De la Vallée Poussin Means for the Hankel Transform

  • Wolfgang zu Castell
  • Frank Filbir
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 142)


We give a construction of a de la Vallée Poussin kernel for the Hankel transform based on the convolution structure on the space L 1 (R +, µ v ). In contrast to the classical way to define such a kernel, our construction directly leads to an approximate identity for the underlying space.


Bessel Function Orthogonal Polynomial Approximate Identity Convolution Theorem Underlying Space 
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  1. [1]
    Andrews, G., R. Askey & R. Roy, Special functions, Cambridge University Press, Cambridge, 1999.zbMATHGoogle Scholar
  2. [2]
    Betancor, J.J. & L. Rodriguez-Mesa, L 1 -convergence and strong summability of Hankel transforms, Publ. Math. Debrecen 55 No. 3–4 (1999), 437–451.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Butzer, P.L. & R.J. Nessel, Fourier analysis and approximation, Vol. I, Birkhäuser, Basel, 1971.zbMATHCrossRefGoogle Scholar
  4. [4]
    Chanillo, S. & B. Muckenhoupt, Weak type estimates for Bochner-Riesz spher-ical summation multipliers,Trans. Amer. Math. Soc. 294 No. 2 (1986), 693— 703.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Colzani, L., G. Travaglini & M. Vignati, Bochner-Riesz means of functions in weak-LP, Mh. Math. 115 (1993), 35–45.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Filbir, F. & W. Themistoclakis, On the construction of de la Vallée Poussin means for orthogonal polynomials using convolution structures,to appear in J. Comput. Anal. Appl.Google Scholar
  7. [7]
    Luke, Y.L., The special functions and their approximations, Vol. I, Academic Press, New York, 1969.zbMATHGoogle Scholar
  8. [8]
    Magnus, W., F. Oberhettinger & R.P. Soni, Formulas and theorems for the special functions of mathematical physics, Springer, Berlin, 1966.zbMATHGoogle Scholar
  9. [9]
    Stein, E.M. & G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, 1971.zbMATHGoogle Scholar
  10. [10]
    Stempak, K., The Littlewood-Paley theory for the Fourier-Bessel transform, University of Wroclaw, Preprint no. 45, 1985.Google Scholar
  11. [11]
    Stempak, K., La théorie de Littlewood-Paley pour la transformation de Fourier-Bessel, C.R. Acad. Sc. Paris 303 Série I (1986), 15–18.Google Scholar
  12. [12]
    Watson, G.N., A treatise on the theory of Bessel functions, reprint of the 2nd edition, Cambridge University press, Cambridge, 1996.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  • Wolfgang zu Castell
    • 1
  • Frank Filbir
    • 1
  1. 1.Institute of Biomathematics and BiometryGSF — National Research Center for Environment and HealthNeuherbergGermany

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