##
**A new approach to universality limits involving orthogonal polynomials.**
*(English)*
Zbl 1176.42022

In the theory of orthogonal polynomials the so called “universality law” has been studied from different viewpoints and with measures satisfying less and less stringent conditions.

If \(\{p_k(x)\}\) are the orthonormal polynomials with respect to a finite positive Borel measure \(\mu\) with compact support on the real line, the simplest case of this law is given in terms of the reproducing kernel \[ K_n(x,y)=\sum_{k=0}^n\,p_k(x)p_k(y),\;\tilde{K}_n(x,y)=w(x)^{1/2}w(y)^{1/2}K_n(x,y), \] and reads \[ \lim_{n\rightarrow\infty}\,{\tilde{K}_n\left(\xi+{a\over\tilde{K}_n(\xi,\xi)}, \xi+{b\over\tilde{K}_n(\xi,\xi)}\right)\over\tilde{K}_n(\xi,\xi)}={\sin{\pi (a-b)}\over\pi (a-b)}. \eqno{(\ast)} \] The measure \(\mu\) has infinitely many points in its support and \[ w={\text{d}\mu\over\text{d}x} \] is the Radon-Nikodym derivative of \(\mu\) and \((\ast)\) holds uniformly in \(\xi\) on a compact subinterval of supp\((\mu)\) and in \(a,b\) in compact subsets of the real line (for \(a=b\): the right hand side is \(1\)).

In this paper, the author studies measures on the real interval \((-1,1)\) and the main results are:

Theorem 1.1 Let \(\mu\) be a finite positive measure on \((-1,1)\), that is regular. Let \(J\subset (-1,1)\) be compact and such that \(\mu\) is absolutely continuous in an open set containing \(J\). Assume, moreover, that \(w\) is positive and continuous at each point of \(J\). Then, uniformly for \(x\in J\) and \(a,\,b\) in compact subsets of the real line, we have:

\[ \lim_{n\rightarrow\infty}\,{\tilde{K}_n\left(x+{a\over\tilde{K}_n(x,x)}, x+{b\over\tilde{K}_n(x,x)}\right)\over\tilde{K}_n(x,x)}={\sin{\pi (a-b)}\over\pi (a-b)}. \]

[and two interesting corollaries that are not cited explicitly here]

Theorem 1.4 Let \(\mu\) be a finite positive measure on \((-1,1)\), that is regular; let \(p>1\). Let \(I\subset (-1,1)\) be a closed interval in which \(\mu\) is absolutely continuous and \(w\) is bounded above and below by positive constants. Then we have:

(a) If \(I'\) is a closed subinterval of \(I^{0}\) \[ \lim_{n\rightarrow\infty}\,\int_{I'}\,\left|{K_n\left(x+{a\over\tilde{K}_n(x,x)}, x+{b\over\tilde{K}_n(x,x)}\right)\over K_n(x,x)}-{\sin{\pi (a-b)}\over\pi (a-b)}\right|^pdx=0, \] uniformly for \(a,\,b\) in compact subsets of the real line.

(b) If, in addition, \(w\) is Riemann integrable in \(I\), we may replace \[ {K_n\left(x+{a\over\tilde{K}_n(x,x)},x+{b\over\tilde{K}_n(x,x)}\right)\over K_n(x,x)}\text{ by } {\tilde{K}_n\left(x+{a\over\tilde{K}_n(x,x)},x+{b\over\tilde{K}_n(x,x)}\right)\over\tilde{K}_n(x,x)} \] in the limit in (a).

The proofs are given in the sections 2 (Christoffel functions), 3 (Localization), 4 (Smoothing), 5 (Universality in \(L_1\)), 6 (Universality in \(L_p\)) and 7 (the two corollaries).

If \(\{p_k(x)\}\) are the orthonormal polynomials with respect to a finite positive Borel measure \(\mu\) with compact support on the real line, the simplest case of this law is given in terms of the reproducing kernel \[ K_n(x,y)=\sum_{k=0}^n\,p_k(x)p_k(y),\;\tilde{K}_n(x,y)=w(x)^{1/2}w(y)^{1/2}K_n(x,y), \] and reads \[ \lim_{n\rightarrow\infty}\,{\tilde{K}_n\left(\xi+{a\over\tilde{K}_n(\xi,\xi)}, \xi+{b\over\tilde{K}_n(\xi,\xi)}\right)\over\tilde{K}_n(\xi,\xi)}={\sin{\pi (a-b)}\over\pi (a-b)}. \eqno{(\ast)} \] The measure \(\mu\) has infinitely many points in its support and \[ w={\text{d}\mu\over\text{d}x} \] is the Radon-Nikodym derivative of \(\mu\) and \((\ast)\) holds uniformly in \(\xi\) on a compact subinterval of supp\((\mu)\) and in \(a,b\) in compact subsets of the real line (for \(a=b\): the right hand side is \(1\)).

In this paper, the author studies measures on the real interval \((-1,1)\) and the main results are:

Theorem 1.1 Let \(\mu\) be a finite positive measure on \((-1,1)\), that is regular. Let \(J\subset (-1,1)\) be compact and such that \(\mu\) is absolutely continuous in an open set containing \(J\). Assume, moreover, that \(w\) is positive and continuous at each point of \(J\). Then, uniformly for \(x\in J\) and \(a,\,b\) in compact subsets of the real line, we have:

\[ \lim_{n\rightarrow\infty}\,{\tilde{K}_n\left(x+{a\over\tilde{K}_n(x,x)}, x+{b\over\tilde{K}_n(x,x)}\right)\over\tilde{K}_n(x,x)}={\sin{\pi (a-b)}\over\pi (a-b)}. \]

[and two interesting corollaries that are not cited explicitly here]

Theorem 1.4 Let \(\mu\) be a finite positive measure on \((-1,1)\), that is regular; let \(p>1\). Let \(I\subset (-1,1)\) be a closed interval in which \(\mu\) is absolutely continuous and \(w\) is bounded above and below by positive constants. Then we have:

(a) If \(I'\) is a closed subinterval of \(I^{0}\) \[ \lim_{n\rightarrow\infty}\,\int_{I'}\,\left|{K_n\left(x+{a\over\tilde{K}_n(x,x)}, x+{b\over\tilde{K}_n(x,x)}\right)\over K_n(x,x)}-{\sin{\pi (a-b)}\over\pi (a-b)}\right|^pdx=0, \] uniformly for \(a,\,b\) in compact subsets of the real line.

(b) If, in addition, \(w\) is Riemann integrable in \(I\), we may replace \[ {K_n\left(x+{a\over\tilde{K}_n(x,x)},x+{b\over\tilde{K}_n(x,x)}\right)\over K_n(x,x)}\text{ by } {\tilde{K}_n\left(x+{a\over\tilde{K}_n(x,x)},x+{b\over\tilde{K}_n(x,x)}\right)\over\tilde{K}_n(x,x)} \] in the limit in (a).

The proofs are given in the sections 2 (Christoffel functions), 3 (Localization), 4 (Smoothing), 5 (Universality in \(L_1\)), 6 (Universality in \(L_p\)) and 7 (the two corollaries).

Reviewer: Marcel G. de Bruin (Haarlem)

### MSC:

42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |

40A30 | Convergence and divergence of series and sequences of functions |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |

### Keywords:

positive Borel measure; orthogonal polynomials; reproducing kernel; universality law; \(L_p\) spaces
PDF
BibTeX
XML
Cite

\textit{D. S. Lubinsky}, Ann. Math. (2) 170, No. 2, 915--939 (2009; Zbl 1176.42022)

### References:

[1] | J. Baik, T. Kriecherbauer, K. D. T-R. McLaughlin, and P. D. Miller, ”Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: announcement of results,” Int. Math. Res. Notices, vol. 2003, iss. 15, pp. 821-858, 2003. · Zbl 1036.42023 |

[2] | R. A. DeVore and G. G. Lorentz, Constructive Approximation, New York: Springer-Verlag, 1993. · Zbl 0797.41016 |

[3] | P. A. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, New York: New York University Courant Institute of Mathematical Sciences, 1999. · Zbl 0997.47033 |

[4] | 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,” Comm. Pure Appl. Math., vol. 52, iss. 11, pp. 1335-1425, 1999. · Zbl 0944.42013 |

[5] | A. B. J. Kuijlaars and M. Vanlessen, ”Universality for eigenvalue correlations from the modified Jacobi unitary ensemble,” Int. Math. Res. Not., vol. 2002, iss. 30, pp. 1575-1600, 2002. · Zbl 1122.30303 |

[6] | E. Levin and D. S. Lubinsky, ”Universality limits for exponential weights,” Constr. Approx., vol. 29, iss. 2, pp. 247-275, 2009. · Zbl 1169.42313 |

[7] | A. Máté, P. Nevai, and V. Totik, ”Szeg\Ho’s extremum problem on the unit circle,” Ann. of Math., vol. 134, iss. 2, pp. 433-453, 1991. · Zbl 0752.42015 |

[8] | M. L. Mehta, Random Matrices, second ed., Boston, MA: Academic Press Inc., 1991. · Zbl 1107.15019 |

[9] | P. Nevai, ”Orthogonal polynomials,” Mem. Amer. Math. Soc., vol. 18, iss. 213, p. v, 1979. · Zbl 0405.33009 |

[10] | P. Nevai, ”Géza Freud, orthogonal polynomials and Christoffel functions. A case study,” J. Approx. Theory, vol. 48, iss. 1, pp. 3-167, 1986. · Zbl 0606.42020 |

[11] | F. Riesz and B. Sz.-Nagy, Functional Analysis, New York: Dover Publications Inc., 1990. · Zbl 0732.47001 |

[12] | B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, Providence, RI: A.M.S., 2005, vol. 54. · Zbl 1082.42021 |

[13] | H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge: Cambridge Univ. Press, 1992. · Zbl 0791.33009 |

[14] | G. SzegHo, Orthogonal Polynomials, Fourth ed., Providence, R.I.: A.M.S., 1975, vol. 23. · Zbl 0305.42011 |

[15] | V. Totik, ”Asymptotics for Christoffel functions for general measures on the real line,” J. Anal. Math., vol. 81, pp. 283-303, 2000. · Zbl 0966.42017 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.