Christoffel functions and universality limits for orthogonal rational functions (Q2909064)

Consider a nested sequence of spaces \(\{\mathcal{L}_n:n=0,1,\ldots\}\) of rational functions of degree \(n\) with prescribed poles outside \([-1,1]\) with an orthonormal basis \(\{\phi_k:k=0,1,\ldots\}\) for some Borel measure \(\mu\) on \([-1,1]\). The reproducing kernel for \(\mathcal{L}_{n-1}\) is \(K_n(x,y)=\sum_{k=0}^{n-1}\phi_k(x)\overline{\phi_k(y)}\) and the Christoffel function is \(\lambda_n(x)=1/K_n(x,x)\). The poles satisfy some asymptotic distribution characterized by a measure \(\nu\) which is the weak limit of the counting measure \(\nu_n=\frac{1}{n}(\delta_\infty+\sum_{j=1}^{n-1}\delta_{\alpha_j})\) as \(n\to\infty\). The following asymptotic formula is proved for \(x\in(-1,1)\), \(s\in[-r,r]\) and \(r>0\): NEWLINE\[NEWLINE\lim_{n\to\infty}n\lambda_n(x+\frac{s}{n})=\mu'(x)\pi\sqrt{1-x^2}/R_\nu(x)NEWLINE\]NEWLINE with \(R_\nu(x)=\int\mathrm{Re}\{\frac{\sqrt{t^2-1}}{t-x}\}d\nu(t)\). This generalizes the classical polynomial result, i.e., when all \(\alpha_k=\infty\) and \(R_\nu=1\). As a consequence, a universality limit for the reproducing kernels is obtained: Set \(\xi_c(x)=x+c/[\mu'(x)K_n(x,x)]\), \(K_n^{a,b}(x)=K_n(\xi_a(x),\xi_b(x))\) and \(\omega_{n-1}^{a,b}(x)=\mathrm{arg}\{\pi_{n-1}(\xi_a(x))\}-\mathrm{arg}\{\pi_{n-1}(\xi_b(x))\}\) with \(\pi_{n-1}(x)=\prod_{k=1}^n(1-\frac{x}{\alpha_k})\), then, if \(x\in (-1,1)\) and \(\mu'\) positive and continuous in a neighbourhood of \(x\), NEWLINE\[NEWLINE\lim_{n\to\infty}\frac{K_n(\xi_a(x),\xi_b(x))}{K_n(x,x)}E_n^{a,b}(x)=\frac{\sin\pi(a-b)}{\pi(a-b)}NEWLINE\]NEWLINE where \(E_n^{a,b}(x)=\exp\{i\omega_{n-1}^{a,b}(x)\}\) and \(a,b\) are finite real numbers. Again, this generalizes the classical polynomial result where \(E_n^{a,b}(x)=1\).NEWLINENEWLINEThe proof is given under the technical assumption that the poles stay away from \([-1,1]\). By including the denominators of the rational functions into the measure, which then depends on \(n\), the numerators become polynomials that can be orthogonalized with respect to this varying measure. This is used to lift some polynomial results to the rational case. In the special case of the Chebyshev weight of the second kind, Bernstein-Szegő polynomials are used to obtain the asymptotic expression for the Christoffel function. The limit in the rational case then follows by comparing Christoffel functions for different measures and two different measures, one of which is the Chebyshev measure. The universality limit then follows readily.