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Scaling Limits of Polynomials and Entire Functions of Exponential Type

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Approximation Theory XV: San Antonio 2016 (AT 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 201))

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Abstract

The connection between polynomials and entire functions of exponential type is an old one, in some ways harking back to the simple limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\left( 1+\frac{z}{n}\right) ^{n}=e^{z}. \end{aligned}$$

On the left-hand side, we have \(P_{n}\left( \frac{z}{n}\right) \), where \(P_{n}\) is a polynomial of degree n, and on the right, an entire function of exponential type. We discuss the role of this type of scaling limit in a number of topics: Bernstein’s constant for approximation of \(\left| x\right| \); universality limits for random matrices; asymptotics of \(L_{p}\) Christoffel functions and Nikolskii inequalities; and Marcinkiewicz–Zygmund inequalities. Along the way, we mention a number of unsolved problems.

Research supported by NSF grant DMS1362208.

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References

  1. G. Akemann, J. Baik, P. Di Francesco (eds.), The Oxford Handbook of Random Matrix Theory (Oxford University Press, Oxford, 2011)

    MATH  Google Scholar 

  2. G. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics, 118 (Cambridge University Press, Cambridge, 2010)

    Google Scholar 

  3. J. Baik, T. Kriecherbauer, K. McLaughlin, P. Miller, Uniform Asymptotics for Polynomials Orthogonal with Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles. Annals of Math. Studies, vol. 164 (Princeton University Press, Princeton, 2007)

    Google Scholar 

  4. J. Baik, L. Li, T. Kriecherbauer, K. McLaughlin, C. Tomei, Proceedings of the Conference on Integrable Systems, Random Matrices and Applications. Contemporary Mathematics, vol. 458 (American Mathematical Society, Providence, 2008)

    Google Scholar 

  5. P. Bleher, A. Its, Random Matrix Models and their Applications (Cambridge University Press, Cambridge, 2001)

    MATH  Google Scholar 

  6. S.N. Bernstein, Sur la meilleure approximation de \(\left|x\right|\) par des polynômes de degré donnés. Acta Math. 37, 1–57 (1913)

    Article  MATH  Google Scholar 

  7. S.N. Bernstein, Sur la meilleure approximation de \(\left|x\right|^{p}\) par des polynô mes de degrés très élevés. Bull. Acad. Sc. USSR, Ser. Math., 2 181–190 (1938)

    Google Scholar 

  8. R.P. Boas, Entire Functions (Academic Press, New York, 1954)

    MATH  Google Scholar 

  9. A.J. Carpenter, R.S. Varga, Some Numerical Results on Best Uniform Polynomial Approximation of \(x^{\alpha }\) on \(\left[0,1\right] \). Springer Lecture Notes in Mathematics, vol. 1550 (1993), pp. 192–222

    Google Scholar 

  10. P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Institute Lecture Notes, vol. 3 (New York University Press, New York, 1999)

    Google Scholar 

  11. P. Deift, D. Gioev, Random Matrix Theory: Invariant Ensembles and Universality, Courant Institute Lecture Notes, vol. 18 (New York University Press, New York, 2009)

    MATH  Google Scholar 

  12. P. Deift, T. Kriecherbauer, K. McLaughlin, S. Venakides, X. Zhou, Uniform Asymptotics for Polynomials Orthogonal with respect to Varying Exponential Weights and Applications to Universality Questions in Random Matrix Theory. Communications on Pure and Applied Mathematics, vol. L11 (1999), pp. 1335–1425

    Google Scholar 

  13. R. DeVore G. Lorentz, Constructive Approximation, vol. 1 (Springer, Berlin, 1993)

    Google Scholar 

  14. F. Filbir, H.N. Mhaskar, Marcinkiewicz-Zygmund measures on manifolds. J. Complex. 27, 568–596 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Erdős, Universality of Wigner random matrices: a survey of recent results. Russian Math. Surv. 66, 507–626 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Eremenko, P. Yuditskii, Polynomials of the best uniform approximation to sgn \(x\) on two intervals. J. d’Analyse Mathématique 114, 285–315 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. S.R. Finch, Mathematical Constants (Cambridge University Press, Cambridge, 2003)

    Book  MATH  Google Scholar 

  18. E. Findley, Universality for regular measures satisfying Szegő’s condition. J. Approx. Theory 155, 136–154 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. A.F. Moreno, A. Martinez-Finkelshtein, V. Sousa, Asymptotics of orthogonal polynomials for a weight with a jump on \(\left[-1,1\right] \). Constr. Approx. 33, 219–263 (2011)

    Google Scholar 

  20. P. Forrester, Log-Gases and Random Matrices (Princeton University Press, Princeton, 2010)

    MATH  Google Scholar 

  21. M. Ganzburg, Limit Theorems and Best Constants of Approximation Theory (in), Handbook on Analytic Computational Methods in Applied Mathematics, ed. by G. Anastassiou (CRC Press, Boca Raton, FL 2000)

    Google Scholar 

  22. M. Ganzburg, The Bernstein constant and polynomial interpolation at the Chebyshev nodes. J. Approx. Theory 119, 193–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Ganzburg, Limit Theorems of Polynomial Approximation. Memoirs Am. Math. Soc. 192(897) (2008)

    Google Scholar 

  24. M. Ganzburg, Polynomial interpolation and asymptotic representations for zeta functions. Dissertationes Math. (Rozprawy Mat.) 496, 117 (2013)

    Google Scholar 

  25. M. Ganzburg, D.S. Lubinsky, Best approximating entire functions to \(\left|x\right|^{\alpha }\) in \(L_{2}\). Contemp. Math. 455, 93–107 (2008)

    Article  MATH  Google Scholar 

  26. J. Korevaar, An inequality for entire functions of exponential type. Nieuw. Arch. Wiskunde 23, 55–62 (1949)

    MathSciNet  MATH  Google Scholar 

  27. A. Kuijlaars, Universality, Chapter 6 in “The Oxford Handbook on Random Matrix Theory, ed. by G. Akemann, J. Baik, P. Di Francesco (Oxford University Press, Oxford, 2011), pp. 103–134

    Google Scholar 

  28. A. Kuijlaars, M. Vanlessen, Universality for eigenvalue correlations at the origin of the spectrum. Commun. Math. Phys. 243, 163–191 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Kuijlaars, K.T.-R. McLaughlin, W. Van Assche, M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [–1,1]. Adv. Math. 188, 337–398 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. B. Ja Levin, Lectures on Entire Functions, Translations of Mathematical Monographs (American Mathematical Society, Providence, 1996)

    Google Scholar 

  31. E. Levin, D.S. Lubinsky, Universality Limits at the Soft Edge of the Spectrum via Classical Complex Analysis. International Maths. Research Notices (2010), https://doi.org/10.1093/imrn/rnq185

  32. E. Levin, D.S. Lubinsky, Asymptotic behavior of Nikolskii constants for polynomials on the unit circle. Comput. Methods Funct. Theory 15, 459–468 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. E. Levin, D.S. Lubinsky, \(L_{p}\) Christoffel Functions, \(L_{p}\) Universality, and Paley-Wiener Spaces. J. d’Analyse Mathématique, 125 243–283 (2015)

    Google Scholar 

  34. D.S. Lubinsky, Marcinkiewicz-Zygmund Inequalities: Methods and Results, (in) Recent Progress in Inequalities, ed. by G.V. Milovanovic et al. (Kluwer Academic Publishers, Dordrecht, 1998), pp. 213–240

    Google Scholar 

  35. D.S. Lubinsky, On the Bernstein constants of polynomial approximation. Constr. Approx. 25, 303–366 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. D. S. Lubinsky, Universality Limits at the Hard Edge of the Spectrum for Measures with Compact Support, International Maths. Research Notices, International Maths. Research Notices (2008), Art. ID rnn 099, 39 pp

    Google Scholar 

  38. D.S. Lubinsky, A new approach to universality limits involving orthogonal polynomials. Ann. Math. 170, 915–939 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. D.S. Lubinsky, Bulk universality holds in measure for compactly supported measures. J d’Analyse Mathématique 116, 219–253 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. D.S. Lubinsky, On sharp constants in Marcinkiewicz-Zygmund and Plancherel-Polya inequalities. Proc. Am. Math. Soc. 142, 3575–3584 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Maté, P. Nevai, V. Totik, Szegő’s extremum problem on the unit circle. Ann. Math. 134, 433–453 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  42. K.T.-R. McLaughlin, P.D. Miller, The \(\bar{\partial }\) -Steepest Descent Method and the Asymptotic Behavior of Polynomials Orthogonal on the Unit Circle with Fixed and Exponentially Varying Nonanalytic Weights, International Maths. Research Notices (2006), Article ID 48673, pp. 1–78

    Google Scholar 

  43. F. Nazarov, F. Peherstorfer, A. Volberg, P. Yuditskii, Asymptotics of the best polynomial approximation of \( \vert x\vert ^ p\) and of the best Laurent polynomial approximation of sgn(x) on two symmetric intervals. Constr. Approx. 29, 23–39 (2009)

    Google Scholar 

  44. P. Nevai, Orthogonal Polynomials. Memoirs of the AMS, vol. 213 (1979)

    Google Scholar 

  45. P. Nevai, Geza Freud, orthogonal polynomials and Christoffel functions: a case study. J. Approx. Theory 48, 3–167 (1986)

    Google Scholar 

  46. S.M. Nikolskii, On the best mean approximation by polynomials of the functions \(\left|x-c\right|^{s}\). Izvestia Akad. Nauk SSSR 11, 139–180 (1947). (in Russian)

    Google Scholar 

  47. M. Plancherel, G. Polya, Fonctions entierers et integrales de Fourier multiples. Comment. Math. Helvet. 10, 110–163 (1937)

    Google Scholar 

  48. R.A. Raitsin, S. N. Bernstein limit theorem for the best approximation in the mean and some of its applications. Izv. Vysch. Uchebn. Zaved. Mat. 12 81–86(1968)

    Google Scholar 

  49. R.A. Raitsin, On the best approximation in the mean by polynomials and entire functions of finite degree of functions having an algebraic singularity. Izv. Vysch. Uchebn. Zaved. Mat. 13, 59–61 (1969)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  51. B. Simon, Two extensions of Lubinsky’s universality theorem. Journal d’Analyse de Mathématique 105, 345–362 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  52. B. Simon, Weak convergence of CD kernels and applications. Duke Math. J. 146, 305–330 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. B. Simon, Szegö’s theorem and its Descendants: Spectral Theory for \(L_{2}\) Perturbations of Orthogonal Polynomials (Princeton University Press, Princeton, 2011)

    Google Scholar 

  54. H. B. Stahl, Best Uniform Rational Approximations of \(\left|x\right|\) on \(\left[ -1,1 \right] \), Mat. Sb. 183(1992), 85–118. (Translation in Russian Acad. Sci. Sb. Math., 76(1993), 461–487)

    Google Scholar 

  55. H.B. Stahl, Best uniform rational approximation of \(x^{\alpha }\) on \(\left[0,1\right] \). Acta Math. 190, 241–306 (2003)

    Article  MathSciNet  Google Scholar 

  56. H. Stahl, V. Totik, General Orthogonal Polynomials (Cambridge University Press, Cambridge, 1992)

    Book  MATH  Google Scholar 

  57. T. Tao, Topics in Random Matrix Theory, Graduate Studies in Mathematics, vol. 132 (American Mathematical Society, Providence, 2012)

    Book  Google Scholar 

  58. A.F. Timan, Theory of Approximation of Functions of a Real Variable (translated by J Berry) (Dover, New York, 1994)

    Google Scholar 

  59. V. Totik, Asymptotics for Christoffel functions for general measures on the real line. J. d’Analyse Mathématique 81, 283–303 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  61. V. Totik, Universality under Szegő’s condition. Canad. Math. Bull. 59, 211–224 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  62. V. Totik, Metric properties of harmonic measure. Memoirs Am. Math. Soc. 184 867 (2006)

    Google Scholar 

  63. R.K. Vasiliev, Chebyshev Polynomials and Approximation, Theory on Compact Subsets of the Real Axis (Saratov University Publishing House, 1998)

    Google Scholar 

  64. R.S. Varga, Scientific Computation on Mathematical Problems and Conjectures. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Vermont (1990)

    Google Scholar 

  65. R.S. Varga, A.J. Carpenter, On the Bernstein conjecture in approximation theory. Constr. Approx. 1, 333–348 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  66. A. Zygmund, Trigonometric Series, vols. I, II (Cambridge University Press, Cambridge, II, 1990)

    Google Scholar 

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Acknowledgements

The author would like to thank the organizers of Approximation Theory XV for the opportunity to take part in a very successful and stimulating conference. In addition, the author thanks the referees for their thorough reports.

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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 Applied Mathematics and Statistics, Colorado School of Mines, Golden, Colorado, USA

    Gregory E. Fasshauer

  2. Department of Mathematics, Vanderbilt University , Nashville, Tennessee, USA

    Larry L. Schumaker

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Lubinsky, D.S. (2017). Scaling Limits of Polynomials and Entire Functions of Exponential Type. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XV: San Antonio 2016. AT 2016. Springer Proceedings in Mathematics & Statistics, vol 201. Springer, Cham. https://doi.org/10.1007/978-3-319-59912-0_11

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