Mathematische Zeitschrift

, 262:895 | Cite as

Continuous wavelets on compact manifolds

  • Daryl GellerEmail author
  • Azita Mayeli


Let M be a smooth compact oriented Riemannian manifold, and let Δ M 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 K t (x, y) denote the kernel of f (t 2 Δ M ). We show that K t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t 2Δ) on \({\mathbb {R}^n}\) . We define continuous \({\mathcal {S}}\)-wavelets on M, in such a manner that K t (x, y) satisfies this definition, because of its localization near the diagonal. Continuous \({\mathcal {S}}\)-wavelets on M are analogous to continuous wavelets on \({\mathbb {R}^n}\) in \({\mathcal {S}}\) (\({\mathbb {R}^n}\)). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous \({\mathcal {S}}\)-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus \({\mathbb T^2}\) or the sphere S 2, and f (s)  =  se s (the “Mexican hat” situation), we obtain two explicit approximate formulas for K t , one to be used when t is large, and one to be used when t is small.


Frames Wavelets Continuous Wavelets Spectral Theory Schwartz Functions Time-Frequency Analysis Manifolds Sphere Torus Pseudodifferential Operators Hölder Spaces 

Mathematics Subject Classification (2000)

42C40 42B20 58J40 58J35 35P05 


  1. 1.
    Antoine P.-J., Vandergheynst P.: Wavelets on the n-sphere and related manifolds. J. Math. Phys. 39, 3987–4008 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D.: Asymptotics for Spherical Needlets. Ann. Stat. (in press) arxiv:math/0606599Google Scholar
  3. 3.
    Bogdanova I., Vandergheynst P., Antoine P.-J., Jacques L., Morvidone M.: Stereographic wavelet frames on the sphere. Appl. Comput. Harmon. Anal. 19, 223–252 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cahn S.R., Wolf A.J.: Zeta functions and their asymptotics expansions for compact symmetric spaces of rank one. Comment. Math. Helv. 51, 1–21 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Camporesi R.: Harmonic analysis and propagators on homogeneous spaces. Phys. Rep. 196, 1–134 (1990)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Coifman R., Maggioni M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21, 53–94 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Coifman R.R., Weiss G.: Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. #242. Springer, Berlin (1971)Google Scholar
  8. 8.
    Dahlke, S.: Multiresolution analysis, Haar bases and wavelets on Riemannian manifolds, in Wavelets: Theory, Algorithms and Applications, Taormina, 33–52 (1993)Google Scholar
  9. 9.
    Dahmen W., Schneider R.: Wavelets on manifolds I: construction and domain decomposition. Siam J. Math. Anal. 31, 184–230 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Daubechies I.: Ten Lectures on Wavelets. Pennsylvania, Philadelphia (1992)zbMATHGoogle Scholar
  11. 11.
    Frazier M, Jawerth B: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Frazier M., Jawerth B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Geller D.: The Laplacian and the Kohn Laplacian for the sphere. J. Differ. Geom. 15, 417–455 (1980)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Freeden W., Gervens T., Schreiner M.: Constructive Approximation on the Sphere with Applications to Geomathematics. Clarendon Press, Oxford (1998)zbMATHGoogle Scholar
  15. 15.
    Freeden W., Volker M.: Multiscale Potential Theory with Applications to Geoscience. Birkhauser, Boston (2004)zbMATHGoogle Scholar
  16. 16.
    Geller D., Mayeli A.: Continuous wavelets and frames on stratified Lie groups I. J. Fourier Anal. Appl. 12, 543–579 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Geller, D., Mayeli, A.: Nearly Tight Frames and Space-Frequency Analysis on Compact Manifolds. Math. Z. (2008) (to appear)Google Scholar
  18. 18.
    Geller, D., Mayeli, A.: Besov spaces and frames on compact manifolds (2008), available on arXivGoogle Scholar
  19. 19.
    Gilkey P.: The spectral geometry of a Riemannian manifold. J. Differ. Geom. 10, 601–618 (1975)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Gilkey P.: Invariance Theory, the Heat Equation and the Atiyah–Singer Index Theorem. Publish or Perish Inc., Wilmington (1984)zbMATHGoogle Scholar
  21. 21.
    Guilloux, F., Faÿ, G., Cardoso J-F.: Practical wavelet design on the sphere. arXiv:0706.2598 (18 June 2007)Google Scholar
  22. 22.
    Han S.Y.: Calderón-type reproducing formula and the Tb theorem. Rev. Mat. Ibero. 10, 51–91 (1994)zbMATHGoogle Scholar
  23. 23.
    Han Y.: Discrete Calderón-type reproducing formula. Acta Math. Sin. 16, 277–294 (2006)CrossRefGoogle Scholar
  24. 24.
    Holschneider, M., Tchamitchian , Ph.: Régularité locale de la fonction ‘non-différentiable’ de Riemann. In: Les ondelettes en 1989, Lecture Notes in Mathematics, no. 1438, pp. 102–124. Springer, BerlinGoogle Scholar
  25. 25.
    Hörmander L.: The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kannai Y.: Off diagonal short term asymptotics for fundamental solutions of diffusion equations. Comm. PDE 2, 781–830 (1977)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Lan, X., Marinucci, D.: On the dependence structure of wavelet coefficiets for spherical random fields (2008) (preprint)Google Scholar
  28. 28.
    Mayeli, A.: Asymptotic Uncorrelation for Mexican Needlets. (2008) (preprint)Google Scholar
  29. 29.
    Maggioni, M., Davis, L.G., Warner, J.F., Geshwind, B.F., Coppi, C.A., DeVerse, R.A., Coifman, R.R.: Spectral analysis of normal and malignant tissue sections using a novel micro-optoelectricalmechanical system. Mod. Pathol 17(Suppl 1), 358AGoogle Scholar
  30. 30.
    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 (5 July 2007)Google Scholar
  31. 31.
    Mayeli, A.: Discrete and continuous wavelet transformation on the Heisenberg group, PhD thesis, Technische Universität München (2005)Google Scholar
  32. 32.
    Minakshisundaram S., Plegel A.: Some properties of the eignenfunctions of the Laplace-operator on Riemannian manifolds. Can. J. Math 1, 242–256 (1949)zbMATHGoogle Scholar
  33. 33.
    Milnor J.: Morse Theory, Ann. of Math. Studies, vol. 51. Princeton University Press, Princeton (1963)Google Scholar
  34. 34.
    Narcowich J.F., Petrushev P., Ward J.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Narcowich J.F., Petrushev P., Ward J.: Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Func. Anal. 238, 530–564 (2006)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Polterovich I.: Heat invariants of Riemannian manifolds. Israel J. Math 119, 239–252 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Polterovich I.: Combinatorics of the heat trace on spheres. Canad. J. Math 54, 1086–1099 (2002)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Schröder, P., Sweldens, W.: Spherical wavelets: efficiently representing functions on a sphere, Computer Graphics Proceedings (SIGGRAPH 95), 161–172 (1985)Google Scholar
  39. 39.
    Seeger A., Sogge D.C.: On the boundedness of functions of (pseudo-) differential operators on compact manifolds. Duke Math. J. 59, 709–736 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Sogge C.: Fourier Integrals in Classical Analysis. Cambridge University Press, (1993)Google Scholar
  41. 41.
    Stein E.M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)zbMATHGoogle Scholar
  42. 42.
    Strichartz R.: A functional calculus for elliptic pseudodifferential operators. Amer. J. Math 94, 711–722 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Tao T.: Weak-type endpoint bounds for Riesz means. Proc. AMS 124, 2797–2805 (1996)zbMATHCrossRefGoogle Scholar
  44. 44.
    Taylor M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA

Personalised recommendations