Asymptotics for orthogonal rational functions

Abstract

Let{αn}\{ {\alpha _n}\}be a sequence of (not necessarily distinct) points in the open unit disk, and let\[B0=1,Bn(z)=∏m=1nαm¯|αm|(αm−z)(1−αm¯z),n=1,2,…,{B_0} = 1,\quad {B_n}(z) = \prod \limits _{m = 1}^n {\frac {{\overline {{\alpha _m}} }} {{|{\alpha _m}|}}\frac {{({\alpha _m} - z)}} {{(1 - \overline {{\alpha _m}} z}}),\qquad n = 1,2, \ldots ,}\](αn¯|αn|=−1\frac {{\overline {{\alpha _n}} }} {{|{\alpha _n}|}} = - 1whenαn=0{\alpha _n} = 0). Letμ\mube a finite (positive) Borel measure on the unit circle, and let{φn(z)}\{ {\varphi _n}(z)\}be orthonormal functions obtained by orthogonalization of{Bn:n=0,1,2,…}\{ {B_n}:n = 0,1,2, \ldots \}with respect toμ\mu. Boundedness and convergence properties of the reciprocal orthogonal functionsφn∗(z)=Bn(z)φn(1/z¯)¯\varphi _n^*(z) = {B_n}(z)\overline {{\varphi _n}(1/\overline z )}and the reproducing kernelskn(z,w)=∑m=0nφm(z)φm(w)¯{k_n}(z,w) = \sum \nolimits _{m = 0}^n {{\varphi _m}(z)\overline {{\varphi _m}(w)} }are discussed in the situation|αn|⩽R>1|{\alpha _n}| \leqslant R > 1for allnn, in particular their relationship to the Szegö condition∫−ππln⁡μ′(θ)dθ>−∞\int _{ - \pi }^\pi {\ln \mu ’(\theta )d\theta > - \infty }and noncompleteness inL2(μ){L_2}(\mu )of the system{φn(z):n=0,1,2,…}\{ {\varphi _n}(z):n = 0,1,2, \ldots \}. Limit functions ofφn∗(z)\varphi _n^{\ast }(z)andkn(z,w){k_n}(z,w)are obtained. In particular, if a subsequence{αn(s)}\{ {\alpha _{n(s)}}\}converge toα\alpha, then the subsequence{φn(s)∗(z)}\{ \varphi _{n(s)}^{\ast }(z)\}converges to\[eiλ1−|α|21−α¯z1σμ(z),λ∈R,{e^{i\lambda }}\frac {{\sqrt {1 - |\alpha {|^2}} }} {{1 - \overline \alpha z}}\frac {1} {{{\sigma _{\mu (z)}}}},\qquad \lambda \in {\mathbf {R}},\]where\[σμ(z)=2πexp⁡[14π∫−ππeiθ+zeiθ−zln⁡μ′(θ)dθ].{\sigma _\mu }(z) = \sqrt {2\pi } \exp \left [ {\frac {1} {{4\pi }}\int _{ - \pi }^\pi {\frac {{{e^{i\theta }} + z}} {{{e^{i\theta }} - z}}} \ln \mu ’(\theta )d\theta } \right ].\]The kernels{kn(z,w)}\{ {k_n}(z,w)\}converge to1/(1−zw¯)σμ(z)σμ(w)¯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 situationαn=0{\alpha _n} = 0for allnn).