Asymptotics for orthogonal rational functions

Abstract:

Let $\{ {\alpha _n}\}$ be a sequence of (not necessarily distinct) points in the open unit disk, and let \[ {B_0} = 1,\quad {B_n}(z) = \prod \limits _{m = 1}^n {\frac {{\overline {{\alpha _m}} }} {{|{\alpha _m}|}}\frac {{({\alpha _m} - z)}} {{(1 - \overline {{\alpha _m}} z}}),\qquad n = 1,2, \ldots ,} \] ($\frac {{\overline {{\alpha _n}} }} {{|{\alpha _n}|}} = - 1$ when ${\alpha _n} = 0$). Let $\mu$ be a finite (positive) Borel measure on the unit circle, and let $\{ {\varphi _n}(z)\}$ be orthonormal functions obtained by orthogonalization of $\{ {B_n}:n = 0,1,2, \ldots \}$ with respect to $\mu$. Boundedness and convergence properties of the reciprocal orthogonal functions $\varphi _n^*(z) = {B_n}(z)\overline {{\varphi _n}(1/\overline z )}$ and the reproducing kernels ${k_n}(z,w) = \sum \nolimits _{m = 0}^n {{\varphi _m}(z)\overline {{\varphi _m}(w)} }$ are discussed in the situation $|{\alpha _n}| \leqslant R < 1$ for all $n$, in particular their relationship to the Szegö condition $\int _{ - \pi }^\pi {\ln \mu ’(\theta )d\theta > - \infty }$ and noncompleteness in ${L_2}(\mu )$ of the system $\{ {\varphi _n}(z):n = 0,1,2, \ldots \}$. Limit functions of $\varphi _n^{\ast }(z)$ and ${k_n}(z,w)$ are obtained. In particular, if a subsequence $\{ {\alpha _{n(s)}}\}$ converge to $\alpha$, then the subsequence $\{ \varphi _{n(s)}^{\ast }(z)\}$ converges to \[ {e^{i\lambda }}\frac {{\sqrt {1 - |\alpha {|^2}} }} {{1 - \overline \alpha z}}\frac {1} {{{\sigma _{\mu (z)}}}},\qquad \lambda \in {\mathbf {R}},\] where \[ {\sigma _\mu }(z) = \sqrt {2\pi } \exp \left [ {\frac {1} {{4\pi }}\int _{ - \pi }^\pi {\frac {{{e^{i\theta }} + z}} {{{e^{i\theta }} - z}}} \ln \mu ’(\theta )d\theta } \right ].\] The kernels $\{ {k_n}(z,w)\}$ converge to $1/(1 - z\overline w ){\sigma _\mu }(z)\overline {{\sigma _\mu }(w)}$. The results generalize corresponding results from the classical Szegö theory (concerned with the polynomial situation ${\alpha _n} = 0$ for all $n$).