Mean periodic functions on phase space and the Pompeiu problem with a twist

Sundaram Thangavelu

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 4, page 1007-1035
  • ISSN: 0373-0956

Abstract

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We show that when f is a mean periodic function of tempered growth on the reduced Heisenberg group then the closed translation and rotation invariant subspace generated by f contains an elementary spherical function. Using a Paley-Wiener theorem for the Fourier-Weyl transform we formulate a conjecture for arbitrary mean periodic functions.

How to cite

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Thangavelu, Sundaram. "Mean periodic functions on phase space and the Pompeiu problem with a twist." Annales de l'institut Fourier 45.4 (1995): 1007-1035. <http://eudml.org/doc/75142>.

@article{Thangavelu1995,
abstract = {We show that when $f$ is a mean periodic function of tempered growth on the reduced Heisenberg group then the closed translation and rotation invariant subspace generated by $f$ contains an elementary spherical function. Using a Paley-Wiener theorem for the Fourier-Weyl transform we formulate a conjecture for arbitrary mean periodic functions.},
author = {Thangavelu, Sundaram},
journal = {Annales de l'institut Fourier},
keywords = {mean periodic functions; spherical functions; representations; Paley- Wiener theorem; Weyl transform; Pompeiu problem; reduced Heisenberg group; Fourier-Weyl transform},
language = {eng},
number = {4},
pages = {1007-1035},
publisher = {Association des Annales de l'Institut Fourier},
title = {Mean periodic functions on phase space and the Pompeiu problem with a twist},
url = {http://eudml.org/doc/75142},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Thangavelu, Sundaram
TI - Mean periodic functions on phase space and the Pompeiu problem with a twist
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 4
SP - 1007
EP - 1035
AB - We show that when $f$ is a mean periodic function of tempered growth on the reduced Heisenberg group then the closed translation and rotation invariant subspace generated by $f$ contains an elementary spherical function. Using a Paley-Wiener theorem for the Fourier-Weyl transform we formulate a conjecture for arbitrary mean periodic functions.
LA - eng
KW - mean periodic functions; spherical functions; representations; Paley- Wiener theorem; Weyl transform; Pompeiu problem; reduced Heisenberg group; Fourier-Weyl transform
UR - http://eudml.org/doc/75142
ER -

References

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  1. [1]M. AGRANOVSKY, C. BERENSTEIN, D.C. CHANG and D. PASCUAS, Injectivity of the Pompeiu transform in the Heisenberg group, J. Analyse Math., 63 (1994), 131-173. Zbl0808.43002MR95d:43007
  2. [2]S.C. BAGCHI and A. SITARAM, Spherical mean periodic functions on semisimple Lie groups, Pacific J. Math., 84 (1979), 241-250. Zbl0442.43017MR81j:43021
  3. [3]S.C. BAGCHI and A. SITARAM, The Pompeiu problem revisited, L'Enseignement Math., 36 (1990), 67-91. Zbl0722.43009MR91k:43013
  4. [4]L. BROWN, B.M. SCHREIBER and B.A. TAYLOR, Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier, Grenoble, 23-3 (1973), 125-154. Zbl0265.46044MR50 #4979
  5. [5]G. FOLLAND, Harmonic Analysis in phase space, Ann. Math. Stud. N° 122, Princeton Univ. Press, Princeton (1989). Zbl0682.43001MR92k:22017
  6. [6]D.I. GUREVICH, Counterexamples to a problem of L. Schwartz, Funct. Anal. Appl., 197 (1975), 116-120. Zbl0326.46020
  7. [7]L. SCHWARTZ, Theorie générale des functions moyenne-periodique, Ann. Math., 48 (1947), 857-928. Zbl0030.15004MR9,428c
  8. [8]S. THANGAVELU, On Paley-Wiener theorems for the Heisenberg group, J. Funct. Anal., Vol. 115, N° 1 (1993), 24-44. Zbl0793.43006MR94m:43017
  9. [9]S. THANGAVELU, Lectures on Hermite and Laguerre expansions, Math. Notes. 42, Princeton Univ. Press, Princeton, 1993. Zbl0791.41030MR94i:42001
  10. [10]S. THANGAVELU, Regularity of twisted spherical means and special Hermite expansions, Proc. Ind. Acad. Sc., Vol. 103, N° 3 (1993), 303-320. Zbl0821.43005MR95c:33015
  11. [11]S. THANGAVELU, A Paley-Wiener theorem for step two nilpotent Lie groups, Revist. Math. Ibero, Vol. 10, N° 1 (1994), 177-187. Zbl0820.43004MR95b:22019
  12. [12]S. THANGAVELU, Spherical means and C.R. functions on the Heisenberg group, J. Analyse Math., Vol. 63 (1994), 255-286. Zbl0822.43001MR95c:43008
  13. [13]Y. WEIT, On Schwartz theorem for the motion group, Ann. Inst. Fourier, Grenoble, 30-1 (1980), 91-107. Zbl0407.43008MR81h:43007
  14. [14]L. ZALCMAN, A bibliographic survey of the Pompeiu problem, in “Approximation by solutions of P.D.E.” (B. Fuglede et al., Eds), 177-186. Kluwer Academic, (1992). Zbl0830.26005

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