PDE and Analysis Seminar: Grzegorz Swiderski (Wroclaw)

Thu, 27/04/201713:00-14:00
Ross 70
Title: Asymptotics of Christoffel functions in an unbounded setting
Consider a measure $\mu$ supported on the real line with all moments finite.
Let $(p_n : n \geq 0)$ be the corresponding sequence of orthonormal
polynomials. This sequence satisfies the three-term recurrence relation
a_{n-1} p_{n-1}(x) + b_n p_n(x) a_n p_{n+1}(x) = x p_n(x) \quad (n > 0)
for some sequences $a$ and $b$.
One defines the $n$th Christoffel function by
\lambda_n(x) = \left[ \sum_{k=0}^n p_k(x)^2 \right]^{-1}.
In the talk, under some regularity hypotheses on $a$ and $b$, we show
exact asymptotics of the sequence $(\lambda_n(x) : n \geq 0)$ in the case
when $\textrm{supp}(\mu) = \mathbb{R}$.