Advances in Computational Mathematics

, Volume 21, Issue 1–2, pp 147–180 | Cite as

Coorbit Spaces and Banach Frames on Homogeneous Spaces with Applications to the Sphere

  • Stephan Dahlke
  • Gabriele Steidl
  • Gerd Teschke
Article

Abstract

This paper is concerned with the construction of generalized Banach frames on homogeneous spaces. The major tool is a unitary group representation which is square integrable modulo a certain subgroup. By means of this representation, generalized coorbit spaces can be defined. Moreover, we can construct a specific reproducing kernel which, after a judicious discretization, gives rise to atomic decompositions for these coorbit spaces. Furthermore, we show that under certain additional conditions our discretization method generates Banach frames. We also discuss nonlinear approximation schemes based on the atomic decomposition. As a classical example, we apply our construction to the problem of analyzing and approximating functions on the spheres.

square integrable group representations time-frequency analysis atomic decompositions frames homogeneous spaces coorbit spaces modulation spaces nonlinear approximation spheres 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975).Google Scholar
  2. [2]
    S.T. Ali, J.-P. Antoine, J.-P. Gazeau and U.A. Mueller, Coherent states and their generalizations: A mathematical overview, Rev. Math. Phys. 39 (1998) 3987–4008.Google Scholar
  3. [3]
    J.-P. Antoine, L. Jaques and P. Vandergheynst, Wavelets on the sphere: Implementation and approximation, Preprint, Universitá Catholique de Louvain (2000).Google Scholar
  4. [4]
    J.-P. Antoine and P. Vandergheynst, Wavelets on the n-sphere and other manifolds, J. Math. Phys. 7 (1995) 1013–1104.Google Scholar
  5. [5]
    J.-P. Antoine and P. Vandergheynst, Wavelets on the 2-sphere: A group theoretical approach, Appl. Comput. Harmon. Anal. 7 (1999) 1–30.Google Scholar
  6. [6]
    A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations-Convergence rates, Math. Comp. 70 (2001) 27–75.Google Scholar
  7. [7]
    A. Cohen, I. Daubechies and J. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992) 485–560.Google Scholar
  8. [8]
    S. Dahlke and R. DeVore, Besov regularity for elliptic boundary value problems, Comm. Partial Differential Equations 22(1/2) (1997) 1–16.Google Scholar
  9. [9]
    I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61 (SIAM, Philadelphia, PA, 1992).Google Scholar
  10. [10]
    R. DeVore, Nonlinear approximation, Acta Numerica 7 (1998) 51–150.Google Scholar
  11. [11]
    R. DeVore, B. Jawerth and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992) 737–785.Google Scholar
  12. [12]
    R. DeVore and V.N. Temlyakov, Some remarks on greedy algorithms, Adv. in Comput. Math. 5 (1996) 173–187.Google Scholar
  13. [13]
    H.G. Feichtinger, Minimal Banach spaces and atomic decompositions, Publ. Math. Debrecen 33 (1986) 167–168 and 34 (1987) 231-240.Google Scholar
  14. [14]
    H.G. Feichtinger, Atomic characterization of modulation spaces through Gabor-type representations, in: Proc. of Conf. “Constructive Function Theory”, Edmonton, 1986, Rocky Mount. J. Math. 19 (1989) 113–126.Google Scholar
  15. [15]
    H.G. Feichtinger and K. GrÖchenig, A unified approach to atomic decompositions via integrable group representations, in: Proc. of Conf. “Function Spaces and Applications”, Lund, 1986, Lecture Notes in Mathematics, Vol. 1302 (Springer, New York, 1988) pp. 52–73.Google Scholar
  16. [16]
    H.G. Feichtinger and K. GrÖchenig, Banach spaces related to integrable group representations and their atomic decomposition I, J. Funct. Anal. 86 (1989) 307–340.Google Scholar
  17. [17]
    H.G. Feichtinger and K. GrÖchenig, Banach spaces related to integrable group representations and their atomic decomposition II, Monatsh. Math. 108 (1989) 129–148.Google Scholar
  18. [18]
    H.G. Feichtinger and K. GrÖchenig, Non-orthogonal wavelet and Gabor expansions and group representations, in: Wavelets and Their Applications, eds. M.B. Ruskai et al. (Jones and Bartlett, Boston, 1992) pp. 353–376.Google Scholar
  19. [19]
    G.B. Folland, Real Analysis (Wiley, New York, 1984).Google Scholar
  20. [20]
    D. Gabor, Theory of communication, J. Inst. Elect. Engrg. 93 (1946) 429–457.Google Scholar
  21. [21]
    R. Gilmore, Geometry of symmetrisized states, Ann. Phys. (NY) 74 (1972) 391–463.Google Scholar
  22. [22]
    R. Gilmore, On properties of coherent states, Rev. Mex. Fis. 23 (1974) 143–187.Google Scholar
  23. [23]
    K. GrÖchenig, Describing functions: Atomic decomposition versus frames, Monatsh. Math. 112 (1991) 1–42.Google Scholar
  24. [24]
    K. GrÖchenig, Foundations of Time-Frequency Analysis (Birkhäuser, Basel, 2001).Google Scholar
  25. [25]
    K. GrÖchenig and S. Samarah, Nonlinear approximation with local Fourier bases, Constr. Approx. 16 (2000) 317–331.Google Scholar
  26. [26]
    A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984) 723–736.Google Scholar
  27. [27]
    A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representations, II. Examples, Ann. Inst. H. Poincará 45 (1986) 293–309.Google Scholar
  28. [28]
    E. Hernandez and G. Weiss, A First Course on Wavelets (CRC Press, Boca Raton, FL, 1996).Google Scholar
  29. [29]
    J.A. Hogan and J.D. Lakey, Extensions of the Heisenberg group by dilations and frames, Appl. Comput. Harmon. Anal. 2(2) (1995) 174–199.Google Scholar
  30. [30]
    A.K. Louis, P. Maass and A. Rieder, Wavelets. Theory and Applications (Wiley, Chichester, 1997).Google Scholar
  31. [31]
    S. Mallat, A Wavelet Tour of Signal Processing(Academic Press, San Diego, 1999).Google Scholar
  32. [32]
    Y. Meyer, Wavelets and Operators, Cambridge Studies in Advanced Mathematics, Vol. 37 (Cambridge Univ. Press, Cambridge, 1992).Google Scholar
  33. [33]
    A. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin, 1978).Google Scholar
  34. [34]
    W. Schempp and B. Dreseler, EinfÜhrung in die harmonische Analyse (B.G. Teubner, Stuttgart, 1980).Google Scholar
  35. [35]
    B. Torresani, Position-frequency analysis for signals defined on spheres, Signal Process. 43(3) (1995) 341–346.Google Scholar
  36. [36]
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978).Google Scholar
  37. [37]
    P. Wojtasczyk, A Mathematical Introduction to Wavelets (Cambridge Univ. Press, Cambridge, 1997).Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Stephan Dahlke
    • 1
  • Gabriele Steidl
    • 2
  • Gerd Teschke
    • 3
  1. 1.Fachbereich Mathematik und InformatikUniversität MarburgMarburgGermany
  2. 2.Fakultät für Mathematik und InformatikUniversität MannheimMannheimGermany
  3. 3.Universität BremenBremenGermany

Personalised recommendations