Lubinsky, D. S. Scaling limits of polynomials and entire functions of exponential type. (English) Zbl 1393.30028 Fasshauer, Gregory E. (ed.) et al., Approximation theory XV: San Antonio 2016. Selected papers based on the presentations at the international conference, San Antonio, TX, USA, May 22–25, 2016. Cham: Springer (ISBN 978-3-319-59911-3/hbk; 978-3-319-59912-0/ebook). Springer Proceedings in Mathematics & Statistics 201, 219-238 (2017). In the paper, the role of the scaling limit of the type \[ \lim\limits_{n\to+\infty}\left(1+\frac zn\right)^n=e^z \] in a number of topics is discussed. Bernstein’s constant \(\Lambda_1\) for approximation on \([-1,1]\) of \(|x|\) by polynomials of degree less than or equal \(n\) and the limit \(\Lambda_\alpha\) as the error in approximation on the whole real axis of \(|x|^\alpha\) by entire functions of exponential type are discussed. Universality limits for random matrices, asymptotics of \(L_p\) Christoffel functions and Nikolskii inequalities, and Marcinkiewicz-Zygmund inequalities are also discussed. Along the way, a number of unsolved problems are mentioned.For the entire collection see [Zbl 1378.65012]. Reviewer: Konstantin Malyutin (Sumy) MSC: 30E10 Approximation in the complex plane 30D15 Special classes of entire functions of one complex variable and growth estimates Keywords:approximation by entire functions; Christoffel functions; Bernstein constant; Nikolskii inequalities; Marcinkiewicz-Zygmund inequalities PDFBibTeX XMLCite \textit{D. S. Lubinsky}, Springer Proc. Math. Stat. 201, 219--238 (2017; Zbl 1393.30028) Full Text: DOI References: [1] G. Akemann, J. Baik, P. Di Francesco (eds.), The Oxford Handbook of Random Matrix Theory (Oxford University Press, Oxford, 2011) · Zbl 1225.15004 [2] G. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics, 118 (Cambridge University Press, Cambridge, 2010) · Zbl 1184.15023 [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) · Zbl 1036.42023 [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) [5] P. Bleher, A. Its, Random Matrix Models and their Applications (Cambridge University Press, Cambridge, 2001) [6] S.N. Bernstein, Sur la meilleure approximation de \(\left|x\right|\) par des polynômes de degré donnés. Acta Math. 37, 1-57 (1913) · JFM 44.0475.01 [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) · Zbl 0022.21601 [8] R.P. Boas, Entire Functions (Academic Press, New York, 1954) · Zbl 0058.30201 [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 · Zbl 0784.65009 [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) · Zbl 0997.47033 [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) · Zbl 1171.15023 [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 · Zbl 0944.42013 [13] R. DeVore G. Lorentz, Constructive Approximation, vol. 1 (Springer, Berlin, 1993) · Zbl 0797.41016 [14] F. Filbir, H.N. Mhaskar, Marcinkiewicz-Zygmund measures on manifolds. J. Complex. 27, 568-596 (2011) · Zbl 1235.58007 [15] L. Erdős, Universality of Wigner random matrices: a survey of recent results. Russian Math. Surv. 66, 507-626 (2011) · Zbl 1230.82032 [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) · Zbl 1241.41005 [17] S.R. Finch, Mathematical Constants (Cambridge University Press, Cambridge, 2003) · Zbl 1054.00001 [18] E. Findley, Universality for regular measures satisfying Szegő’s condition. J. Approx. Theory 155, 136-154 (2008) · Zbl 1171.42015 [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) · Zbl 1213.42090 [20] P. Forrester, Log-Gases and Random Matrices (Princeton University Press, Princeton, 2010) · Zbl 1217.82003 [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) · Zbl 0968.41012 [22] M. Ganzburg, The Bernstein constant and polynomial interpolation at the Chebyshev nodes. J. Approx. Theory 119, 193-213 (2002) · Zbl 1035.41015 [23] M. Ganzburg, Limit Theorems of Polynomial Approximation. Memoirs Am. Math. Soc. 192(897) (2008) [24] M. Ganzburg, Polynomial interpolation and asymptotic representations for zeta functions. Dissertationes Math. (Rozprawy Mat.) 496, 117 (2013) [25] M. Ganzburg, D.S. Lubinsky, Best approximating entire functions to \(\left|x\right|^{\alpha }\) in \(L_2\). Contemp. Math. 455, 93-107 (2008) · Zbl 1153.41310 [26] J. Korevaar, An inequality for entire functions of exponential type. Nieuw. Arch. Wiskunde 23, 55-62 (1949) · Zbl 0031.21202 [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 [28] A. Kuijlaars, M. Vanlessen, Universality for eigenvalue correlations at the origin of the spectrum. Commun. Math. Phys. 243, 163-191 (2003) · Zbl 1041.82007 [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) · Zbl 1082.42017 [30] B. Ja Levin, Lectures on Entire Functions, Translations of Mathematical Monographs (American Mathematical Society, Providence, 1996) [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 · Zbl 1235.42023 [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) · Zbl 1327.42030 [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) · Zbl 1320.30010 [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 · Zbl 0908.41021 [35] D.S. Lubinsky, On the Bernstein constants of polynomial approximation. Constr. Approx. 25, 303-366 (2007) · Zbl 1118.41003 [36] D.S. Lubinsky, Universality limits in the bulk for arbitrary measures on compact sets. J. d’Analyse Mathématique 106, 373-394 (2008) · Zbl 1156.42005 [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 · Zbl 1172.28006 [38] D.S. Lubinsky, A new approach to universality limits involving orthogonal polynomials. Ann. Math. 170, 915-939 (2009) · Zbl 1176.42022 [39] D.S. Lubinsky, Bulk universality holds in measure for compactly supported measures. J d’Analyse Mathématique 116, 219-253 (2012) · Zbl 1279.60015 [40] D.S. Lubinsky, On sharp constants in Marcinkiewicz-Zygmund and Plancherel-Polya inequalities. Proc. Am. Math. Soc. 142, 3575-3584 (2014) · Zbl 1303.30023 [41] A. Maté, P. Nevai, V. Totik, Szegő’s extremum problem on the unit circle. Ann. Math. 134, 433-453 (1991) · Zbl 0752.42015 [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 · Zbl 1133.45001 [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) · Zbl 1176.41032 [44] P. Nevai, Orthogonal Polynomials. Memoirs of the AMS, vol. 213 (1979) [45] P. Nevai, Geza Freud, orthogonal polynomials and Christoffel functions: a case study. J. Approx. Theory 48, 3-167 (1986) · Zbl 0606.42020 [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) · Zbl 0910.42011 [47] M. Plancherel, G. Polya, Fonctions entierers et integrales de Fourier multiples. Comment. Math. Helvet. 10, 110-163 (1937) · Zbl 0018.15204 [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) [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) [50] B. Simon, Orthogonal Polynomials on the Unit Circle, Parts 1 and 2 (American Mathematical Society, Providence, 2005) · Zbl 1082.42020 [51] B. Simon, Two extensions of Lubinsky’s universality theorem. Journal d’Analyse de Mathématique 105, 345-362 (2008) · Zbl 1168.42304 [52] B. Simon, Weak convergence of CD kernels and applications. Duke Math. J. 146, 305-330 (2009) · Zbl 1158.33003 [53] B. Simon, Szegö’s theorem and its Descendants: Spectral Theory for \(L_2\) Perturbations of Orthogonal Polynomials (Princeton University Press, Princeton, 2011) · Zbl 1230.33001 [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) · Zbl 0785.41019 [55] H.B. Stahl, Best uniform rational approximation of \(x^{\alpha }\) on \(\left[0,1\right] \). Acta Math. 190, 241-306 (2003) · Zbl 1077.41012 [56] H. Stahl, V. Totik, General Orthogonal Polynomials (Cambridge University Press, Cambridge, 1992) · Zbl 0791.33009 [57] T. Tao, Topics in Random Matrix Theory, Graduate Studies in Mathematics, vol. 132 (American Mathematical Society, Providence, 2012) · Zbl 1256.15020 [58] A.F. Timan, Theory of Approximation of Functions of a Real Variable (translated by J Berry) (Dover, New York, 1994) [59] V. Totik, Asymptotics for Christoffel functions for general measures on the real line. J. d’Analyse Mathématique 81, 283-303 (2000) · Zbl 0966.42017 [60] V. Totik, Universality and fine zero spacing on general sets. Arkiv för Matematik 47, 361-391 (2009) · Zbl 1180.42017 [61] V. Totik, Universality under Szegő’s condition. Canad. Math. Bull. 59, 211-224 (2016) · Zbl 1342.42027 [62] V. Totik, Metric properties of harmonic measure. Memoirs Am. Math. Soc. 184 867 (2006) · Zbl 1107.31001 [63] R.K. Vasiliev, Chebyshev Polynomials and Approximation, Theory on Compact Subsets of the Real Axis (Saratov University Publishing House, 1998) · Zbl 0955.41020 [64] R.S. Varga, Scientific Computation on Mathematical Problems and Conjectures. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Vermont (1990) [65] R.S. Varga, A.J. Carpenter, On the Bernstein conjecture in approximation theory. Constr. Approx. 1, 333-348 (1985) · Zbl 0648.41013 [66] A. Zygmund, Trigonometric Series, vols. 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