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

  • Kenier Castillo
  • Francisco Marcellán
  • Jorge RiveroEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (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.

Keywords

Orthogonal polynomials on the real line Orthogonal polynomials on the unit circle Zeros Spectral transformations Co-polynomials Szegő transformation 

Mathematics Subject Classification (2010):

42C05 

Notes

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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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Kenier Castillo
    • 1
  • Francisco Marcellán
    • 2
  • Jorge Rivero
    • 2
    • 3
    Email author
  1. 1.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  2. 2.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain
  3. 3.Instituto de Ciencias Matemáticas (ICMAT) Campus de CantoblancoUAMMadridSpain

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