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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 307974, 8 pages
http://dx.doi.org/10.1155/2013/307974
Research Article

-Coherent Pairs on the Unit Circle

1Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, 28045 Colima, COL, Mexico
2Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain

Received 23 July 2013; Accepted 19 September 2013

Academic Editor: Jinde Cao

Copyright © 2013 Luis Garza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna, “On polynomials orthogonal with respect to certain Sobolev inner products,” Journal of Approximation Theory, vol. 65, no. 2, pp. 151–175, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. M. Delgado and F. Marcellán, “Companion linear functionals and Sobolev inner products: a case study,” Methods and Applications of Analysis, vol. 11, no. 2, pp. 237–266, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. G. Meijer, “Determination of all coherent pairs,” Journal of Approximation Theory, vol. 89, no. 3, pp. 321–343, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. N. de Jesus, F. Marcellán, J. Petronilho, and N. C. Pinzón-Cortés, “(M,N)-coherent pairs of order (m,k) and Sobolev orthogonal polynomials,” Journal of Computational and Applied Mathematics, vol. 256, pp. 16–35, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  5. M. N. de Jesus and J. Petronilho, “On linearly related sequences of derivatives of orthogonal polynomials,” Journal of Mathematical Analysis and Applications, vol. 347, no. 2, pp. 482–492, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. N. de Jesus and J. Petronilho, “Sobolev orthogonal polynomials and (M,N)-coherent pairs of measures,” Journal of Computational and Applied Mathematics, vol. 237, no. 1, pp. 83–101, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. F. Marcellán and N. C. Pinzón-Cortés, “Higher order coherent pairs,” Acta Applicandae Mathematicae, vol. 121, pp. 105–135, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. I. Area, E. Godoy, and F. Marcellán, “Classification of all Δ-coherent pairs,” Integral Transforms and Special Functions, vol. 9, no. 1, pp. 1–18, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  9. I. Area, E. Godoy, and F. Marcellán, “q-coherent pairs and q-orthogonal polynomials,” Applied Mathematics and Computation, vol. 128, no. 2-3, pp. 191–216, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  10. I. Area, E. Godoy, and F. Marcellán, “Δ-coherent pairs and orthogonal polynomials of a discrete variable,” Integral Transforms and Special Functions, vol. 14, no. 1, pp. 31–57, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. F. Marcellán and N. C. Pinzón-Cortés, “(1,1)-Dω-coherent pairs,” Journal of Difference Equations and Applications, 2013. View at Publisher · View at Google Scholar
  12. F. Marcellán and N. C. Pinzón-Cortés, “(1,1)-q-coherent pairs,” Numerical Algorithms, vol. 60, no. 2, pp. 223–239, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. Álvarez-Nodarse, J. Petronilho, N. C. Pinzón-Cortés, and R. Sevinik-Adgüzel, “On linearly related sequences of difference derivatives of discrete orthogonal polynomials,” in progress.
  14. A. Branquinho, A. F. Moreno, F. Marcellán, and M. N. Rebocho, “Coherent pairs of linear functionals on the unit circle,” Journal of Approximation Theory, vol. 153, no. 1, pp. 122–137, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. A. Branquinho and M. N. Rebocho, “Structure relations for orthogonal polynomials on the unit circle,” Linear Algebra and Its Applications, vol. 436, no. 11, pp. 4296–4310, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Ya. L. Geronimus, Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, vol. 18, Consultants Bureau, New York, NY, USA, 1961. View at MathSciNet
  17. Ya. L. Geronimus, Polynomials Orthogonal on a Circle and Their Applications, vol. 3 of American Mathematical Society Translations, Providence, RI, USA, 1962.
  18. G. Szegő, Orthogonal Polynomials, vol. 23 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 4th edition, 1975.
  19. B. Simon, Orthogonal Polynomials on the Unit Circle, vol. 54 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 2005.
  20. A. Máté and P. G. Nevai, “Remarks on E. A. Rakhmanov's paper: “The asymptotic behavior of the ratio of orthogonal polynomials”,” Journal of Approximation Theory, vol. 36, no. 1, pp. 64–72, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  21. A. Máté, P. Nevai, and V. Totik, “Extensions of Szegő's theory of orthogonal polynomials. II,” Constructive Approximation, vol. 3, no. 1, p. 51–72, 73–96, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  22. F. Peherstorfer and R. Steinbauer, “Characterization of orthogonal polynomials with respect to a functional,” Journal of Computational and Applied Mathematics, vol. 65, no. 1–3, pp. 339–355, 1995. View at Publisher · View at Google Scholar
  23. K. Castillo, L. Garza, and F. Marcellán, “Linear spectral transformations, Hessenberg matrices, and orthogonal polynomials,” Rendiconti Circolo Matematico di Palermo, vol. 2, supplement 82, pp. 3–26, 2010.
  24. F. Peherstorfer, “A special class of polynomials orthogonal on the unit circle including the associated polynomials,” Constructive Approximation, vol. 12, no. 2, pp. 161–185, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet