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On Christoffel Functions and Related Quantities for Compactly Supported Measures

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Approximation Theory XIII: San Antonio 2010

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 13))

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Abstract

Let μ be a compactly supported positive measure on the real line, with associated orthogonal polynomials p n . Without any global restrictions such as regularity, we discuss convergence in measure for

We also establish convergence a.e.for sufficiently sparse subsequences of Christoffel function ratios.

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References

  1. A. Avila, Y. Last, and B. Simon, Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectrum Analysis & PDE, 3(2010), 81–108.

    Google Scholar 

  2. J. Baik, T. Kriecherbauer, K. T-R. McLaughlin, P.D. Miller, Uniform Asymptotics for Polynomials Orthogonal with respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles, Princeton Annals of Mathematics Studies, 2006.

    Google Scholar 

  3. P. Bleher and A. Its, Random matrix models and their applications Cambridge University Press, Cambridge, 2001.

    MATH  Google Scholar 

  4. J. Breuer, Y. Last, B. Simon, The Nevai Condition, Constr. Approx., 32 (2010), 221–254.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach Courant Institute Lecture Notes, Vol. 3, New York University Pres, New York, 1999.

    Google Scholar 

  6. P. Deift, D. Gioev, Random Matrix Theory: Invariant Ensembles and UniversalityCourant Institute Lecture Notes, Vol. 18, New York University Pres, New York, 2009.

    Google Scholar 

  7. P. Deift, T. Kriecherbauer, K. T-R. McLaughlin, S. Venakides and X. Zhou, Uniform Asymptotics for Polynomials Orthogonal with respect to Varying Exponential Weights and Applications to Universality Questions in Random Matrix Theory, Communications in Pure and Applied Maths., 52(1999), 1335–1425.

    Article  MATH  Google Scholar 

  8. E. Findley, Universality for Regular Measures satisfying Szegő’s Condition J. Approx. Theory, 155 (2008), 136–154.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. J. Forrester, Log-gases and Random matrices, online book, http://www.ms.unimelb.edu.au/symbol126matpjf/matpjf.html

  10. Eli Levin and D.S. Lubinsky, Universality Limits in the Bulk for Varying MeasuresAdvances in Mathematics, 219(2008), 743–779.

    Google Scholar 

  11. G. Lopez, Relative Asymptotics for Polynomials Orthogonal on the Real Axis, Math. USSR. Sbornik, 65(1990), 505–529.

    Article  MATH  Google Scholar 

  12. D.S. Lubinsky, Universality limits in the bulk for arbitrary measures on compact sets, J. d’Analyse Mathematique, 106 (2008), 373–394.

    Article  MathSciNet  MATH  Google Scholar 

  13. D.S. Lubinsky, A New Approach to Universality Limits involving Orthogonal Polynomials, Annals of Mathematics, 170(2009), 915–939.

    Article  MathSciNet  MATH  Google Scholar 

  14. D.S. Lubinsky, Universality Limits for Random Matrices and de Branges Spaces of Entire Functions, Journal of Functional Analysis, 256(2009), 3688–3729.

    Article  MathSciNet  MATH  Google Scholar 

  15. D.S. Lubinsky, A Maximal Function Approach to Christoffel Functions and Nevai’s Operators, to appear in Constr. Approx.

    Google Scholar 

  16. D.S. Lubinsky, Bulk Universality Holds in Measure for Compactly Supported Measures, to appear in J. d’Analyse Mathematique.

    Google Scholar 

  17. A. Maté, P. Nevai, V. Totik, Extensions of Szegő’s Theory of Orthogonal Polynomials, II, Constr. Approx., 3(1987), 51–72.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Mate, P. Nevai, V. Totik, Szegő’s Extremum Problem on the Unit Circle, Annals of Mathematics, 134(1991), 433–453.

    Article  MathSciNet  MATH  Google Scholar 

  19. M.L. Mehta, Random Matrices, 2nd edn., Academic Press, Boston, 1991.

    MATH  Google Scholar 

  20. P. Nevai, Orthogonal Polynomials, Memoirs of the AMS no. 213 (1979).

    Google Scholar 

  21. P. Nevai, V.Totik, and J. Zhang, Orthogonal Polynomials: their growth relative to their Sums, J. Approx. Theory, 67(1991), 215–234.

    Article  MathSciNet  MATH  Google Scholar 

  22. T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  23. E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, New York, 1997.

    MATH  Google Scholar 

  24. B. Simon, Orthogonal Polynomials on the Unit Circle, Parts 1 and 2, American Mathematical Society, Providence, 2005.

    Google Scholar 

  25. B. Simon, Two Extensions of Lubinsky’s Universality Theorem, J. d’Analyse Mathematique, 105 (2008), 345–362.

    Article  MATH  Google Scholar 

  26. B. Simon, The Christoffel-Darboux Kernel, (in) Perspectives in PDE, Harmonic Analysis and Applications, a volume in honor of V.G. Maz’ya’s 70th birthday, Proceedings of Symposia in Pure Mathematics, 79 (2008), 295–335.

    Google Scholar 

  27. B. Simon, Weak Convergence of CD Kernels and Applications, Duke Math. J., 146 (2009) 305–330.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Soshnikov, Universality at the Edge of the Spectrum in Wigner Random Matrices Comm. Math. Phys., 207(1999), 697–733.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992.

    Book  MATH  Google Scholar 

  30. T. Tao, V. Vu, From the Littlewood-Offord Problem to the Circular Law: Universality of the Spectral Distribution of Random Matrices, Bull. Amer. Math. Soc., 46 (2009), 377–396.

    Article  MathSciNet  MATH  Google Scholar 

  31. V. Totik, Asymptotics for Christoffel Functions for General Measures on the Real Line, J. d’ Analyse de Mathematique, 81(2000), 283–303.

    Article  MathSciNet  MATH  Google Scholar 

  32. V. Totik, Universality and fine zero spacing on general sets, Arkiv för Matematik, 47(2009), 361–391.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Research supported by NSF grant DMS0700427 and US-Israel BSF grant 2004353.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    D. S. Lubinsky

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  1. D. S. Lubinsky
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Correspondence to D. S. Lubinsky .

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Editors and Affiliations

  1. , Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, 37240, Tennessee, USA

    Marian Neamtu

  2. , Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, 37240, Tennessee, USA

    Larry Schumaker

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Lubinsky, D.S. (2012). On Christoffel Functions and Related Quantities for Compactly Supported Measures. In: Neamtu, M., Schumaker, L. (eds) Approximation Theory XIII: San Antonio 2010. Springer Proceedings in Mathematics, vol 13. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0772-0_12

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