The Journal of Geometric Analysis

, Volume 14, Issue 1, pp 101–121

Poincaré-Type inequalities and reconstruction of Paley-Wiener functions on manifolds



The main goal of the article is to show that Paley-Wiener functions ƒ ∈ L2(M) of a fixed band width to on a Riemannian manifold of bounded geometry M completely determined and can be reconstructed from a set of numbers Φi (ƒ), i ∈ ℕwhere Φiis a countable sequence of weighted integrals over a collection of “small” and “densely” distributed compact subsets. In particular, Φi, i ∈ ℕ,can be a sequence of weighted Dirac measures δxi, xiM.

It is shown that Paley-Wiener functions on M can be reconstructed as uniform limits of certain variational average spline functions.

To obtain these results we establish certain inequalities which are generalizations of the Poincaré-Wirtingen and Plancherel-Polya inequalities.

Our approach to the problem and most of our results are new even in the one-dimensional case.

Math Subject Classifications

42C05 41A17 41A65 43A85 46C99 

Key Words and Phrases

Poincaré inequality on manifolds Laplace-Beltrami operator band-limited functions on manifolds splines on manifolds sampling theorem on manifolds 


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  1. [1]
    Benedetto, J.J.Irregular Sampling and Frames, Academic Press, Boston, 445–507, (1992).Google Scholar
  2. [2]
    Cheeger, J., Gromov, M., and Taylor, M. Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds,J. Diff. Geom.,17, 15–53, (1982).MathSciNetMATHGoogle Scholar
  3. [3]
    Feichtinger, H. and Grochenig, K. Theory and practice of irregular sampling. Wavelets: mathematics and applications,Stud. Adv. Math., 305–363, CRC Press, Boca Raton, FL, (1994).Google Scholar
  4. [4]
    Hebey, E.Sobolev Spaces on Riemannian Manifolds, Springer-Verlag, Berlin, Heidelberg, (1996).MATHGoogle Scholar
  5. [5]
    Krein, S. and Pesenson, I.Interpolation Spaces and Approximation on Lie Groups, The Voronezh State University, Voronezh, (1990), (Russian).Google Scholar
  6. [6]
    Lions, J.L. and Magenes, E.Non-Homogeneous Boundary Value Problem and Applications, Springer-Verlag, (1975).Google Scholar
  7. [7]
    Maheux, P. and Saloff-Coste, L. Analyse sur les boules d’un operateur sous-elliptique,Math. Annal.,303, 301–324, (1995).CrossRefMathSciNetGoogle Scholar
  8. [8]
    Pesenson, I. The best approximation in a representation space of a Lie group,Dokl. Acad. Nauk USSR,302(5), 1055–1059, (1988), English transl. inSoviet Math. Dokl.,38(2), 384–388, (1989).Google Scholar
  9. [9]
    Pesenson, I. The Bernstein inequality in the space of representation of Lie group,Dokl. Acad. Nauk USSR,313, 86–90, (1990); English transl. inSoviet Math. Dokl.,42, (1991).Google Scholar
  10. [10]
    Pesenson, I.Lagrangian Splines, Spectral Entire Functions and Shannon-Whittaker Theorem on Manifolds, Temple University, Research Report, 95-87, 1–28, (1995).Google Scholar
  11. [11]
    Pesenson, I. Reconstruction of Paley-Wiener functions on the Heisenberg group,Am. Math. Soc. Transl.,184(2), 207–216, (1998).MathSciNetGoogle Scholar
  12. [12]
    Pesenson, I. Sampling of Paley-Wiener functions on stratified groups,J. Fourier Anal. Appl.,4(3), 271–281, (1998).CrossRefMathSciNetMATHGoogle Scholar
  13. [13]
    Pesenson, I. Reconstruction of band limited functions inL 2(R d),Proceed. of AMS,127(12), 3593–3600, (1999).CrossRefMathSciNetMATHGoogle Scholar
  14. [14]
    Pesenson, I. A sampling theorem on homogeneous manifolds,Trans. AMS,352(9), 4257–4270, (2000).CrossRefMathSciNetMATHGoogle Scholar
  15. [15]
    Pesenson, I. Sampling of band limited vectors,J. Fourier Anal. Appl.,7(1), 93–100, (2001).CrossRefMathSciNetMATHGoogle Scholar
  16. [16]
    Roe, J. An index theorem on open manifolds,J. Diff. Geom.,27, 87–113; 115–136, (1988).MathSciNetGoogle Scholar
  17. [17]
    Schoenberg, I. Cardinal spline interpolation,CBMS,12, SIAM, Philadelphia, (1973).Google Scholar
  18. [18]
    Strichartz, R. Analysis of the Laplacian on the complete Riemannian manifold,J. Funct. Anal.,52, 48–79, (1983).CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2004

Authors and Affiliations

  1. 1.Department of MathematicsTemple UniversityPhiladelphia

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