CHRISTOFFEL FUNCTIONS AND UNIVERSALITY LIMITS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO MEASURES WITH NON-COMPACT SUPPORT

by GRZEGORZ ŚWIDERSKI (POLISH ACADEMY OF SCIENCES)

Location: 200B.02.18, Leuven
Time: Thursday November 8, 2018 at 14:00

Let $\mu$ be a probability measure on the real line with all moments finite. Let $(p_n : n \geq 0)$ be the corresponding sequence of orthonormal polynomials. The associated Christoffel-Darboux kernel is defined by [ K_n(x,y) = \sum_{k=0}^n p_k(x) p_k(y). ] It turns out that in the case when the support of $\mu$ is compact, the limits [ \lim_{n \to \infty} \frac{1}{n} K_n(x, x), \qquad \lim_{n \to \infty} \frac{K_n(x + a/K_n(x,x), x + b/K_n(x,x))}{K_n(x,x)} ] usually exist and its values are pretty universal under weak assumptions on $\mu$. In particular, one can compute the limits even when precise asymptotics of the polynomials is not available. In the case of $\mu$ with non-compact support much less is known.

We are going to present some elementary tools to prove the existence of the aforementioned limits under information on the asymptotics of the polynomials. The required information is much weaker than what one usually obtains by using, e.g. Riemann-Hilbert method.

This is a joint work with Bartosz Trojan (Polish Academy of Sciences).