Abstract
Iserles et al. (J. Approx. Theory 65:151–175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt (J. Approx. Theory 114:115–140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt, when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bavinck, H., Meijer, H.G.: Orthogonal polynomials with respect to a symmetric inner product involving derivatives. Appl. Anal. 33, 103–117 (1989)
Bavinck, H., Meijer, H.G.: On orthogonal polynomials with respect to an inner product involving derivatives: zeros and recurrence relations. Indag. Math. 1, 7–14 (1990)
Berti, A.C., Sri Ranga, A.: Companion orthogonal polynomials: some applications. Appl. Numer. Math. 39, 127–149 (2001)
Bracciali, C.F., Berti, A.C., Sri Ranga, A.: Orthogonal polynomials associated with related measures and Sobolev orthogonal polynomials. Numer. Algorithms 34, 203–216 (2003)
Bracciali, C.F., Dimitrov, D.K., Sri Ranga, A.: Chain sequences and symmetric generalized orthogonal polynomials. J. Comput. Appl. Math. 143, 95–106 (2002)
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Mathematics and its Applications Series. Gordon and Breach, New York (1978)
Delgado, A.M., Marcellán, F.: On an extension of symmetric coherent pairs of orthogonal polynomials. J. Comput. Appl. Math. 178, 155–168 (2005)
de Bruin, M.G., Groenevelt, W.G.M., Meijer, H.G.: Zeros of Sobolev orthogonal polynomials of Hermite type. Appl. Math. Comput. 132, 135–166 (2002)
Groenevelt, W.G.M.: Zeros of Sobolev orthogonal polynomials of Gegenbauer type. J. Approx. Theory 114, 115–140 (2002)
Iserles, A., Koch, P.E., Nørsett, S.P., Sanz-Serna, J.M.: On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65, 151–175 (1991)
Marcellán, F., Moreno-Balcazar, J.J.: Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded support. Acta Appl. Math. 94, 163–192 (2006)
Marcellán, F., Pérez, T.E., Piñar, M.A.: Gegenbauer-Sobolev orthogonal polynomials. In: Cuyt, A. (ed.) Nonlinear Numerical Methods and Rational Approximation. Math. Appl., vol. 296, pp. 71–82. (1994)
Meijer, H.G.: Determination of all coherent pairs. J. Approx. Theory 89, 321–343 (1997)
Meijer, H.G., de Bruin, M.G.: Zeros of Sobolev orthogonal polynomials following from coherent pairs. J. Comput. Appl. Math. 139, 253–274 (2002)
Szegő, G.: Orthogonal Polynomials, 4th edn. Amer. Math. Soc. Colloq. Publ., vol. 23. Am. Math. Soc., Providence (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by grants from CNPq and FAPESP.
Rights and permissions
About this article
Cite this article
de Andrade, E.X.L., Bracciali, C.F. & Sri Ranga, A. Zeros of Gegenbauer-Sobolev Orthogonal Polynomials: Beyond Coherent Pairs. Acta Appl Math 105, 65–82 (2009). https://doi.org/10.1007/s10440-008-9265-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-008-9265-8
Keywords
- Gegenbauer polynomials
- Zeros of Gegenbauer-Sobolev orthogonal polynomials
- Symmetrically coherent pairs