Journal of Geometric Analysis

, Volume 21, Issue 2, pp 334–371 | Cite as

Band-Limited Localized Parseval Frames and Besov Spaces on Compact Homogeneous Manifolds



In the last decade, methods based on various kinds of spherical wavelet bases have found applications in virtually all areas where analysis of spherical data is required, including cosmology, weather prediction, and geodesy. In particular, the so-called needlets (= band-limited Parseval frames) have become an important tool for the analysis of Cosmic Microwave Background (CMB) temperature data. The goal of the present paper is to construct band-limited and highly localized Parseval frames on general compact homogeneous manifolds. Our construction can be considered as an analogue of the well-known φ-transform on Euclidean spaces.


Compact homogeneous manifold Wavelets Laplace operator Eigenfunctions 

Mathematics Subject Classification (2000)

43A85 42C40 41A17 41A10 


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  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) MATHGoogle Scholar
  2. 2.
    Antoine, J.-P., Vandergheynst, P.: Wavelets on the sphere: a group-theoretic approach. Appl. Comput. Harmon. Anal. 7, 262–291 (1999) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Antoine, J.-P., Vandergheynst, P.: Wavelets on the sphere and other conic sections. J. Fourier Anal. Appl. 13, 369–386 (2007) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D.: Asymptotics for spherical needlets. Ann. Stat. 37(3), 1150–1171 (2009). MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D.: Subsampling needlet coefficients on the sphere. Bernoulli 15, 438–463 (2009). arxiv:0706.4169 MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Butzer, P., Berens, H.: Semi-Groups of Operators and Approximation. Springer, Berlin (1967) MATHGoogle Scholar
  7. 7.
    Cabella, P., Hansen, F.K., Marinucci, D., Pagano, D., Vittorio, N.: Search for non-Gaussianity in pixel, harmonic, and wavelet space: compared and combined. Phys. Rev. D 69, 063007 (2004) CrossRefGoogle Scholar
  8. 8.
    Cruz, M., Cayon, L., Martinez-Gonzalez, E., Vielva, P., Jin, J.: The non-Gaussian cold spot in the 3-year WMAP data. Astrophys. J. 655, 11–20 (2007) CrossRefGoogle Scholar
  9. 9.
    Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Freeden, W.: Multiscale Modelling of Spaceborne Geodata. European Consortium for Mathematics in Industry. B.G. Teubner, Stuttgart (1999). 351 pp. ISBN: 3-519-02600-7 MATHGoogle Scholar
  11. 11.
    Freeden, W., Michel, V., Nutz, H.: Satellite-to-satellite tracking and satellite gravity gradiometry (advanced techniques for high-resolution geopotential field determination). J. Eng. Math. 43(1), 19–56 (2002) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Garrigs, G., Eugenio, H., Sikic, H., Soria, F., Weiss, G., Wilson, E.: Connectivity in the set of tight frame wavelets (TFW). Glasg. Mat. Ser. III 38(58), 75–98 (2003). no. 1 CrossRefGoogle Scholar
  13. 13.
    Geller, D., Marinucci, D.: Spin wavelets on the sphere (2008). arxiv:0811.2935
  14. 14.
    Geller, D., Mayeli, A.: Continuous wavelets and frames on stratified Lie groups I. J. Fourier Anal. Appl. 12, 543–579 (2006) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Geller, D., Mayeli, A.: Continuous wavelets on compact manifolds. Math. Z. 262, 895–927 (2009) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Geller, D., Mayeli, A.: Nearly tight frames and space-frequency analysis on compact manifolds. Math. Z. 263, 235–264 (2009) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Geller, D., Mayeli, A.: Besov spaces and frames on compact manifolds. Indiana Univ. Math. J. 58, 2003–2042 (2009) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Geller, D., Mayeli, A.: Nearly tight frames of spin wavelets on the sphere (2009). arxiv:0907.3164
  19. 19.
    Geller, D., Hansen, F.K., Marinucci, D., Kerkyacharian, G., Picard, D.: Spin needlets for cosmic microwave background polarization data analysis. Phys. Rev. D 78, 123533 (2008) CrossRefGoogle Scholar
  20. 20.
    Geller, D., Lan, X., Marinucci, D.: Spin needlets spectral estimation. Electron. J. Stat. 3, 1497–1530 (2009) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gorski, K.M., Lilje, P.B.: Foreground subtraction of cosmic microwave background maps using WI-FIT (Wavelet based high resolution fitting of internal templates). Astrophys. J. 648, 784–796 (2006) CrossRefGoogle Scholar
  22. 22.
    Gorski, K.M., Banday, A.J., Hivon, E., Wandelt, B.D.: HEALPix a framework for high resolution, fast analysis on the sphere. In: ADASS XI, p. 107 (2002) Google Scholar
  23. 23.
    Jin, J., Starck, J.-L., Donoho, D.L., Aghanim, N., Forni, O.: Cosmological non-Gaussian signature detection: comparing performance of different statistical tests. EURASIP J. Appl. Signal Process. 2470–2485 Google Scholar
  24. 24.
    Helgason, S.: Differential Geometry and Symmetric Spaces. Academic Press, New York (1962) MATHGoogle Scholar
  25. 25.
    Helgason, S.: Groups and Geometric Analysis. Academic Press, New York (1984) MATHGoogle Scholar
  26. 26.
    Marinucci, D., Pietrobon, D., Balbi, A., Baldi, P., Cabella, P., Kerkyacharian, G., Natoli, P., Picard, D., Vittorio, N.: Spherical needlets for CMB data analysis. Mon. Not. R. Astron. Soc. 383, 539–545 (2008) Google Scholar
  27. 27.
    McEwen, J.D., Hobson, M.P., Lasenby, A.N., Mortlock, D.J.: A high-significance detection of non-Gaussianity in the WMAP 3-year data using directional spherical wavelets. Mon. Not. R. Astron. Soc. 371(123002), L50–L54 (2006) Google Scholar
  28. 28.
    McEwen, J.D., Vielva, P., Hobson, M.P., Martinez-Gonzalez, E., Lasenby, A.N.: Detection of the integrated Sachs–Wolfe effect and corresponding dark energy constraints made with directional spherical wavelets. Mon. Not. R. Astron. Soc. 376(3), 1211–1226 (2007) CrossRefGoogle Scholar
  29. 29.
    Mhaskar, H.N.: Eignets for function approximation on manifolds. Appl. Comput. Harmon. Anal. (2010). doi:10.1016/j.acha.2009.08.006
  30. 30.
    Narcowich, F.J., Petrushev, P., Ward, J.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Narcowich, F.J., Petrushev, P., Ward, J.: Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal. 238, 530–564 (2006) MathSciNetMATHGoogle Scholar
  32. 32.
    Paluszynski, M., Sikic, H., Weiss, G., Xiao, S.: Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties. Frames. Adv. Comput. Math. 18(2–4), 297–327 (2003) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Peetre, J.: New Thoughts on Besov Spaces. Duke Univ. Math. Series, vol. 1. Dept. Math., Duke Univ., Durham (1976) MATHGoogle Scholar
  34. 34.
    Pesenson, I.: On interpolation spaces on Lie groups. Dokl. Akad. Nauk USSR 246, 1298–1303 (1979); English transl. in Soviet Math. Dokl. 20 (1979) MathSciNetGoogle Scholar
  35. 35.
    Pesenson, I.: Approximations in the representation space of a Lie group. Izv. Vyssh. Uchebn. Zaved. Mat. 7, 43–50 (1990) (Russian); translation in Soviet Math. (Iz. VUZ) 34(7), 49–57 (1990) MathSciNetGoogle Scholar
  36. 36.
    Pesenson, I.: The Nikol’skii-Besov spaces in representations of Lie groups. Dokl. Acad. Nauk, USSR 273(1), 45–49 (1983); Engl. transl. in Soviet Math. Dokl. 28 (1983) MathSciNetGoogle Scholar
  37. 37.
    Pesenson, I.: Abstract theory of Nikol’skii-Besov spaces, Izv. VUZ, Math. 59–70 (1988). Engl. Transl. in Soviet Math. 32(6) (1988) Google Scholar
  38. 38.
    Pesenson, I.: A sampling theorem on homogeneous manifolds. Trans. Am. Math. Soc. 352(9), 4257–4269 (2000) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Pesenson, I.: An approach to spectral problems on Riemannian manifolds. Pac. J. Math. 215(1), 183–199 (2004) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Pesenson, I.: Poincaré-type inequalities and reconstruction of Paley–Wiener functions on manifolds. J. Geom. Anal. 14(1), 101–121 (2004) MathSciNetMATHGoogle Scholar
  41. 41.
    Pesenson, I.: Bernstein–Nikolski inequality and Riesz interpolation formula on compact homogeneous manifolds. J. Approx. Theory 150(2), 175–198 (2008) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Pesenson, I.: Paley–Wiener approximations and multiscale approximations in Sobolev and Besov spaces on manifolds. J. Geom. Anal. 19(2), 390–419 (2009) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Seeley, R.T.: Complex powers of an elliptic operator. Proc. Symp. Pure Math. 10, 288–307 (1968) Google Scholar
  44. 44.
    Seeger, A., Sogge, C.D.: On the boundedness of functions of (pseudo-) differential operators on compact manifolds. Duke Math. J. 59, 709–736 (1989) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Sikic, H., Speegle, D., Weiss, G.: Structure of the set of dyadic PFW’s. In: Frames and Operator Theory in Analysis and Signal Processing. Contemp. Math., vol. 451, pp. 263–291. Amer. Math. Soc., Providence (2008) Google Scholar
  46. 46.
    Sogge, C.: Fourier Integrals in Classical Analysis. Cambridge University Press, Cambridge (1993) MATHCrossRefGoogle Scholar
  47. 47.
    Strichartz, R.: A functional calculus for elliptic pseudodifferential operators. Am. J. Math. 94, 711–722 (1972) MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981) MATHGoogle Scholar
  49. 49.
    Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983) CrossRefGoogle Scholar
  50. 50.
    Triebel, H.: Spaces of Besov–Hardy–Sobolev type on complete Riemannian manifolds. Ark. Mat. 24, 299–337 (1986) MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Triebel, H.: Theory of Function Spaces II. Monographs in Mathematics, vol. 84. Birkhäuser, Basel (1992) MATHCrossRefGoogle Scholar
  52. 52.
    Vielva, P., Martínez-González, E., Gallegos, J.E., Toffolatti, L., Sanz, J.L.: Point source detection using the spherical Mexican hat wavelet on simulated all-sky Planck maps. Mon. Not. R. Astron. Soc. 344(1), 89–104 (2003) CrossRefGoogle Scholar
  53. 53.
    Vielva, P., Martínez-González, E., Barreiro, B., Sanz, J., Cayon, L.: Detection of non-Gaussianity in the WMAP first year data using spherical wavelets. Astrophys. J. 609, 22–34 (2004) CrossRefGoogle Scholar
  54. 54.
    Vilenkin, N.: Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs, vol. 22. American Mathematical Society, Providence (1968). x+613 pp. MATHGoogle Scholar
  55. 55.
    Wiaux, Y., McEwen, J.D., Vandergheynst, P., Blanc, O.: Exact reconstruction with directional wavelets on the sphere. Mon. Not. R. Astron. Soc. 388(2), 770–788 (2008) CrossRefGoogle Scholar
  56. 56.
    Wiaux, Y., McEwen, J.D., Vielva, P.: Complex data processing: fast wavelet analysis on the sphere. J. Fourier Anal. Appl. 13, 477–494 (2007) MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Zelobenko, D.: Compact Lie Groups and Their Representations. Translations of Mathematical Monographs, vol. 40. American Mathematical Society, Providence (1973). viii+448 pp. MATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA

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