Skip to main content
SpringerLink
Log in

Efficient computation of Fourier transforms on compact groups

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

This article genralizes the fast Fourier transform algorithm to the computation of Fourier transforms on compact Lie groups. The basic technique uses factorization of group elements and Gel'fand-Tsetlin bases to simplify the computations, and may be extended to treat the computation of Fourier transforms of finitely supported distributions on the group. Similar transforms may be defined on homogeneous spaces; in that case we show how special function properties of spherical functions lead to more efficient algorithms. These results may all be viewed as generalizations of the fast Fourier transform algorithms on the circle, and of recent results about Fourier transforms on finite groups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atkinson, M. (1977). The complexity of group algebra computations,Theor. Comp. Sci. 5, 205–209.

    Article  Google Scholar 

  2. Alpert, B. and Rokhlin, V. (1991). A fast algorithm for the evaluation of Legendre transforms,SIAM J. Sci. Statist. Comput. 12, 158–179.

    Article  MATH  MathSciNet  Google Scholar 

  3. Baum, U. (1991). Existence and efficient construction of fast Fourier transforms for supersolvable groups,Comput. Complexity,1, 235–256.

    Article  MATH  MathSciNet  Google Scholar 

  4. Beth, T. (1984).Verfahren der schnellen Fourier-Transformation, Teubner Studien-bücher, Stuttgart.

    MATH  Google Scholar 

  5. Bratelli, O. (1972). Inductive limits of finite dimensionalC *-algebras.Trans. Am. Math. Soc.:171, 195–234.

    Article  Google Scholar 

  6. Clausen, M. (1989). Fast generalized Fourier transforms,Theoret. Comput. Sci. 67, 55–63.

    Article  MATH  MathSciNet  Google Scholar 

  7. Clausen, M. (1988).Beiträge zum Entwurf schneller Spektraltransformationen, Habilitations-schrift Fakultät für Informatik der Universität Karlsruhe (TH).

  8. Clausen, M. and Baum, U. (1993).Fast Fourier Transforms, Wissenschafts-Verlag, Manheim.

    MATH  Google Scholar 

  9. Cooley, J. and Tukey, J. (1965). An algorithm for machine calculation of complex Fourier series,Math. Comp. 19, 297–301.

    Article  MATH  MathSciNet  Google Scholar 

  10. Diaconis and Rockmore, D. (1990). Efficient computation of the Fourier transform on finite groups,J. Am. Math. Soc. 3(2), 297–332.

    Article  MATH  MathSciNet  Google Scholar 

  11. Driscoll, J.R. and Healy, Jr., D. (1994). Computing Fourier transforms and convolutions on the 2-sphere, (extended abstract) Proc. 34th IEEE FOCS, (1989), 344–349;Adv. in Appl. Math.,15, 202–250.

  12. Driscoll, J.R., Healy, Jr., D., and Rockmore, D. Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs.SIAM J. Comput. (to appear).

  13. Gel'fand, I.M. and Tsetlin, M. (1950). Finite dimensional representations of the group of unimodular matrices,Dokl. Akad. Nauk SSSR 71, 825–828 (Russian).

    MATH  Google Scholar 

  14. Goodman, F., de la Harpe, P., and Jones, V. (1989).Coxeter Graphs and Towers of Algebras, Springer-Verlag, New York.

    MATH  Google Scholar 

  15. Healy, Jr., D.Evaluation of phase polynomials on non-uniform sample sets in the plane, with applications in fast MRI, in preparation.

  16. Healy, Jr., D., Maslen, D., Moore, S., Rockmore, D., and Taylor, M.Applications of a fast convolution algorithm on the 2-sphere, in preparation.

  17. Healy, Jr., D., Moore, S., and Rocknore, D. (1994).Efficiency and reliability issues in a fast Fourier transform on the 2-sphere, Technical Report, Department of Mathamatics and Computer Science, Dartmouth College.

    Google Scholar 

  18. Karpovsky, M. (1977). Fast Fourier transforms on finite nonabelian groups,IEEE Trans. Comput. 26(10), 1028–1030.

    MATH  MathSciNet  Google Scholar 

  19. Klimyk, A. (1979).Matrix Elements and Clebsch-Gordan Coefficients of Group Representations, Naukova Dumka, Kiev, (Russian).

    Google Scholar 

  20. Klimyk, A. and Vilenkin, N. (1993).Representation of Lie Groups and Special Functions, vol. 2, Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer, Boston.

    Google Scholar 

  21. Klimyk, A. and Vilenkin, N. (1993).Representation of Lie Groups and Special Functions, vol. 3, Mathematics and its Application (Soviet Series), vol. 75, Kluwer, Boston.

    Google Scholar 

  22. Lambina, E. (1965). Matrix elements of irreducible representations of the groupK n of orthogonal matrices,Dokl. Akad. Nauk Byelorussian SSR 9, 77–81 (Russian).

    MATH  MathSciNet  Google Scholar 

  23. Lang, S. (1984).Algebra, 2nd ed. Addison-Wesley, Reading, MA.

    Google Scholar 

  24. Malsen, D. (1993).East transforms and sampling for compact groups, Ph.D. Thesis. Department of Mathematics, Harvard University, Cambridge, MA.

    Google Scholar 

  25. Maslen, D.Sampling of functions and sections for compact groups, preprint.

  26. Maslen, D. (1995).A polynomial approach to orthogonal polynomial transforms of the symmetric group, preprint IHES/M/96/52, andMath. Comp. (to appear).

  27. Maslen, D. and Rockmore, D. (1996). Generalized FFTs-a survey of some recent results,DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Groups and Computation, II, L. Finkelstein and W. Kantor, Eds., 183–237.

  28. Maslen, D. and Rockmore, D. (1997). Separation of variables and the efficient computation of Fourier transforms on finite groups, I,J. Am. Math. Soc. 10(1), 169–214.

    Article  MATH  MathSciNet  Google Scholar 

  29. Maslen, D. and Rockmore, D.Separation of variables and the efficient computation of Fourier transforms on finite groups, II, in preparation.

  30. Moore, S. (1994).Efficient stabilization methods for fast polynomial transforms, Ph.D. Thesis, Dartmouth College, NH.

  31. Moore, S., Healy, Jr., D., and Rockmore, D. (1993). Symmetry stabilization for polynomial evaluation and interpolation,Linear Algebra Appl. 192, 249–299.

    Article  MATH  MathSciNet  Google Scholar 

  32. Rockmore, D. (1990). Fast Fourier analysis for abelian group extensions,Adv. in Appl. Math. 11, 164–204.

    Article  MATH  MathSciNet  Google Scholar 

  33. Rockmore, D. (1996). Some applications of generalized FFTs,DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Groups and Computation, II, L. Finkelstein and W. Kantor, Eds., 329–369.

  34. Vilenkin, N. (1968).Special functions and the theory of group representations, Transl. Math. Monog. 22, AMS, Providence RI.

    MATH  Google Scholar 

  35. Wallach, N. (1973).Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, New York.

    MATH  Google Scholar 

  36. Willsky, A.S. (1978). On the algebraic structure of certain partially observable finite-state Markov processes,Inform. Contr. 38, 179–212.

    Article  MATH  Google Scholar 

  37. Želobenko, D. (1973). Compact Lie groups and their representations,Transl. Math. Monog. 40, AMS, Providence RI.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Univesiteit Utrecht, 3584 CD, Utrecht, Netherlands

    David K. Maslen

Authors
  1. David K. Maslen
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Daniel Rockmore

Acknowledgements and Notes. This paper was written while the author was supported by the Max-Planck-Institut für Mathematik, Bonn, Germany.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maslen, D.K. Efficient computation of Fourier transforms on compact groups. The Journal of Fourier Analysis and Applications 4, 19–52 (1998). https://doi.org/10.1007/BF02475926

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02475926

Math Subject Classifications

Keywords and Phrases

Search

Navigation