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On Co-polynomials on the Real Line and the Unit Circle

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Operations Research, Engineering, and Cyber Security

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 113))

Abstract

In this paper, we present an overview about algebraic and analytic aspects of orthogonal polynomials on the real line when finite modifications of the coefficients of the three-term recurrence relation they satisfy, the so-called co-polynomials on the real line, are considered. We investigate the behavior of their zeros, mainly interlacing and monotonicity properties. Furthermore, using a transfer matrix approach we obtain new structural relations, combining theoretical and computational advantages. In the case of orthogonal polynomials on the unit circle, we analyze the effects of finite modifications of Verblunsky coefficients on Szegő recurrences. More precisely, we study the structural relations and the corresponding \(\mathcal{C}\)-functions of the orthogonal polynomials with respect to these modifications from the initial ones. By using the Szegő’s transformation we deduce new relations between the recurrence coefficients for orthogonal polynomials on the real line and the Verblunsky parameters of orthogonal polynomials on the unit circle as well as the relation between the corresponding \(\mathcal{S}\)-functions and \(\mathcal{C}\)-functions is studied.

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Acknowledgements

The authors wish to express their thanks to Th. M. Rassias and N. J. Daras for the invitation to participate in this volume. The research of the first author is supported by the Portuguese Government through the FCT under the grant SFRH/BPD/101139/2014 and partially supported by the Brazilian Government through the CNPq under the project 470019/2013-1. The research of the first and second author is supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, grant MTM2012-36732-C03-01.

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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    Kenier Castillo

  2. Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911, Leganés, Madrid, Spain

    Francisco Marcellán & Jorge Rivero

  3. Instituto de Ciencias Matemáticas (ICMAT) Campus de Cantoblanco, UAM, 28049, Madrid, Spain

    Jorge Rivero

Authors
  1. Kenier Castillo
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  2. Francisco Marcellán
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  3. Jorge Rivero
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Corresponding author

Correspondence to Jorge Rivero .

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Editors and Affiliations

  1. Department of Mathematics, Hellenic Military Academy, Vari Attikis, Greece

    Nicholas J. Daras

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Castillo, K., Marcellán, F., Rivero, J. (2017). On Co-polynomials on the Real Line and the Unit Circle. In: Daras, N., Rassias, T. (eds) Operations Research, Engineering, and Cyber Security. Springer Optimization and Its Applications, vol 113. Springer, Cham. https://doi.org/10.1007/978-3-319-51500-7_4

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