Sampling of Paley-Wiener functions on stratified groups

  • Isaac Pesenson
Article

Abstract

We consider a generalization of entire functions of spherical exponential type on stratified groups. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a stratified group is uniquely determined by its values on some discrete subgroups. The main result of the article is reconstruction formula of spectral entire functions from their values on discrete subgroups using Lagrangian splines.

Math Subject Classifications

Primary 43A80 secondary 41A15, 41A17 Keywords and Phrases Stratified group sub-Laplacian Bernstein inequality Lagrangian spline 

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References

  1. [1]
    Benedetto, J. (1992). Irregular sampling and frames, inWavelets: A Tutorial in Theory and Applications, Chui, C.K. Ed., Academic Press, Boston, 445–507.Google Scholar
  2. [2]
    Ter Elst, A.F.M. and Robinson, D.W. (1994). Subelliptic operators on Lie groups: regularity,J. Austral. Math. Soc., (Series A), 57, 179–229.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Folland, G. (1975). Subelliptic estimates and function spaces on nilpotent Lie groups,Ark. Mat., 13, 161–207.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Krein, S. and Pesenson, I. (1990)Interpolation Spaces and Approximation on Lie Groups, The Voronezh State University, Voronezh, (in Russian), 200.Google Scholar
  5. [5]
    Lemarie, P. (1990). Bases d'ondelettes sur les groupes stratifies,Bull. Soc. Math. France, 177, 211–232.MathSciNetGoogle Scholar
  6. [6]
    Lions, J.-L. and Magenes, E. (1975),Non-Homogeneous Boundary Value Problem and Applications, Springer-Verlag.Google Scholar
  7. [7]
    Meyer, Y. (1990).Ondelettes et Operateurs, 2 volumes, Hermann, Paris.MATHGoogle Scholar
  8. [8]
    Pesenson, I. (1983). The Nikol'skii-Besov Spaces in Representations of Lie Groups.Dokl. Acad. Nauk USSR, 273(1), 45–49; English translation inSoviet Math. Dokl., 28, (1983).MATHMathSciNetGoogle Scholar
  9. [9]
    Pesenson, I. (1990). The Bernstein Inequality in the Space of Representation of Lie group.Dokl. Acad. Nauk USSR, 313(4), 86–90; English translation inSoviet Math. Dokl., 42, (1991).Google Scholar
  10. [10]
    Pesenson, I. (1990). Approximation in Representation Space of a Lie Group.Izvestiya VUZ, Mathematika, 34(7), 43–50; English translation inSoviet Math., 34(7), (1991).MATHMathSciNetGoogle Scholar
  11. [11]
    Pesenson, I. (1995).Lagrangian Splines, Spectral Entire Functions and Shannon-Whittaker Theorem on Manifolds, Temple University Research Report 95-87, 1–28.Google Scholar
  12. [12]
    Pesenson, I. (1996). A sampling theorem on homogeneous manifolds, submitted.Google Scholar

Copyright information

© Birkhäuser Boston 1998

Authors and Affiliations

  • Isaac Pesenson
    • 1
  1. 1.Department of Mathematics 038-16Temple UniversityPhiladelphia

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