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On Approximation Methods by Using Orthogonal Polynomial Expansions

  • Rupert Lasser
  • Detlef H. Mache
  • Josef Obermaier
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 142)

Abstract

The following investigations start from a general point of view. Therefore let (P n )n∈N 0 be an orthogonal polynomial sequence (OPS) on the real line with respect to a probability measure π with compact support S and card(S) = ∞. The polynomials P n are assumed to be real valued with deg(P n ) = n.

Keywords

Chebyshev Polynomial Approximate Identity Positive Linear Operator Orthogonal Expansion Durrmeyer Operator 
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.

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References

  1. [1]
    Bloom, W.R., Heyer, H.,Harmonic Analysis of Probability Measures on Hypergroups, de Gruyter, Berlin—New York, (1995).zbMATHCrossRefGoogle Scholar
  2. [2]
    Butzer, P.L., Nessel, R.J., Fourier analysis and approximation, Birkhäuser, Basel—Stuttgart, (1971).zbMATHCrossRefGoogle Scholar
  3. [3]
    Chihara, T.S., An introduction to orthogonal polynomials, Gordon and Breach, New York, (1978).zbMATHGoogle Scholar
  4. [4]
    Gasper, G., Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math., VOL. 95, (1972), 261–280.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Laine, T.P., The product formula and convolution structure for the generalized Chebyshev polynomials, SIAM J. Math. Anal., VOL. 11, (1980), 133–146.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Lasser, R., Orthogonal polynomials and hypergroups, Rend. Mat., VOL. 3, (1983), 185–209.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Lasser, R., Introduction to Fourier Series, Marcel Dekker, New York, (1996).zbMATHGoogle Scholar
  8. [8]
    Lasser, R., Obermaier, J., On Fejér means with respect to orthogonal polynomials: A hypergroup theoretic approach, in: Progress of Approximation Theorie, Academic Press, Boston, (1991), 551–565.Google Scholar
  9. [9]
    Lasser, R., Obermaier, J., On the convergence of weighted Fourier expansions, Acta. Sci. Math., VOL. 61, (1995), 345–355.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Lasser, R., Obermaier, J., Orthogonal Expansions for LP and C-spaces, in: Special Functions, Proceedings of the International Workshop, World Scientific Publishing, Singapore–London, (2000), 194–206.Google Scholar
  11. [11]
    Lenze, B., On Lipschitz-type maximal functions and their smoothness spaces, Proc. Netherl. Acad. Sci. A 91 (1988), 53–63.MathSciNetGoogle Scholar
  12. [12]
    Lubinsky, D.S. and Mache, D.H. (C, 1) Means of Orthonormal Expansions for Exponential Weights, Journal of Approximation Theory, VOL. 103, No. 1, (2000), 151–182.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Lupas, A. & Mache, D.H.; The Θ-Transformation of Certain Positive Linear Operators,in: International Journal of Mathematics and Mathematical Sciences,VOL. 19, No. 4, (1996), 667–678.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Mache, D.H.; Generalized Methods using Integral Transforms and Convolution Structures with Jacobi Orthogonal Polynomials, Journal Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 68 (2002), 625–640.MathSciNetGoogle Scholar
  15. [15]
    Mache, D.H.; Summation of Durrmeyer Operators with Chebyshev Weights, in: Approximation and Optimization, Proceedings of the Int. Conf. on Approximation and Optimization–ICAOR, Cluj–Napoca (1996), VOL. I, 307–318.Google Scholar
  16. [16]
    Mache, D.H.; Optimale Konvergenzordnung positiver Summationsverfahren der Dui r ineyer Operatoren mit Jacobi-Gewichtungen; Habilitationsschrift, Dortmund (1997).Google Scholar
  17. [17]
    Mache, D.H. & Zhou, D.X.; Characterization Theorems for the Approximation by a Family of Operators,in: Journal of Approximation Theory,VOL. 84, No. 2, (1996), 145–161.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Obermaier, J., The de la Vallée Poussin Kernel for Orthogonal Polynomial Systems, Analaysis, VoL. 21, (2001), 277–288.MathSciNetzbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 2002

Authors and Affiliations

  • Rupert Lasser
    • 1
  • Detlef H. Mache
    • 1
  • Josef Obermaier
    • 2
  1. 1.Institut f. Biomathematik und BiometrieGSF-Forschungszentrum für Umwelt und GesundheitNeuherbergGermany
  2. 2.Institut f. Angewandte MathematikUniversität DortmundDortmundGermany

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