Mathematical Society Monographs Series, 34, Princeton University Press, Princeton, NJ, 2010.Google Scholar
[6] Kroó, A., Extremal properties of multivariate polynomials on sets with analytic parametrizatio. East J. Approx. 7(2001), no. 1, 27–40.Google Scholar
[7] Kroó, A. and Swetits, J. J., On density of interpolation points, a Kadec-type theorem, and Saff ‘s principle of contamination in Lp approximation. Constr. Approx. 8(1992), no. 1, 87–103. http://dx.doi.org/10.1007/BF01208908 Google Scholar
[8] Lubinsky, D. S., A new approach to universality limits involving orthogonal polynomials. Ann. of Math. 170(2009), no. 2, 915–939. http://dx.doi.org/10.4007/annals.2009.170.915Google Scholar
[9] Lubinsky, D. S., Bulk universality holds in measure for compactly supported measures. J. Anal. Math. 116(2012), 219–253. http://dx.doi.org/10.1007/s11854-012-0006-6 Google Scholar
[10] Mató, A., Nevai, P., and Totik, V., Szegő's extremum problem on the unit circle. Ann. of Math. 134(1991), no. 2, 433–453. http://dx.doi.org/10.2307/2944352 Google Scholar
[11] Nevai, P. G., Orthogonal polynomials. Mem. Amer. Math. Soc. 18(1979), no. 213.Google Scholar
[12] Nevai, P. G., Géza Freud, orthogonal polynomials and Christoffel functions: A case study. J. Approx. Theory 48(1986), no. 1.Google Scholar
[13] 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
[14] Simon, B., Szego's theorem and its descendants. Spectral theory for L2 perturbations of orthogonal polynomials. M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011.Google Scholar
[15] Stahl, H. and V.|Totik, General orthogonal polynomials. Encyclopedia of Mathematics and its Applications, 43, Cambridge University Press, Cambridge, 1992 Google Scholar
[16] Szegő, G., Orthogonal polynomials. Fourth ed., American Mathematical Society, Providence, RI, 1975.Google Scholar
[17] 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
[18] Totik, V., Universality and fine zero spacing on general sets. Ark. Mat. 47(2009), no. 2, 361–391. http://dx.doi.org/10.1007/s11512-008-0071-3 Google Scholar
[19] Y. Xu, , Christoffel functions and Fourier series for multivariate orthogonal polynomials. J. Approx. Theory 82(1995), no. 2, 205–239. http://dx.doi.org/10.1006/jath.1995.1075 Google Scholar
[20] Y. Xu, , Asymptotics for orthogonal polynomials and Christoffel functions on a ball. Methods Appl. Anal. 3(1996), no. 2, 257–272.Google Scholar
[21] Y. Xu, , Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in Rd. Proc. Amer. Math. Soc. 128(1998), no. 10, 3027–3036. http://dx.doi.org/10.1090/S0002-9939-98-04415-3 Google Scholar
[22] Y. Xu, , Summability of Fourier orthogonal series for Jacobi weight on a ball in Rd. Trans. Amer. Math. Soc. 351(1999), no. 6, 2439–2458. http://dx.doi.org/10.1090/S0002-9947-99-02225-4 Google Scholar
[23] Xu, Y., Asymptotics of the Christoffel functions on a simplex in Rd. J. Approx. Theory 99(1999), no. 1, 122–133. http://dx.doi.org/10.1006/jath.1998.3312 Google Scholar