Twisted spherical means in annular regions in n and support theorems
[Moyennes sphériques tordues dans des anneaux de n et théorèmes de support]
Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2509-2523.

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.

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.

DOI : 10.5802/aif.2498
Classification : 43A85, 44A35, 53C65
Keywords: Heisenberg group, twisted spherical means, twisted convolution, spherical harmonics, support theorems
Mot clés : groupe d’Heisenberg, moyennes sphériques tordues, convolution tordue, harmoniques sphériques, théorèmes de supports
Rawat, Rama 1 ; Srivastava, R.K. 2

1 Indian Institute of Technology Department of Mathematics and Statistics, Kanpur 208 016 (India)
2 Indian Institute of Technology Department of Mathematics and Statistics Kanpur 208 016 (India)
@article{AIF_2009__59_6_2509_0,
     author = {Rawat, Rama and Srivastava, R.K.},
     title = {Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems},
     journal = {Annales de l'Institut Fourier},
     pages = {2509--2523},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {6},
     year = {2009},
     doi = {10.5802/aif.2498},
     mrnumber = {2640928},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2498/}
}
TY  - JOUR
AU  - Rawat, Rama
AU  - Srivastava, R.K.
TI  - Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 2509
EP  - 2523
VL  - 59
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2498/
DO  - 10.5802/aif.2498
LA  - en
ID  - AIF_2009__59_6_2509_0
ER  - 
%0 Journal Article
%A Rawat, Rama
%A Srivastava, R.K.
%T Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems
%J Annales de l'Institut Fourier
%D 2009
%P 2509-2523
%V 59
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2498/
%R 10.5802/aif.2498
%G en
%F AIF_2009__59_6_2509_0
Rawat, Rama; Srivastava, R.K. Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems. Annales de l'Institut Fourier, Tome 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., Volume 5 (1999) no. 4, pp. 363-372 | DOI | MR | Zbl

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

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

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

[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, Volume 56 (2006) no. 2, pp. 459-473 | DOI | Numdam | MR | Zbl

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

[7] Thangavelu, S. An introduction to the uncertainty principle, Prog. Math., Volume 217, Birkhauser, Boston, 2004 | MR

Cité par Sources :