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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).

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.)
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References:

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