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The Mathematical Legacy of Leon Ehrenpreis pp 61–94Cite as

Deconvolution for the Pompeiu Problem on the Heisenberg Group, I

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Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 16))

Abstract

We consider variations on the Pompeiu transform for the Heisenberg group H n and focus on cases where the transform is known to be injective; in particular the cases of averages over a sphere and a ball, or two balls of appropriate radii. In these cases we develop a method which provides for the reconstruction of the function f from its integrals.

In addition, we consider these issues in connection with the Weyl calculus and group Fourier transform. We furthermore explore issues of convergence for this method of deconvolution and related issues of size of the Gelfand transform near the zero sets. Finally, given a set of deconvolvers which work for Euclidean space C n, we consider the problem of how to extend the deconvolution to the Heisenberg group, and we provide the extension in special cases.

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References

  1. Agranovsky, M., Berenstein, C., Chang, D.C., Pascuas, D.: Injectivity of the Pompeiu transform in the Heisenberg group. J. Anal. Math. 63, 131–173 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benson, C., Jenkins, J., Ratcliffe, G.: The spherical transform of a Schwartz function on the Heisenberg group. J. Funct. Anal. 154, 379–423 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berenstein, C.A., Gay, R.: A local version of the two circles theorem. Isr. J. Math. 55, 267–288 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berenstein, C.A., Taylor, B.A., Yger, A.: Sur quelques formules explicites de déconvolution. J. Opt. (Paris) 14, 75–82 (1983)

    Article  Google Scholar 

  5. Berenstein, C., Chang, D.C., Tie, J.: Laguerre Calculus and Its Applications on the Heisenberg Group. AMS/IP Series in Advanced Mathematics, vol. 22. International Press, Cambridge (2001)

    MATH  Google Scholar 

  6. Berenstein, C., Chang, D.C., Eby, W.: L p results for the Pompeiu problem with moments on the Heisenberg group. J. Fourier Anal. Appl. 10, 545–571 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berenstein, C., Yger, A.: Le problème de déconvolution. J. Funct. Anal. 54, 113–160 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Casey, S.D., Berenstein, C.A., Walnut, D.F.: Exact multichannel deconvolution on radial domains. In: IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’97), vol. 3, pp. 1865–1868 (1997)

    Google Scholar 

  9. Chang, D.C., Eby, W.: Tauberian type theorem for the m-spherical transform on the Heisenberg group. Math. Nachr. 280, 815–837 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eby, W.: Moment results for the Heisenberg group interpreted using the Weyl calculus. In: Recent Progress on Some Problems in Several Complex Variables and Partial Differential Equations. Contemp. Math., vol. 400, pp. 95–105 (2006)

    Chapter  Google Scholar 

  11. Eby, W.: Aspects of the Pompeiu problem in the setting of the Heisenberg group. In: Birman, J., Catto, S., Nicolescu, B. (eds.) Group Theoretical Methods in Physics, pp. 220–225. Canopus publishing, Bristol (2006)

    Google Scholar 

  12. Ehrenpreis, L.: Solution of some problems of division, I, Division by a polynomial of derivation. Am. J. Math. 76, 883–903 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ehrenpreis, L.: Solution of some problems of division, II, Division by a punctual distribution. Am. J. Math. 77, 286–292 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ehrenpreis, L.: Solution of some problems of division, III, Division in the spaces , , , . Am. J. Math. 78, 685–715 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  15. Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)

    Google Scholar 

  16. Folland, G.: Harmonic Analysis in Phase Space. Annals of Mathematical Studies, vol. 122. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  17. Hörmander, L.: Generators for some rings of analytic functions. Bull. Am. Math. Soc. 73, 943–949 (1967)

    Article  MATH  Google Scholar 

  18. Zalcman, L.: Mean values and differential equations. Isr. J. Math. 14, 339–352 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Part of the research work for this paper was completed during the visits of the three authors at the National Center for Theoretical Sciences in Hsinchu, Taiwan. They would like to thank NCTS (Hsinchu) for partial support of this research. They take great pleasure in expressing their thanks to Professor Winnie Li, the director of NCTS, for the invitation and the warm hospitality during their visits in Taiwan. The second author would also like to use this opportunity to thank the Academia Sinica in Taipei for the invitation during the summer of 2007. He would also like to thank the colleagues at the Sinica for the warm hospitality during his visit in Taipei.

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Authors and Affiliations

  1. Department of Mathematics and Department of Computer Science, Georgetown University, Washington D.C., 20057, USA

    Der-Chen Chang

  2. Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Wayne Eby

  3. Department of Mathematics, University of Massachusetts Boston, Boston, MA, 02125, USA

    Eric Grinberg

Authors
  1. Der-Chen Chang
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  2. Wayne Eby
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  3. Eric Grinberg
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Corresponding author

Correspondence to Der-Chen Chang .

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Editors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, Milano, 20133, Italy

    Irene Sabadini

  2. Schmid College of Science and Technology, Chapman University, University Dr. 1, Orange, 92868, California, USA

    Daniele C Struppa

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Chang, DC., Eby, W., Grinberg, E. (2012). Deconvolution for the Pompeiu Problem on the Heisenberg Group, I. In: Sabadini, I., Struppa, D. (eds) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol 16. Springer, Milano. https://doi.org/10.1007/978-88-470-1947-8_6

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