Abstract
The main goal of the article is to show that Paley-Wiener functions ƒ ∈ L 2(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 Φi is 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, xi ∈M.
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.
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References
Benedetto, J.J.Irregular Sampling and Frames, Academic Press, Boston, 445–507, (1992).
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).
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).
Hebey, E.Sobolev Spaces on Riemannian Manifolds, Springer-Verlag, Berlin, Heidelberg, (1996).
Krein, S. and Pesenson, I.Interpolation Spaces and Approximation on Lie Groups, The Voronezh State University, Voronezh, (1990), (Russian).
Lions, J.L. and Magenes, E.Non-Homogeneous Boundary Value Problem and Applications, Springer-Verlag, (1975).
Maheux, P. and Saloff-Coste, L. Analyse sur les boules d’un operateur sous-elliptique,Math. Annal.,303, 301–324, (1995).
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).
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).
Pesenson, I.Lagrangian Splines, Spectral Entire Functions and Shannon-Whittaker Theorem on Manifolds, Temple University, Research Report, 95-87, 1–28, (1995).
Pesenson, I. Reconstruction of Paley-Wiener functions on the Heisenberg group,Am. Math. Soc. Transl.,184(2), 207–216, (1998).
Pesenson, I. Sampling of Paley-Wiener functions on stratified groups,J. Fourier Anal. Appl.,4(3), 271–281, (1998).
Pesenson, I. Reconstruction of band limited functions inL 2(R d),Proceed. of AMS,127(12), 3593–3600, (1999).
Pesenson, I. A sampling theorem on homogeneous manifolds,Trans. AMS,352(9), 4257–4270, (2000).
Pesenson, I. Sampling of band limited vectors,J. Fourier Anal. Appl.,7(1), 93–100, (2001).
Roe, J. An index theorem on open manifolds,J. Diff. Geom.,27, 87–113; 115–136, (1988).
Schoenberg, I. Cardinal spline interpolation,CBMS,12, SIAM, Philadelphia, (1973).
Strichartz, R. Analysis of the Laplacian on the complete Riemannian manifold,J. Funct. Anal.,52, 48–79, (1983).
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Communicated by Guido Weiss
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Pesenson, I. Poincaré-Type inequalities and reconstruction of Paley-Wiener functions on manifolds. J Geom Anal 14, 101–121 (2004). https://doi.org/10.1007/BF02921868
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DOI: https://doi.org/10.1007/BF02921868