Mathematische Zeitschrift

, Volume 263, Issue 2, pp 235–264 | Cite as

Nearly tight frames and space-frequency analysis on compact manifolds

Article

Abstract

Let M be a smooth compact oriented Riemannian manifold of dimension n without boundary, and let Δ be the Laplace–Beltrami operator on M. Say \({0 \neq f \in \mathcal{S}(\mathbb R^+)}\) , and that f (0)  =  0. For t  >  0, let Kt(x, y) denote the kernel of f (t2 Δ). Suppose f satisfies Daubechies’ criterion, and b  >  0. For each j, write M as a disjoint union of measurable sets Ej,k with diameter at most baj, and measure comparable to \({(ba^j)^n}\) if baj is sufficiently small. Take xj,kEj,k. We then show that the functions \({\phi_{j,k}(x)=\mu(E_{j,k})^{1/2} \overline{K_{a^j}}(x_{j,k},x)}\) form a frame for (I  −  P)L2(M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ (I  −  P)L2 is in space and in frequency, we can describe which terms in the summation \({F \sim SF = \sum_j \sum_k \langle F,\phi_{j,k} \rangle \phi_{j,k}}\) are so small that they can be neglected. If n  =  2 and M is the torus or the sphere, and f (s)  =  ses (the “Mexican hat” situation), we obtain two explicit approximate formulas for the φj,k, one to be used when t is large, and one to be used when t is small.

Keywords

Frames Wavelets Continuous wavelets Spectral theory Schwartz functions Time–frequency analysis Manifolds Sphere Torus Pseudodifferential operators 

Mathematics Subject Classification (2000)

42C40 42B20 58J40 58J35 35P05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Marinucci, D., Pietrobon, D., Balbi, A., Baldi, P., Cabella, P., Kerkyacharian, G., Natoli, P., Picard, D., Vittorio, N.: Spherical needlets for CMB data analysis, arXiv:0707.0844 (2007)Google Scholar
  2. 2.
    Daubechies I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)MATHGoogle Scholar
  3. 3.
    Fabes E., Mitrea I., Mitrea M.: On the boundedness of singular integrals. Pac. J. Math. 189, 21–29 (1999)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    David G., Journé J.-L.: A boundedness criterion for generalized Calderón–Zygmund operators. Ann. Math. 120, 371–397 (1984)CrossRefGoogle Scholar
  5. 5.
    Doroshkevich A.G., Naselsky P.D., Verkhodanov O.V., Novikov D.I., Turchaninov I.V., Novikov I.D., Christensen P.R., Chiang L.-Y.: Gauss–Legendre sky pixelization (GLESP) for CMB maps. Int. J. Mod. Phys. D. 14, 275–290 (2005)MATHCrossRefGoogle Scholar
  6. 6.
    Frazier M., Jawerth B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Frazier M., Jawerth B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–70 (1990)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Geller D., Mayeli A.: Continuous wavelets and frames on stratified Lie groups I. J. Fourier Anal. Appl. 12, 543–579 (2006)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Geller, D., Mayeli, A.: Continuous wavelets on manifolds. Math. Z. (2008) (to appear)Google Scholar
  10. 10.
    Geller, D., Mayeli, A.: Besov Spaces and Frames on Compact Manifolds, available on arXivGoogle Scholar
  11. 11.
    Gilbert J.E., Han Y.S., Hogan J.A., Lakey J.D., Weiland D., Weiss G.: Smooth molecular decompositions of functions and singular integral operators. Memoirs AMS 156, 742 (2002)MathSciNetGoogle Scholar
  12. 12.
    Guilloux, F., Faÿ, G., Cardoso, J.-F.: Practical wavelet design on the sphere, arXiv:0706.2598 (2007)Google Scholar
  13. 13.
    Han Y.: Discrete Calderón-type reproducing formula. Acta Math. Sinica 16, 277–294 (2000)MATHCrossRefGoogle Scholar
  14. 14.
    Narcowich F.J., Petrushev P., Ward J.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Narcowich F.J., Petrushev P., Ward J.: Decomposition of Besov and Triebel–Lizorkin spaces on the sphere. J. Func. Anal. 238, 530–564 (2006)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA

Personalised recommendations