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On Some Recent Results on Asymptotic Behavior of Orthogonal Polynomials on the Unit Circle and Inserting Point Masses

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Computation, Cryptography, and Network Security

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Abstract

In the present paper, we formulate and reprove, in a brief and self-contained presentation, some recent results concerning the asymptotic behavior of orthogonal polynomials on the unit circle by inserting point masses recently obtained by the authors and co-workers. In a first part, we deal with a spectral transformation of a Hermitian linear functional by the addition of the first derivative of a complex Dirac linear functional supported either in a point on the unit circle or in two symmetric points with respect to the unit circle. In this case, outer relative asymptotics for the new sequences of orthogonal polynomials in terms of the original ones are obtained. Necessary and sufficient conditions for the quasi-definiteness of the new linear functionals are given. The relation between the corresponding sequence of orthogonal polynomials in terms of the original one is presented. The second part is devoted to the study of a relevant family of orthogonal polynomials associated with perturbations of the original orthogonality measure by means of mass points: discrete Sobolev orthogonal polynomials. We compare the discrete Sobolev orthogonal polynomials with the initially ones. Finally, we analyze the behavior of their zeros.

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Acknowledgements

The authors wish to express their thanks to Nicholas J. Daras and Michael Th. Rassias for the invitation to participate in this volume. The research of the first author is supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/ 101139/2014. This author also acknowledges the financial support by the Brazilian Government through the CNPq under the project 470019/2013-1. The research of the first and second author is supported by the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project 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. Instituto de Ciencias Matemáticas (ICMAT) and Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911, Leganés, Spain

    Francisco Marcellán

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  1. Kenier Castillo
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  2. Francisco Marcellán
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Correspondence to Kenier Castillo .

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Editors and Affiliations

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

    Nicholas J. Daras

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Castillo, K., Marcellán, F. (2015). On Some Recent Results on Asymptotic Behavior of Orthogonal Polynomials on the Unit Circle and Inserting Point Masses. In: Daras, N., Rassias, M. (eds) Computation, Cryptography, and Network Security. Springer, Cham. https://doi.org/10.1007/978-3-319-18275-9_4

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