Acta Applicandae Mathematica

, Volume 94, Issue 2, pp 163–192 | Cite as

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

  • Francisco Marcellán
  • Juan José Moreno Balcázar
Article

Abstract

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.

Key words

Sobolev orthogonal polynomials asymptotics zeros 

Mathematics Subject Classifications (2000)

Primary 42C05 Secondary 33C45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alfaro, M., López, G., Rezola, M.L.: Some properties of zeros of Sobolev-type orthogonal polynomials. J. Comput. Appl. Math. 69, 171–179 (1996)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alfaro, M., Martínez-Finkelshtein, A., Rezola, M.L.: Asymptotics properties of balanced extremal Sobolev polynomials: coherent case. J. Approx. Theory 100, 44–59 (1999)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Alfaro, M., Moreno-Balcázar, J.J., Peña, A., Rezola, M.L.: Sobolev orthogonal polynomials: how to balance and asymptotics (submitted for publication).Google Scholar
  4. 4.
    Alfaro, M., Moreno-Balcázar, J.J., Pérez, T.E., Piñar, M.A., Rezola, M.L.: Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs. J. Comput. Appl. Math. 133, 141–150 (2001)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Alfaro, M., Moreno-Balcázar, J.J., Rezola, M.L.: Laguerre–Sobolev orthogonal polynomials: asymptotics for coherent pairs of type II. J. Approx. Theory 122, 79–96 (2003)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Alvarez-Nodarse, R., Moreno–Balcázar, J.J.: Asymptotic properties of generalized Laguerre orthogonal polynomials. Indag. Math. (N.S.) 15, 151–165 (2004)MATHMathSciNetGoogle Scholar
  7. 7.
    Andrews, G.E., Askey, R.: Classical orthogonal polynomials. In: Brezinski, C. et al. (eds.) Polynômes Orthogonaux et Applications, Lecture Notes in Mathematics, vol. 1171, pp. 36–62. Springer, Berlin Heidelberg New York (1985)Google Scholar
  8. 8.
    Area, I., Godoy, E., Marcellán, F.: Classification of all Δ-coherent pairs. Integral Transform. Spec. Funct. 9(1), 1–18 (2000)MATHMathSciNetGoogle Scholar
  9. 9.
    Area, I., Godoy, E., Marcellán, F., Moreno–Balcázar, J.J.: Δ-Sobolev orthogonal polynomials of Meixner type: asymptotics and limit relation. J. Comput. Appl. Math. 178, 21–36 (2005)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Cachafeiro, A., Marcellán, F., Moreno-Balcázar, J.J.: On asymptotic properties of Freud–Sobolev orthogonal polynomials. J. Approx. Theory 125, 26–41 (2003)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Castaño-García, L., Moreno–Balcázar, J.J.: A Mehler–Heine type formula for Hermite–Sobolev orthogonal polynomials. J. Comput. Appl. Math. 150, 25–35 (2003)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    de Bruin, M.G., Groenevelt, W.G.M., Marcellán, F., Meijer, H.G., Moreno–Balcázar, J.J.: Asymptotics and zeros of symmetrically coherent pairs of Hermite type (submitted for publication).Google Scholar
  13. 13.
    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)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Ditzian, Z., Lubinsky, D.S.: Jackson and smoothness theorems for Freud weights in L p (0 < p ≤ ∞). Constr. Approx. 13, 99–152 (1997)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Gautschi, W., Kuijlaars, A.B.J.: Zeros and critical points of Sobolev orthogonal polynomials. J. Approx. Theory 91, 117–137 (1997)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Iserles, A., Koch, P.E., Nørsett, S.P., Sanz–Serna, J.M.: Orthogonality and approximation in a Sobolev space. In: Mason, J.C., Cox, M.G. (eds.) Algorithms for Approximation, pp. 117–124. Chapman & Hall, London, UK (1990)Google Scholar
  17. 17.
    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)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Geronimo, J.S., Lubinsky, D.S., Marcellán, F.: Asymptotics for Sobolev orthogonal polynomials for exponential weights. Constr. Approx. 22, 309–346 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Geronimo, J.S., Van Assche, W.: Relative asymptotics for orthogonal polynomials with unbounded recurrence coefficients. J. Approx. Theory 62, 47–69 (1990)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Koekoek, R., Meijer, H.G.: A generalization of Laguerre polynomials. SIAM J. Math. Anal. 24(3), 768–782 (1993)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Levin, E., Lubinsky, D.S.: Orthogonal Polynomials for Exponential Weights. Springer, Berlin Heidelberg New York (2001)MATHGoogle Scholar
  22. 22.
    Lewis, D.C.: Polynomial least square approximations. Amer. J. Math. 69, 273–278 (1947)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    López-Lagomasino, G., Marcellán, F., Pijeira, H.: Logarithmic asymptotic of contracted Sobolev extremal polynomials on the real line. J. Approx. Theory (in press)Google Scholar
  24. 24.
    López-Lagomasino, G., Pijeira, H.: Zero location and n-th root asymptotics de Sobolev orthogonal polynomials. J. Approx. Theory 99, 30–43 (1999)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    López-Lagomasino, G., Pijeira, H., Pérez, I.: Sobolev orthogonal polynomials in the complex plane. J. Comput. Appl. Math. 127, 219–230 (2001)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Marcellán, F., Alfaro, M., Rezola, M.L.: Orthogonal polynomials on Sobolev spaces: old and new directions. J. Comput. Appl. Math. 48, 113–131 (1993)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Marcellán, F., Meijer, H.G., Pérez, T.E., Piñar, M.A.: An asymptotic result for Laguerre–Sobolev orthogonal polynomials. J. Comput. Appl. Math. 87, 87–94 (1997)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Marcellán, F., Martínez-Finkelshtein, A., Moreno–Balcázar, J.J.: k-coherence of measures with non-classical weights. In: Español, L., Varona, J.L.(eds.) MargaritaMathematica en memoria de José Javier (Chicho) Guadalupe Hernández, pp. 77–83. Servicio de Publicaciones Universitario de La Rioja, Spain (2001)Google Scholar
  29. 29.
    Marcellán, F., Moreno-Balcázar, J.J.: Strong and Plancherel–Rotach asymptotics of nondiagonal Laguerre–Sobolev orthogonal polynomials. J. Approx. Theory 110, 54–73 (2001)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Marcellán, F., Ronveaux, A.: A bibliography of Sobolev orthogonal polynomials. Internal Report. Universidad Carlos III de Madrid, Spain (July 2005)Google Scholar
  31. 31.
    Martínez-Finkelshtein, A.: Asymptotic properties of Sobolev orthogonal polynomials. J. Comput. Appl. Math. 99, 491–510 (1998)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Martínez-Finkelshtein, A.: Analytic aspects of Sobolev orthogonal polynomials revisited. J. Comput. Appl. Math. 127, 255–266 (2001)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Martínez-Finkelshtein, A., Moreno–Balcázar, J.J., Pérez, T.E., Piñar, M.A.: Asymptotics of Sobolev orthogonal polynomials for coherent pairs of measures. J. Approx. Theory 92, 280–293 (1998)CrossRefGoogle Scholar
  34. 34.
    Meijer, H.G.: A short history of orthogonal polynomials in a Sobolev space. I. The. non-discrete case. Nieuw Arch. Wisk. 14, 93–112 (1996)MATHMathSciNetGoogle Scholar
  35. 35.
    Meijer, H.G.: Determination of all coherent pairs. J. Approx. Theory 89, 321–343 (1997)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Meijer, H.G., de Bruin, M.G.: Zeros of Sobolev orthogonal polynomials following from coherent pairs. J. Comput. Appl. Math. 139, 253–274 (2002)MATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Meijer, H.G., Pérez, T.E., Piñar, M.A.: Asymptotics of Sobolev orthogonal polynomials for coherent pairs of Laguerre type. J. Math. Anal. Appl. 245, 528–546 (2000)MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Mhaskar, H.N., Saff, E.B.: Extremal problems for polynomials with exponential weights. Trans. Amer. Math. Soc. 285, 204–234 (1984)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Moreno-Balcázar, J.J.: Smallest zeros of some types of orthogonal polynomials: asymptotics. J. Comput. Appl. Math. 179, 289–301 (2005)MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Moreno-Balcázar, J.J.: A note on the zeros of Freud–Sobolev orthogonal polynomials. J. Comput. Appl. Math. (in press)Google Scholar
  41. 41.
    Pan, K.: On Sobolev orthogonal polynomials with coherent pairs: the Laguerre case, type I. J. Math. Anal. Appl. 223, 319–334 (1998)MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Szegő, G.: Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ. 23, 4th edn. Amer. Math. Soc., Providence, Rhode Island (1975)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • Francisco Marcellán
    • 1
  • Juan José Moreno Balcázar
    • 2
    • 3
  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain
  2. 2.Departamento de Estadística y Matemática AplicadaUniversidad de AlmeríaAlmeríaSpain
  3. 3.Instituto Carlos I de Física Teórica y ComputacionalUniversidad de GranadaGranadaSpain

Personalised recommendations