Revista Integración, temas de matemáticas.

Revista Integración, temas de matemáticas

Vol. 34 No. 1 (2016): Revista Integración
Research and Innovation Articles

On a finite moment perturbation of linear functionals and the inverse Szegö transformation

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  • Edinson Fuentes,
  • Luis E. Garza
Edinson Fuentes
Universidad Pedagógica y Tecnológica de Colombia
Luis E. Garza
Universidad de Colima
DOI: https://doi.org/10.18273/revint.v34n1-2016003

Published 2016-05-06

Keywords

  • Orthogonal polynomials on the unit circle,
  • perturbation of moments,
  • inverse Szegö transformation

How to Cite

Fuentes, E., & Garza, L. E. (2016). On a finite moment perturbation of linear functionals and the inverse Szegö transformation. Revista Integración, Temas De matemáticas, 34(1), 39–58. https://doi.org/10.18273/revint.v34n1-2016003

Copyright (c) 2016 Edinson Fuentes, Luis E. Garza

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Given a sequence of moments $\{c_{n}\}_{n\in\ze}$ associated with an Hermitian linear functional $\mathcal{L}$ defined in the space of Laurent polynomials, we study a new functional $\mathcal{L}_{\Omega}$ which is a perturbation of $\mathcal{L}$ in such a way that a finite number of moments are perturbed. Necessary and sufficient conditions are given for the regularity of $\mathcal{L}_{\Omega}$, and a connection formula between the corresponding families of orthogonal polynomials is obtained. On the other hand, assuming $\mathcal{L}_{\Omega}$ is positive definite, the perturbation is analyzed through the inverse Szegö transformation.

To cite this article: E. Fuentes, L.E. Garza, On a finite moment perturbation of linear functionals and the inverse Szegö transformation, Rev. Integr. Temas Mat. 34 (2016), No. 1, 39–58.

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References

  1. Bueno M.I. and Marcellán F., “Polynomial perturbations of bilinear functionals and Hessenberg matrices”, Linear Algebra Appl. 414 (2006), No.1, 64–83.
  2. Castillo K., Garza L.E. and Marcellán F., “Linear spectral transformations, Hessenberg matrices and orthogonal polynomials”, Rend. Circ. Mat. Palermo (2) Suppl. (2010), No.82, 3–26.
  3. Castillo K., Garza L.E. and Marcellán F., “Perturbations on the subdiagonals of Toeplitz matrices”, Linear Algebra Appl. 434 (2011), No. 6, 1563–1579.
  4. Castillo K. and Marcellán F., “Generators of rational spectral transformations for nontrivial C-functions”, Math. Comp. 82 (2013), No. 282,1057–1068.
  5. Fuentes E. and Garza L.E., “Analysis of perturbations of moments associated with orthogonality linear functionals through the Szeg´´o transformation”, Rev. Integr. Temas Mat.33 (2015), No. 1, 61–82.
  6. Garza L., Hernández J. and Marcellán F., “Spectral transformations of measures supported on the unit circle and the Szeg´´o transformation”, Numer. Algorithms. 49 (2008), No. 1-4,169–185.
  7. Grenander U. and Szeg´´o G., Toeplitz forms and their applications, Second ed., Chelsea Publishing Co., New York, 1984.
  8. Kreyszig E., Introductory functional analysis with applications, John Wiley & Sons Inc., New York, 1989.
  9. Peherstorfer F., “A special class of polynomials orthogonal on the circle including the associated polynomials”, Constr. Approx. 12 (1996), No. 2, 161–185.
  10. Simon B., Orthogonal polynomials on the unit circle. Part 1 and 2, American Mathematical Society Colloquium Publications 54, American Mathematical Society, Providence, RI, 2005.
  11. Szeg´´o G., Orthogonal polynomials, Fourth ed., American Mathematical Society Colloquium Publications 23, American Mathematical Society, Providence, RI, 1975.
  12. Zhedanov A., “Rational spectral transformations and orthogonal polynomials”, J. Comput.
  13. Appl. Math. 85 (1997), No. 1, 67–86.