Multiresolution on the Sphere

  • Matthias Conrad
  • Jürgen Prestin
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

In this paper we study some basic tools for the construction of multi-scale systems on the unit sphere. Particularly, we emphasize properties of spherical harmonics and Legendre functions. Based on these orthogonal systems we discuss in some detail the decomposition of the classical Hilbert space on the sphere into subspaces of different level. To this end we explain different bases and frames. In our examples the building blocks consist of polynomials and spherical radial basis functions.

Keywords

Convolution 

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References

  1. 1.
    Å. Björck. Numerical Methods for Least Squares Problems. SIAM, Philadelphia, 1996.MATHCrossRefGoogle Scholar
  2. 2.
    W. Cheney. Approximation and interpolation on spheres. Approximation Theory, Wavelets and Applications, S. P. Singh (ed.), Kluwer Academic Publishers, Dordrecht, 1995, 47–53.Google Scholar
  3. 3.
    W. Cheney and W. Light. A Course in Approximation Theory. Brooks/Cole, Pacific Grove, 2000.Google Scholar
  4. 4.
    S. Dahlke, W. Dahmen, E. Schmitt, and I. Weinreich. Multiresolution analysis and wavelets on S 2 and S 3. Numer. Punc. Anal. Optim. 16, 1995, 19–41.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    B. Fischer and J. Prestin. Wavelets based on orthogonal polynomials. Math. Comp. 66, 1997, 1593–1618.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    G. B. Folland. Fourier Analysis and its Applications. Brooks/Cole, Pacific Grove, 1992.MATHGoogle Scholar
  7. 7.
    W. Freeden, T. Gervens, and M. Schreiner. Constructive Approximation on the Sphere: with Applications to Geomathematics. Clarendon Press, Oxford, 1998.MATHGoogle Scholar
  8. 8.
    D. Funaro. Polynomial Approximation of Differential Equations. Springer-Verlag, Berlin, 1992.MATHGoogle Scholar
  9. 9.
    R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985.MATHGoogle Scholar
  10. 10.
    A. Iske. Scattered data modelling using radial basis functions. This volume.Google Scholar
  11. 11.
    Y. W. Koh, S. L. Lee, and H. H. Tan. Periodic orthogonal splines and wavelets. Appl. Comput. Harmonic Anal. 2, 1995, 201–218.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    T.-H. Li. Multiscale representation and analysis of spherical data by spherical wavelets. SIAM J. Scient. Computing 21, 1999, 924–953.MATHCrossRefGoogle Scholar
  13. 13.
    T. Lyche and L. L. Schumaker. A multiresolution tensor spline method for fitting functions on the sphere. SIAM J. Scient. Computing 22, 2000, 724–746.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    V. A. Menegatto. Strict positive definiteness on spheres. Analysis 19, 1999, 217–233.MathSciNetMATHGoogle Scholar
  15. 15.
    H. N. Mhaskar, F. J. Narcowich, and J. D. Ward. Spherical Marcinkiewicz-Zymund inequalities and positive quadrature. Math. Comp. 70, 2001, 1113–1130.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    H. N. Mhaskar, F. J. Narcowich, J. Prestin, and J. D. Ward. Polynomial frames on the sphere. To appear in Advances in Comp. Math.Google Scholar
  17. 17.
    M. J. Mohlenkamp. A fast transform for spherical harmonics. J. Fourier Anal. Appl. 5, 1999, 159–184.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    C. Müller. Spherical Harmonics. Springer-Verlag, Berlin, 1966.MATHGoogle Scholar
  19. 19.
    F. J. Narcowich and J. D. Ward. Wavelets associated with periodic basis functions. Appl. Comput. Harmonic Anal. 3, 1996, 40–56.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    F. J. Narcowich and J. D. Ward. Nonstationary wavelets on the m-sphere for scattered data. Appl. Comput. Harmonic Anal. 2, 1996, 324–336.MathSciNetCrossRefGoogle Scholar
  21. 21.
    G. Plonka and M. Tasche. A unified approach to periodic wavelets. Wavelets: theory, algorithms and applications, C. K. Chui, L. Montefusco, and L. Puccio (eds.), Academic Press, New York, 1994, 137–151.Google Scholar
  22. 22.
    D. Potts, G. Steidl, and M. Tasche. Kernels of spherical harmonics and spherical frames. Advanced Topics in Multivariate Approximation, F. Fontanella, K. Jetter, and P. J. Laurent (eds.), World Scientific, Singapore, 1996, 287–301.Google Scholar
  23. 23.
    D. Potts, G. Steidl, and M. Tasche. Fast and stable algorithms for discrete spherical Fourier transforms. Linear Algebra Appl. 275, 1998, 433–450.MathSciNetCrossRefGoogle Scholar
  24. 24.
    H. Schaeben, J. Prestin, and D. Potts. Wavelet representation of diffraction pole figures. Advances in X-ray Analysis 44, Proceedings of the 49th Denver X-ray Conference, Denver, 2000.Google Scholar
  25. 25.
    I. J. Schoenberg. Positive definite functions on spheres. Duke Math. J. 9, 1942, 98–108.MathSciNetCrossRefGoogle Scholar
  26. 26.
    M. Schreiner. On a new condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc.125, 1997, 531–539.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    P. Schröder and W. Sweldens. Efficiently representing functions on a sphere. Wavelets in the Geosciences, Lect. Notes Earth Sci. 90, R. Klees (ed.), Springer-Verlag, Berlin, 2000, 158–188.Google Scholar
  28. 28.
    G. Szegö. Orthogonal Polynomials. 4th edition, American Mathematical Society, Providence, Rhode Island, 1975.MATHGoogle Scholar
  29. 29.
    Z. X. Wang and D. R. Guo. Special Functions. World Scientific Publishing, Singapore, 1989.CrossRefGoogle Scholar
  30. 30.
    I. Weinreich. A construction of C 1-wavelets on the two-dimensional sphere. Appl. Comput. Harmonic Anal. 10, 2001, 1–26.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Y. Xu and W. Cheney. Strictly positive definite functions on spheres. Proc. Amer. Math. Soc.116, 1992, 977–981.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Matthias Conrad
    • 1
  • Jürgen Prestin
    • 2
  1. 1.FB InformatikUniversity of HamburgGermany
  2. 2.Institute of MathematicsMedical University of LübeckGermany

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