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Transactions of the American Mathematical Society

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Comparative asymptotics for perturbed orthogonal polynomials


Authors: Franz Peherstorfer and Robert Steinbauer
Journal: Trans. Amer. Math. Soc. 348 (1996), 1459-1486
MSC (1991): Primary 42C05
DOI: https://doi.org/10.1090/S0002-9947-96-01498-5
MathSciNet review: 1322954
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Abstract: Let $\{\Phi _n\}_{n\in \mathbb N_0}$ and $\{\widetilde \Phi _n\}_{n\in \mathbb N_0}$ be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials $\widetilde \Phi _n$ behave asymptotically like those of $\Phi _n$. We give, under weak assumptions on the system $\{\Phi _n\}_{n\in \mathbb N_0}$ and the perturbations, comparative asymptotics as for $\widetilde \Phi _n^*(z)/ \Phi _n^*(z)$ etc., $\Phi _n^*(z):= z^n\bar \Phi _n(\frac 1z)$, on the open unit disk and on the circumference mainly off the support of the measure $\sigma$ with respect to which the $\Phi _n$’s are orthonormal. In particular these results apply if the comparative system $\{\Phi _n\} _{n\in \mathbb N_0}$ has a support which consists of several arcs of the unit circumference, as in the case when the recurrence coefficients are (asymptotically) periodic.


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Additional Information

Franz Peherstorfer
Affiliation: Institut für Mathematik, Johannes Kepler Universität Linz, A-4040 Linz, Austria
Email: franz.peherstorfer@jk.uni-linz.ac.at

Robert Steinbauer
Affiliation: Institut für Mathematik, Johannes Kepler Universität Linz, A-4040 Linz, Austria
Email: robert.steinbauer@jk.uni-linz.ac.at

Keywords: Orthogonal polynomials, unit circle, arcs, asymptotics
Received by editor(s): March 5, 1994
Received by editor(s) in revised form: January 5, 1995
Additional Notes: Supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, Projektnummer P9267-PHY
Article copyright: © Copyright 1996 American Mathematical Society