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Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure

Authors: A. Martínez-Finkelshtein, A. Sri Ranga and D. O. Veronese
Journal: Math. Comp.
MSC (2010): Primary 42C05, 33C47; Secondary 65D20, 33C45
Published electronically: April 28, 2017
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Abstract: Given a nontrivial Borel measure $ \mu $ on the unit circle $ \mathbb{T}$, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at $ z=1$ constitute a family of so-called para-orthogonal polynomials, whose zeros belong to $ \mathbb{T}$. With a proper normalization they satisfy a three-term recurrence relation determined by two sequences of real coefficients, $ \{c_n\}$ and $ \{d_n\}$, where $ \{d_n\}$ is additionally a positive chain sequence. Coefficients $ (c_n,d_n)$ provide a parametrization of a family of measures related to $ \mu $ by addition of a mass point at $ z=1$.

In this paper we estimate the location of the extreme zeros (those closest to $ z=1$) of the para-orthogonal polynomials from the $ (c_n,d_n)$-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of $ \mu $ at $ z=1$. These results are easily reformulated in order to find gaps in the support of $ \mu $ at any other $ z\in \mathbb{T}$.

We provide also some examples showing that the bounds are tight and illustrate their computational applications.

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Additional Information

A. Martínez-Finkelshtein
Affiliation: Departamento de Matemáticas, Universidad de Almería, 04120 Almería, and Instituto Carlos I de Física Teórica and Computacional, Granada University, Spain

A. Sri Ranga
Affiliation: Departamento de Matemática Aplicada, IBILCE, UNESP - Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil

D. O. Veronese
Affiliation: ICTE, UFTM - Universidade Federal do Triângulo Mineiro, 38064–200 Uberaba, MG, Brazil

Keywords: Orthogonal polynomials on the unit circle, para-orthogonal polynomials on the unit circle, three term recurrence, positive chain sequences
Received by editor(s): October 27, 2015
Received by editor(s) in revised form: September 2, 2016
Published electronically: April 28, 2017
Article copyright: © Copyright 2017 American Mathematical Society