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Strong asymptotics for Sobolev orthogonal polynomials

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Abstract

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.

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References

  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.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Barrios, G. López and H. Pijeira,The moment problem for a Sobolev inner product, to appear in J. Approx. Theory.

  3. P. Duren,Theory of H p Spaces, Academic Press, New York, 1970.

    MATH  Google Scholar 

  4. W. Gautschi and A.B.J. Kuijlars,Zeros and critical points of Sobolev orthogonal polynomials, J. Approx. Theory91 (1997), 117–137.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  6. G. López and H. Pijeira,Zero location and n-th root asymptotics of Sobolev orthogonal polynomials, to appear in J. Approx. Theory.

  7. F. Marcellán and W. Van Assche,Relative asymptotics for orthogonal polynomials, J. Approx. Theory72 (1993), 193–209.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Martínez Finkelshtein,Asymptotic properties of Sobolev orthogonal polynomials, J. Comput. Appl. Math.99 (1998), 491–510.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Martínez Finkelshtein,Bernstein-Szegö's theorem for Sobolev orthogonal polynomials, to appear in Constr. Approx.

  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.

    MATH  MathSciNet  Google Scholar 

  11. H. Stahl and V. Totik,General Orthogonal Polynomials, Cambridge Univ. Press, Cambridge, 1992.

    MATH  Google Scholar 

  12. G. Szegö,Orthogonal Polynomials, 4th edition, Amer. Math. Soc., Providence, RI, 1975.

    MATH  Google Scholar 

  13. H. Widom,Extremal polynomials associated with a system of curves in the complex plane, Adv. Math.3 (1969), 127–232.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departmento de Estadística y Matemática Aplicada, Universidad de Almería, La Cañada, 04120, Almería, Spain

    Andrei Martínez Finkelshtein & Héctor Pijeira Cabrera

  2. Instituto Carlo I de Física Teórica y Computacional, Universidad de Granada, Spain

    Andrei Martínez Finkelshtein

  3. Departmento de Matemática, Universidad de Matanzas, Matanzas, Cuba

    Héctor Pijeira Cabrera

Authors
  1. Andrei Martínez Finkelshtein
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  2. Héctor Pijeira Cabrera
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Additional information

Research partially supported by a research grant from Dirección General de Enseñanza Superior (DGES) of Spain, project code PB95-1205, by INTAS project 93-219-ext, and by Junta de Andalucía, Grupo de Investigación FQM 0229.

Research carried out under grant from Agencia Española de Cooperación Internacional and Doctoral Program of the Universidad Carlos III de Madrid (Spain).

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Finkelshtein, A.M., Cabrera, H.P. Strong asymptotics for Sobolev orthogonal polynomials. J. Anal. Math. 78, 143–156 (1999). https://doi.org/10.1007/BF02791131

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  • DOI: https://doi.org/10.1007/BF02791131

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