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Asymptotics for orthogonal rational functions


Authors: A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad
Journal: Trans. Amer. Math. Soc. 346 (1994), 307-329
MSC: Primary 42C05; Secondary 30B70, 30D50, 41A20
MathSciNet review: 1272674
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Abstract: Let $ \{ {\alpha _n}\} $ be a sequence of (not necessarily distinct) points in the open unit disk, and let

$\displaystyle {B_0} = 1,\quad {B_n}(z) = \prod\limits_{m = 1}^n {\frac{{\overli... ...alpha _m} - z)}} {{(1 - \overline {{\alpha _m}} z}}),\qquad n = 1,2, \ldots ,} $

( $ \frac{{\overline {{\alpha _n}} }} {{\vert{\alpha _n}\vert}} = - 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 $ \vert{\alpha _n}\vert \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

$\displaystyle {e^{i\lambda }}\frac{{\sqrt {1 - \vert\alpha {\vert^2}} }} {{1 - ... ...ine \alpha z}}\frac{1} {{{\sigma _{\mu (z)}}}},\qquad \lambda \in {\mathbf{R}},$

where

$\displaystyle {\sigma _\mu }(z) = \sqrt {2\pi } \exp \left[ {\frac{1} {{4\pi }}... ...^{i\theta }} + z}} {{{e^{i\theta }} - z}}} \ln \mu '(\theta )d\theta } \right].$

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$).


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1272674-9
Keywords: Orthogonal rational functions, Szegö theory
Article copyright: © Copyright 1994 American Mathematical Society