Twisted spherical means in annular regions in ℂ n and support theorems

Rama Rawat[1]; R.K. Srivastava[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)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 6, page 2509-2523
  • ISSN: 0373-0956

Abstract

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Let Z ( Ann ( r , R ) ) be the class of all continuous functions f on the annulus Ann ( r , R ) in ℂ n with twisted spherical mean f × μ s ( z ) = 0 , whenever z ∈ ℂ 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 ℂ n which improve some of the earlier results.

How to cite

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Rawat, Rama, and Srivastava, R.K.. "Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems." Annales de l’institut Fourier 59.6 (2009): 2509-2523. <http://eudml.org/doc/10462>.

@article{Rawat2009,
abstract = {Let $Z(\{\rm Ann\}(r,R))$ be the class of all continuous functions $f$ on the annulus $\{\rm Ann\}(r,R)$ in $\mathbb\{C\}^n$ with twisted spherical mean $f \times \mu _s(z)=0,$ whenever $z\in \mathbb\{C\}^n$ and $s &gt;0$ satisfy the condition that the sphere $S_s(z)\subseteq \{\rm Ann\}(r, R) $ and ball $B_r(0)\subseteq B_s(z).$ In this paper, we give a characterization for functions in $Z(\{\rm Ann\}(r,R))$ in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in $\mathbb\{C\}^n$ which improve some of the earlier results.},
affiliation = {Indian Institute of Technology Department of Mathematics and Statistics, Kanpur 208 016 (India); Indian Institute of Technology Department of Mathematics and Statistics Kanpur 208 016 (India)},
author = {Rawat, Rama, Srivastava, R.K.},
journal = {Annales de l’institut Fourier},
keywords = {Heisenberg group; twisted spherical means; twisted convolution; spherical harmonics; support theorems},
language = {eng},
number = {6},
pages = {2509-2523},
publisher = {Association des Annales de l’institut Fourier},
title = {Twisted spherical means in annular regions in $\mathbb\{C\}^n$ and support theorems},
url = {http://eudml.org/doc/10462},
volume = {59},
year = {2009},
}

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
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 6
SP - 2509
EP - 2523
AB - Let $Z({\rm Ann}(r,R))$ be the class of all continuous functions $f$ on the annulus ${\rm Ann}(r,R)$ in $\mathbb{C}^n$ with twisted spherical mean $f \times \mu _s(z)=0,$ whenever $z\in \mathbb{C}^n$ and $s &gt;0$ satisfy the condition that the sphere $S_s(z)\subseteq {\rm Ann}(r, R) $ and ball $B_r(0)\subseteq B_s(z).$ In this paper, we give a characterization for functions in $Z({\rm Ann}(r,R))$ in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in $\mathbb{C}^n$ which improve some of the earlier results.
LA - eng
KW - Heisenberg group; twisted spherical means; twisted convolution; spherical harmonics; support theorems
UR - http://eudml.org/doc/10462
ER -

References

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  1. M. L. Agranovsky, Rama Rawat, Injectivity sets for spherical means on the Heisenberg group, J. Fourier Anal. Appl. 5 (1999), 363-372 Zbl0931.43007MR1700090
  2. C. L. Epstein, B. Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math. 46 (1993), 441-451 Zbl0841.31006MR1202964
  3. S. Helgason, The Radon Transform, (1983), Birkhauser Zbl0547.43001
  4. E. K. Narayanan, S. Thangavelu, Injectivity sets for spherical means on the Heisenberg group, J. Math. Anal. Appl. 263 (2001), 565-579 Zbl0995.43003MR1866065
  5. E. K. Narayanan, S. Thangavelu, A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on ℂ n , Ann. Inst. Fourier, Grenoble 56 (2006), 459-473 Zbl1089.43006MR2226023
  6. W. Rudin, Function theory in the unit ball of ℂ n , (1980), Springer-Verlag, New York-Berlin Zbl0495.32001MR601594
  7. S. Thangavelu, An introduction to the uncertainty principle, Prog. Math. 217 (2004), Birkhauser, Boston Zbl1188.43010MR2008480

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