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Asymptotics of derivatives of orthogonal polynomials based on generalized Jacobi weights. Some new theorems and applications

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New Developments in Approximation Theory

Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 132))

Abstract

In this paper we state uniform asymptotic formulae for n → ∞ valid on the unit circle line for derivatives of (complex) orthogonal polynomials based on Jacobi-type weights with a finite number of power type singularities. By these results we settle the corresponding problems for generalized Jacobi polynomials on [-1,1], i.e. for the (real) orthogonal polynomials based on power type weights with inner singularities.

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References

  1. G. Szegö, Orthogonal Polynomials, AMS Coll. Publ., Vol. 23, Providence, RI, 1975 (4th ed).

    Google Scholar 

  2. V. M. Badkov, Uniform asymptotic representation of orthogonal polynomials, Proc. Steklov Inst. Math., 2 (1985), 5–41.

    Google Scholar 

  3. G. Szegő, On bi-orthogonal systems of trigonometric polynomials, MTA Matematikai Kutató Int. Közlemenyei, 8 (1963/64), 255–273.

    Google Scholar 

  4. Ya. L. Geronimus, Orthogonal Polynomials, Pergamon Press, 1960.

    Google Scholar 

  5. G. Freud, Orthogonal Polynomials, Pergamon Press, 1971.

    Google Scholar 

  6. P. G. Nevai, An asymptotic formula for the derivatives of orthogonal polynomials, SIAM J. Math. Anal, 10(3) (1979), 472–477.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Vértesi, One-sided convergence conditions for Lagrange interpolation based on Jacobi nodes, Acta Sci. Math. (Szeged), 45 (1983), 419–428.

    MATH  MathSciNet  Google Scholar 

  8. P. Vértesi, On the zeros of Jacobi polynomials, Studia Sci. Math. Hungar., 25 (1990), 401–405.

    MATH  MathSciNet  Google Scholar 

  9. P. Vértesi, Recent results on Hermite-Fejér interpolations of higher order, Israel Math. Conf., Proc, 4 (1991), 267–271.

    Google Scholar 

  10. N. Bernstein, On polynomials orthogonal on a finite interval. No. 51 in his Collected Works, Vol. 2, pp. 7–106, Izd. AN SSS, 1954 (Russian).

    Google Scholar 

  11. J. Szabados and P. Vértesi, Interpolation of functions, World Scientific Ltd., 1990.

    Google Scholar 

  12. V. M. Badkov, On the asymptotic and extremal properties of orthogonal polynomials, Proc. Steklov Inst. Math., 1994 (1), 37–82.

    MathSciNet  Google Scholar 

  13. G. Mastroianni and P. Vértesi, Some applications of Generalized Jacobi weights, Acta Math. Hungar. 77 (1997), 323–357.

    Article  MATH  MathSciNet  Google Scholar 

  14. P. Vértesi, Uniform asymptotics of derivatives of orthogonal polynomials based on generalized Jacobi weights, Acta Math. Hungar. 85 (1-2), 1999.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, H-1364, Budapest, P.O.B. 127, Hungary

    P. Vértesi

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  1. P. Vértesi
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Editor information

Editors and Affiliations

  1. Lehrstuhl VIII für Mathematik, Universität Dortmund, Vogelpothsweg 87, 44221, Dortmund, Germany

    Manfred W. Müller , Martin D. Buhmann , Detlef H. Mache  & Michael Felten , ,  & 

  2. Numerische Analysis, Ludwig-Maximilians-Universität München, Theresienstrasse 39, 80333, München, Germany

    Detlef H. Mache

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Vértesi, P. (1999). Asymptotics of derivatives of orthogonal polynomials based on generalized Jacobi weights. Some new theorems and applications. In: Müller, M.W., Buhmann, M.D., Mache, D.H., Felten, M. (eds) New Developments in Approximation Theory. ISNM International Series of Numerical Mathematics, vol 132. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8696-3_20

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  • DOI: https://doi.org/10.1007/978-3-0348-8696-3_20

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9733-4

  • Online ISBN: 978-3-0348-8696-3

  • eBook Packages: Springer Book Archive

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