Mean periodic functions on phase space and the Pompeiu problem with a twist
Thangavelu, Sundaram
Annales de l'Institut Fourier, Tome 45 (1995), p. 1007-1035 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Si f est une fonction moyenne périodique, tempérée, sur le groupe d’Heisenberg réduit, alors le sous-espace fermé engendré par f, invariant par translation et rotation, contient une fonction sphérique élémentaire. À l’aide d’un théorème de Paley-Wiener pour la transformation de Fourier-Weyl, nous formulons une conjecture pour les fonctions moyenne périodiques quelconques.

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.

@article{AIF_1995__45_4_1007_0,
     author = {Thangavelu, Sundaram},
     title = {Mean periodic functions on phase space and the Pompeiu problem with a twist},
     journal = {Annales de l'Institut Fourier},
     volume = {45},
     year = {1995},
     pages = {1007-1035},
     doi = {10.5802/aif.1482},
     mrnumber = {96m:43009},
     zbl = {0831.43003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1995__45_4_1007_0}
}

Thangavelu, Sundaram. Mean periodic functions on phase space and the Pompeiu problem with a twist. Annales de l'Institut Fourier, Tome 45 (1995) pp. 1007-1035. doi : 10.5802/aif.1482. http://gdmltest.u-ga.fr/item/AIF_1995__45_4_1007_0/

[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. | MR 95d:43007 | Zbl 0808.43002

[2]S.C. Bagchi and A. Sitaram, Spherical mean periodic functions on semisimple Lie groups, Pacific J. Math., 84 (1979), 241-250. | MR 81j:43021 | Zbl 0442.43017

[3]S.C. Bagchi and A. Sitaram, The Pompeiu problem revisited, L'Enseignement Math., 36 (1990), 67-91. | MR 91k:43013 | Zbl 0722.43009

[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. | Numdam | MR 50 #4979 | Zbl 0265.46044

[5]G. Folland, Harmonic Analysis in phase space, Ann. Math. Stud. N° 122, Princeton Univ. Press, Princeton (1989). | MR 92k:22017 | Zbl 0682.43001

[6]D.I. Gurevich, Counterexamples to a problem of L. Schwartz, Funct. Anal. Appl., 197 (1975), 116-120. | Zbl 0326.46020

[7]L. Schwartz, Theorie générale des functions moyenne-periodique, Ann. Math., 48 (1947), 857-928. | MR 9,428c | Zbl 0030.15004

[8]S. Thangavelu, On Paley-Wiener theorems for the Heisenberg group, J. Funct. Anal., Vol. 115, N° 1 (1993), 24-44. | MR 94m:43017 | Zbl 0793.43006

[9]S. Thangavelu, Lectures on Hermite and Laguerre expansions, Math. Notes. 42, Princeton Univ. Press, Princeton, 1993. | MR 94i:42001 | Zbl 0791.41030

[10]S. Thangavelu, Regularity of twisted spherical means and special Hermite expansions, Proc. Ind. Acad. Sc., Vol. 103, N° 3 (1993), 303-320. | MR 95c:33015 | Zbl 0821.43005

[11]S. Thangavelu, A Paley-Wiener theorem for step two nilpotent Lie groups, Revist. Math. Ibero, Vol. 10, N° 1 (1994), 177-187. | MR 95b:22019 | Zbl 0820.43004

[12]S. Thangavelu, Spherical means and C.R. functions on the Heisenberg group, J. Analyse Math., Vol. 63 (1994), 255-286. | MR 95c:43008 | Zbl 0822.43001

[13]Y. Weit, On Schwartz theorem for the motion group, Ann. Inst. Fourier, Grenoble, 30-1 (1980), 91-107. | Numdam | MR 81h:43007 | Zbl 0407.43008

[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).