Twisted spherical means in annular regions in n and support theorems
Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2509-2523.

Let Z( Ann (r,R)) be the class of all continuous functions f on the annulus Ann (r,R) in n with twisted spherical mean f×μ s (z)=0, whenever z n and s>0 satisfy the condition that the sphere S s (z) Ann (r,R) and ball B r (0)B s (z). In this paper, we give a characterization for functions in Z( Ann (r,R)) in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in n which improve some of the earlier results.

Soit Z( Ann (r,R)) la classe de toutes les fonctions continues sur l’anneau Ann (r,R) de n de moyenne sphérique tordue f×μ s (z)=0, pour tout z n et s>0 tels que la sphère S s (z) Ann (r,R) et la boule B r (0)B s (z). Dans cet article, nous donnons une caractérisation des fonctions dans Z( Ann (r,R)) en termes de leur coefficients dans le développement en harmoniques sphériques. Nous prouvons également des théorèmes de support pour les moyennes sphériques tordues dans n qui améliorent certains résultats antérieurs.

Received:
Revised:
Accepted:
DOI: 10.5802/aif.2498
Classification: 43A85,  44A35,  53C65
Keywords: Heisenberg group, twisted spherical means, twisted convolution, spherical harmonics, support theorems
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     title = {Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems},
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Rawat, Rama; Srivastava, R.K. Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems. Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2509-2523. doi : 10.5802/aif.2498. https://aif.centre-mersenne.org/articles/10.5802/aif.2498/

[1] Agranovsky, M. L.; Rawat, Rama Injectivity sets for spherical means on the Heisenberg group, J. Fourier Anal. Appl., Tome 5 (1999) no. 4, pp. 363-372 | Article | MR: 1700090 | Zbl: 0931.43007

[2] Epstein, C. L.; Kleiner, B. Spherical means in annular regions, Comm. Pure Appl. Math., Tome 46 (1993) no. 3, pp. 441-451 | Article | MR: 1202964 | Zbl: 0841.31006

[3] Helgason, S. The Radon Transform, Birkhauser, 1983 | Zbl: 0547.43001

[4] Narayanan, E. K.; Thangavelu, S. Injectivity sets for spherical means on the Heisenberg group, J. Math. Anal. Appl., Tome 263 (2001) no. 2, pp. 565-579 | Article | MR: 1866065 | Zbl: 0995.43003

[5] Narayanan, E. K.; Thangavelu, S. A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on n , Ann. Inst. Fourier, Grenoble, Tome 56 (2006) no. 2, pp. 459-473 | Article | Numdam | MR: 2226023 | Zbl: 1089.43006

[6] Rudin, W. Function theory in the unit ball of n , Springer-Verlag, New York-Berlin, 1980 | MR: 601594 | Zbl: 0495.32001

[7] Thangavelu, S. An introduction to the uncertainty principle, Prog. Math., Tome 217, Birkhauser, Boston, 2004 | MR: 2008480 | Zbl: pre02039097

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