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Two extensions of Lubinsky’s universality theorem

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Abstract

We extend some remarkable recent results of Lubinsky and Levin-Lubinsky from [−1, 1] to allow discrete eigenvalues outside σ ess and to allow σ ess first to be a finite union of closed intervals and then a fairly general compact set in ℝ (one which is regular for the Dirichlet problem).

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References

  1. A. B. Bogatyrëv, On the efficient computation of Chebyshev polynomials for several intervals, Sb. Math. 190 (1999), 1571–1605; Russian original in Mat. Sb. 190 (1999), no. 11, 15–50.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995.

    MATH  Google Scholar 

  3. M. J. Cantero, L. Moral and L. Velázquez, Measures and para-orthogonal polynomials on the unit circle, East J. Approx. 8 (2002), 447–464.

    MATH  MathSciNet  Google Scholar 

  4. M. J. Cantero, L. Moral and L. Velázquez, Measures on the unit circle and unitary truncations of unitary operators, J. Approx. Theory 139 (2006), 430–468.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Christiansen, B. Simon and M. Zinchenko, Finite gap Jacobi matrices, I. The isospectral torus, in preparation.

  6. J. L. Doob, Measure Theory, Springer-Verlag, New York, 1994.

    MATH  Google Scholar 

  7. G. Faber, Über Tschebyscheffsche Polynome, J. Reine Angew. Math. 150 (1919), 79–106.

    Google Scholar 

  8. M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 17 (1923), 228–249.

    Article  MathSciNet  Google Scholar 

  9. E. Findley, Universality and zero spacing under local Szegő condition, in preparation.

  10. G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford-New York, 1971.

    Google Scholar 

  11. L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96 (2002), 169–186.

    Article  MATH  MathSciNet  Google Scholar 

  12. L. L. Helms, Introduction to Potential Theory, Wiley-Interscience, New York, 1969.

    MATH  Google Scholar 

  13. K. Jacobs, Measure and Integral, Academic Press, New York-London, 1978.

    MATH  Google Scholar 

  14. W. B. Jones, O. Njåstad and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), 113–152.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. B. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Int. Math. Res. Not. 30 (2002), 1575–1600.

    Article  MathSciNet  Google Scholar 

  16. N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin-New York, 1972.

    MATH  Google Scholar 

  17. Y. Last and B. Simon, Fine structure of the zeros of orthogonal polynomials, IV. A priori bounds and clock behavior, Comm. Pure Appl. Math. 61 (2008), 486–538.

    Article  MATH  MathSciNet  Google Scholar 

  18. E. Levin and D. S. Lubinsky, Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials, J. Approx. Theory 150 (2008), 69–95.

    Article  MATH  MathSciNet  Google Scholar 

  19. D. S. Lubinsky, A new approach to universality limits involving orthogonal polynomials, Ann. of Math., to appear.

  20. D. S. Lubinsky, A new approach to universality limits at the edge of the spectrum, Contemp. Math., to appear.

  21. W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces. Vol. I, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1971.

    MATH  Google Scholar 

  22. A. Máté and P. Nevai, Bernstein’s inequality in L p for 0 < p < 1 and (C, 1) bounds for orthogonal polynomials, Ann. of Math. (2) 111 (1980), 145–154.

    Article  MathSciNet  Google Scholar 

  23. A. Máté, P. Nevai and V. Totik, Szegő’s extremum problem on the unit circle, Ann. of Math. (2) 134 (1991), 433–453.

    Article  MathSciNet  Google Scholar 

  24. P. Nevai, Orthogonal Polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213.

  25. F. Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), 180–195.

    Article  MATH  MathSciNet  Google Scholar 

  26. F. Peherstorfer and P. Yuditskii, Asymptotic behavior of polynomials orthonormal on a homogeneous set, J. Anal. Math. 89 (2003), 113–154.

    MATH  MathSciNet  Google Scholar 

  27. E. B. Saff and V. Totik, Logarithmic Potentials With External Fields, Springer-Verlag, Berlin, 1997.

    MATH  Google Scholar 

  28. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, American Mathematical Society, Providence, RI, 2005.

    MATH  Google Scholar 

  29. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, American Mathematical Society, Providence, RI, 2005.

    MATH  Google Scholar 

  30. B. Simon, Fine structure of the zeros of orthogonal polynomials, I. A tale of two pictures, Electron. Trans. Numer. Anal. 25 (2006), 328–268.

    MATH  MathSciNet  Google Scholar 

  31. B. Simon, Fine structure of the zeros of orthogonal polynomials, III. Periodic recursion coefficients, Comm. Pure Appl. Math. 59 (2005) 1042–1062.

    Article  Google Scholar 

  32. B. Simon, Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle, J. Math. Anal. Appl. 329 (2007), 376–382.

    Article  MATH  MathSciNet  Google Scholar 

  33. B. Simon, Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging 1 (2007), 713–772.

    MATH  MathSciNet  Google Scholar 

  34. B. Simon, Szegő’s Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials, in preparation; to be published by Princeton University Press.

  35. B. Simon, Weak convergence of CD kernels and applications, preprint.

  36. M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. 7 (1997), 387–435.

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  38. G. Szegő, Bemerkungen zu einer Arbeit von Herrn M. Fekete: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 21 (1924), 203–208.

    Article  MathSciNet  Google Scholar 

  39. G. Szegő, Orthogonal Polynomials, Amer. Math. Soc., American Mathematical Society, Providence, R.I., 1939; 3rd edition, 1967.

    Google Scholar 

  40. V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math. 81 (2000), 283–303.

    Article  MATH  MathSciNet  Google Scholar 

  41. V. Totik, Polynomial inverse images and polynomial inequalities, Acta Math. 187 (2001), 139–160.

    Article  MATH  MathSciNet  Google Scholar 

  42. V. Totik, Universality and fine zero spacing on general sets, in preparation.

  43. M. Tsuji, Potential Theory in Modern Function Theory, reprinting of the 1959 original, Chelsea, New York, 1975.

    MATH  Google Scholar 

  44. J. L. Ullman, On the regular behaviour of orthogonal polynomials, Proc. London Math. Soc. (3) 24 (1972), 119–148.

    Article  MATH  MathSciNet  Google Scholar 

  45. W. Van Assche, Invariant zero behaviour for orthogonal polynomials on compact sets of the real line, Bull. Soc. Math. Belg. Ser. B 38 (1986), 1–13.

    MATH  MathSciNet  Google Scholar 

  46. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, fourth ed., American Mathematical Society, Providence, RI, 1965.

    MATH  Google Scholar 

  47. H. Widom, Polynomials associated with measures in the complex plane, J. Math. Mech. 16 (1967), 997–1013.

    MATH  MathSciNet  Google Scholar 

  48. H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Advances in Math. 3 (1969), 127–232.

    Article  MATH  MathSciNet  Google Scholar 

  49. M.-W. L. Wong, First and second kind paraorthogonal polynomials and their zeros, J. Approx. Theory 146 (2007), 282–293.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics 253-37, California Institute of Technology, Pasadena, CA, 91125, USA

    Barry Simon

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  1. Barry Simon
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Correspondence to Barry Simon.

Additional information

Supported in part by NSF grant DMS-0140592 and U.S.-Israel Binational Science Foundation (BSF) Grant No. 2002068.

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Simon, B. Two extensions of Lubinsky’s universality theorem. J Anal Math 105, 345–362 (2008). https://doi.org/10.1007/s11854-008-0039-z

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  • DOI: https://doi.org/10.1007/s11854-008-0039-z

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