Skip to main content
SpringerLink
Log in

Uncertainty principles of Ingham and Paley–Wiener on semisimple Lie groups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Classical results due to Ingham and Paley–Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show a unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on Riemmanian symmetric spaces of noncompact type.

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. S. Ben Säıd, S. Thangavelu and V. N. Dogga, Uniqueness of solutions to Schrödinger equations on H-type groups, Journal of the Australian Mathematical Society 95 (2013), 297–314.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Bhowmik, S. K. Ray and S. Sen, Around uncertainty principles of ingham-type on Rn, Tn and two step nilpotent lie groups, preprint, arXiv:1605.09616.

  3. M. Bhowmik and S. Sen, An Uncertainty Principle of Paley and Wiener on Euclidean Motion Group, J. Fourier Anal. Appl. 23 (2017), 1445–1464.

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Bochner, Quasi-analytic functions, Laplace operator, positive kernels, Annals of Mathematics 51 (1950), 68–91.

    MATH  Google Scholar 

  5. S. Bochner and A. E. Taylor, Some theorems on quasi-analyticity for functions of several variables, American Journal of Mathematics 61 (1939), 303–329.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. B. Burckel, An Introduction to Classical Complex Analysis. Vol. 1, Pure and Applied Mathematics, Vol. 82, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York–London, 1979.

  7. S. Chanillo, Uniqueness of solutions to Schrödinger equations on complex semi-simple Lie groups, Indian Academy of Sciences. Proceedings. Mathematical Sciences 117 (2007), 325–331.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. B. Conway, Functions of One Complex Cariable. II, Graduate Texts in Mathematics, Vol. 159, Springer-Verlag, New York, 1995.

  9. M. Cowling and A. Nevo, Uniform estimates for spherical functions on complex semisimple Lie groups, Geometric and Functional Analysis 11 (2001), 900–932.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Cowling, A. Sitaram and M. Sundari, Hardy’s uncertainty principle on semisimple groups, Pacific Journal of Mathematics 192 (2000), 293–296.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Eguchi, Asymptotic expansions of Eisenstein integrals and Fourier transform on symmetric spaces, Journal of Functional Analysis 34 (1979), 167–216.

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of Schrödinger equations, Communications in Partial Differential Equations 31 (2006), 1811–1823.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, Uniqueness properties of solutions to Schrödinger equations, Bulletin of the American Mathematical Society 49 (2012), 415–442.

    Article  MATH  Google Scholar 

  14. G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, Journal of Fourier Analysis and Applications 3 (1997), 207–238.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Gangolli and V. S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 101, Springer- Verlag, Berlin, 1988.

  16. V. Havin and B. Jöricke, The Uncertainty Principle in Harmonic Analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 28, Springer-Verlag, Berlin, 1994.

  17. S. Helgason, Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs, Vol. 39, American Mathematical Society, Providence, RI, 1994.

  18. S. Helgason, Groups and Geometric Analysis, Mathematical Surveys and Monographs, Vol. 83, American Mathematical Society, Providence, RI, 2000.

  19. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, American Mathematical Society, Providence, RI, 2001.

  20. I. I. Hirschman, Jr., On the behaviour of Fourier transforms at infinity and on quasianalytic classes of functions, American Journal of Mathematics 72 (1950), 200–213.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. E. Ingham, A note on Fourier transforms, Journal of the London Mathematical Society 9 (1934), 29–32.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. W. Knapp, Representation Theory of Semisimple Groups, Princeton Mathematical Series, Vol. 36, Princeton University Press, Princeton, NJ, 1986.

  23. P. Koosis, The Logarithmic Integral. I, Cambridge Studies in Advanced Mathematics, Vol. 12, Cambridge University Press, Cambridge, 1998.

  24. N. Levinson, On a class of non-vanishing functions, Proceedings of the London Mathematical Society 41 (1936), 393–407.

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, Vol. 26, American Mathematical Society, New York, 1940.

  26. J. Ludwig and D. Müller, Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups, Proceedings of the American Mathematical Society 142 (2014), 2101–2118.

    Article  MathSciNet  MATH  Google Scholar 

  27. R. E. A. C. Paley and N. Wiener, Notes on the theory and application of Fourier transforms. I, II, Transactions of the American Mathematical Society 35 (1933), 348–355.

    MathSciNet  MATH  Google Scholar 

  28. R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications, Vol. 19, American Mathematical Society, Providence, RI, 1987.

  29. A. Pasquale and M. Sundari, Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type, Université de Grenoble. Annales de l’Institut Fourier 62 (2012), 859–886.

    Article  MATH  Google Scholar 

  30. W. Rudin, Real and Complex Analysis, third ed., McGraw-Hill Book Co., New York, 1987.

    MATH  Google Scholar 

  31. R. P. Sarkar and J. Sengupta, Beurling’s theorem and characterization of heat kernel for Riemannian symmetric spaces of noncompact type, Canadian Mathematical Bulletin 50 (2007), 291–312.

    Article  MathSciNet  MATH  Google Scholar 

  32. E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis, Vol. 2, Princeton University Press, Princeton, NJ, 2003.

  33. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, Vol. 32, Princeton University Press, Princeton, NJ, 1971.

  34. A. Terras, Harmonic Analysis on Symmetric Spaces and Applications. II, Springer-Verlag, Berlin, 1988.

    Book  MATH  Google Scholar 

  35. S. Thangavelu, An Introduction to the Uncertainty Principle, Progress in Mathematics, Vol. 217, Birkhäuser Boston, Inc., Boston, MA, 2004.

Download references

Author information

Authors and Affiliations

  1. Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, 700108, India

    Mithun Bhowmik & Suparna Sen

Authors
  1. Mithun Bhowmik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suparna Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mithun Bhowmik.

Additional information

This work was supported by the Indian Statistical Institute, India (Research Fellowship to Mithun Bhowmik) and Department of Science and Technology, India (INSPIRE Faculty Award to Suparna Sen).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bhowmik, M., Sen, S. Uncertainty principles of Ingham and Paley–Wiener on semisimple Lie groups. Isr. J. Math. 225, 193–221 (2018). https://doi.org/10.1007/s11856-018-1662-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-018-1662-8

Search

Navigation