Skip to main content

Handling the Inverse Spherical Fourier Transform

  • Chapter
  • 832 Accesses

Part of the book series: Progress in Mathematics ((PM,volume 101))

Abstract

We use the standard notation and refer to [GV], [H] for more details. Let X = G/K be a Riemannian symmetric space of the noncompact type.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.-Ph. Anker, Le noyau de la chaleur sur les espaces symetriques U(p, q)/U(p) X V(q), Harmonic analysis, Luxembourg 1987, Lecture Notes Math., Vol 1359 (P. Eymard, J.-P. Pier, eds.), Springer-Verlag, 1988, pp. 60-82.

    Google Scholar 

  2. J.-Ph. Anker, The spherical Fourier transform of rapidly decreasing functions — a simple proof of a characterization due to Harish-Chandra, Helgason, Trombi and Varadarajan, J. Funct. Anal. 96 (1991), 331-349.

    Article  MathSciNet  MATH  Google Scholar 

  3. J.-Ph. Anker, Lp Fourier multipliers on Riemannian symmetric spaces of the noncom-pact type, Ann. Math. 132 (1990), 597–628.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, preprint (1990).

    Google Scholar 

  5. J.-Ph. Anker and N. Lohoué, Multiplicateurs sur certains espaces symétriques, Amer. J. Math. 108 (1986), 1303–1354.

    Article  Google Scholar 

  6. D. Bakry, Etude des transformations de Riesz dans les variétes Riemmanniennes à courbure de Ricci minorée, Séminaire de probabilites XXI, Lecture Notes Math., Vol. 1247 (J. Azéma, P. A. Meyer, and M. Yor, eds.), Springer Verlag, 1987, pp. 137–172

    Google Scholar 

  7. M. Cowling, G. Gaudry, S. Giulini, and G. Mauceri, Weak type (1,1) estimates for heat kernel maximal functions on Lie groups, Trans. Amer. Math. Soc. 323 (1991), 637–649.

    MathSciNet  MATH  Google Scholar 

  8. J.-L. Clerc and E. M. Stein, L P - multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. USA 71 (1974), 3911–3912.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Eguchi, Asymptotic expansions of Eisenstein integrals and Fourier transform on symmetric spaces, J. Funct. Anal. 34 (1979), 167–216.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Flensted-Jensen, Spherical functions on a real semisimple Lie group; a method of reduction to the complex case, J. Funct. Anal. 30 (1978), 106–146.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Gangolli and V. S. Varadarajan, Harmonic analysis of spherical functions on real reductive groups, Ergebnisse Math. Grenzgeb. 101, Springer Verlag, 1988.

    Google Scholar 

  12. S. Helgason, Groups and geometric analysis — integral geometry, invariant differential operators, and spherical functions, Academic Press, 1984.

    MATH  Google Scholar 

  13. T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple groups, Special functions — group theoretical aspects & applications (R. A. Askey, T. H. Koornwinder, and W. Schempp, eds.), D. Reidel Publ. Co., 1984, pp. 1–85.

    Google Scholar 

  14. N. Lohoué, Comparaison des champs de vecteurs et des puissances du Laplacien sur les varietes Riemanniennes a courbure non positive, J. Funct. Anal. 61 (1985), 164–201.

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Lohoué, Transformées de Riesz et fonctions de Littlewood-Paley sur les groupes non moyennables, C. R. Acad. Sci. Paris Serie I, 306 (1988), 327–330.

    Google Scholar 

  16. N. Lohoué and Zhu Fu Liu, Estimation faible de certaines fonctions maximales, C. R. Acad. Sci. Paris Serie I, 302 (1986), 303–305.

    MATH  Google Scholar 

  17. M. A. Semenov-Tjan-Sanskff, Harmonic analysis on Riemannian symmetric spaces of negative curvature and scattering theory, Math. USSR Izvestija 10 (1976), 535–563.

    Article  Google Scholar 

  18. M. M. Shahshahani, Invariant hyperbolic systems on symmetric spaces, Differential Geometry, Progress in Math. 32 (R. Brooks, A. Gray, and B. L. Reinhart, eds.), Birkhäuser, 1983, pp. 203–233.

    Google Scholar 

  19. R. J. Stanton and P. A. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978), 251–276.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. M. Stein, Topics in harmonic analysis related to the Littlewood - Paley theory, Ann. Math. Stud. 63, Princeton Univ. Press, 1970.

    Google Scholar 

  21. M. E. Taylor, L P estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), 773–793.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. C. Trombi and V. S. Varadarajan, Spherical transforms on semisimple Lie groups, Ann. Math. 94 (1971), 246–303.

    Article  MathSciNet  MATH  Google Scholar 

  23. L. Vretare, On a recurrence formula for elementary spherical functions on symmetric spaces and its applications to multipliers for the spherical Fourier transform, Math. Scand. 41 (1977), 99–112.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer Science+Business Media New York

About this chapter

Cite this chapter

Anker, JP. (1991). Handling the Inverse Spherical Fourier Transform. In: Barker, W.H., Sally, P.J. (eds) Harmonic Analysis on Reductive Groups. Progress in Mathematics, vol 101. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0455-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0455-8_2

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6768-3

  • Online ISBN: 978-1-4612-0455-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics