Journal d’Analyse Mathématique

, Volume 78, Issue 1, pp 143–156 | Cite as

Strong asymptotics for Sobolev orthogonal polynomials

  • Andrei Martínez Finkelshtein
  • Héctor Pijeira Cabrera


In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product\(\left\langle {f,g} \right\rangle s = \sum\limits_{k - 0}^m {\int\limits_{\Delta _k } {f^{\left( k \right)} \left( x \right)g^{\left( k \right)} \left( x \right)d\mu \kappa } } \left( x \right)\) where\(\left\{ {\mu _\kappa } \right\}_{k = 0}^m ,m \in \mathbb{Z}_ + \), are measures supported on [−1,1] which satisfy Szegö's condition.


Compact Subset Orthogonal Polynomial Moment Problem Lebesgue Dominate Convergence Theorem Strong ASYMPTOTICS 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Alfaro, F. Marcellán and M.L. Rezola,Orthogonal polynomials on Sobolev spaces: old and new directions, J. Comput. Appl. Math.48 (1993), 113–132.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    D. Barrios, G. López and H. Pijeira,The moment problem for a Sobolev inner product, to appear in J. Approx. Theory.Google Scholar
  3. [3]
    P. Duren,Theory of H p Spaces, Academic Press, New York, 1970.zbMATHGoogle Scholar
  4. [4]
    W. Gautschi and A.B.J. Kuijlars,Zeros and critical points of Sobolev orthogonal polynomials, J. Approx. Theory91 (1997), 117–137.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    G. López, F. Marcellán and W. Van Assche,Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product, Constr. Approx.11 (1995), 107–137.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    G. López and H. Pijeira,Zero location and n-th root asymptotics of Sobolev orthogonal polynomials, to appear in J. Approx. Theory.Google Scholar
  7. [7]
    F. Marcellán and W. Van Assche,Relative asymptotics for orthogonal polynomials, J. Approx. Theory72 (1993), 193–209.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    A. Martínez Finkelshtein,Asymptotic properties of Sobolev orthogonal polynomials, J. Comput. Appl. Math.99 (1998), 491–510.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    A. Martínez Finkelshtein,Bernstein-Szegö's theorem for Sobolev orthogonal polynomials, to appear in Constr. Approx.Google Scholar
  10. [10]
    H.G. Meijer,A short history of orthogonal polynomials in Sobolev space I. The non-discrete case, Nieuw Archief voor Wiskunde14 (1996), 93–113.zbMATHMathSciNetGoogle Scholar
  11. [11]
    H. Stahl and V. Totik,General Orthogonal Polynomials, Cambridge Univ. Press, Cambridge, 1992.zbMATHGoogle Scholar
  12. [12]
    G. Szegö,Orthogonal Polynomials, 4th edition, Amer. Math. Soc., Providence, RI, 1975.zbMATHGoogle Scholar
  13. [13]
    H. Widom,Extremal polynomials associated with a system of curves in the complex plane, Adv. Math.3 (1969), 127–232.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Magnes Press, The Hebrew University 1999

Authors and Affiliations

  • Andrei Martínez Finkelshtein
    • 1
    • 2
  • Héctor Pijeira Cabrera
    • 1
    • 3
  1. 1.Departmento de Estadística y Matemática AplicadaUniversidad de AlmeríaAlmeríaSpain
  2. 2.Instituto Carlo I de Física Teórica y ComputacionalUniversidad de GranadaSpain
  3. 3.Departmento de MatemáticaUniversidad de MatanzasMatanzasCuba

Personalised recommendations