A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on ℂ n

E. K. Narayanan[1]; S. Thangavelu

  • [1] Indian Institute of Science Department of Mathematics Bangalore 560 012 (India)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 2, page 459-473
  • ISSN: 0373-0956

Abstract

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We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on ℂ n . If f ( z ) e 1 4 | z | 2 is a Schwartz class function we show that f is supported in a ball of radius B in ℂ n if and only if f × μ r ( z ) = 0 for r > B + | z | for all z ∈ ℂ n . This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When n = 1 we show that the two conditions f × μ r ( z ) = μ r × f ( z ) = 0 for r > B + | z | imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter result does not generalize to higher dimensions.

How to cite

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Narayanan, E. K., and Thangavelu, S.. "A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $\mathbb{C}^n$." Annales de l’institut Fourier 56.2 (2006): 459-473. <http://eudml.org/doc/10153>.

@article{Narayanan2006,
abstract = {We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on $\mathbb\{C\}^n.$ If $f(z) e^\{\frac\{1\}\{4\}|z|^2\}$ is a Schwartz class function we show that $f$ is supported in a ball of radius $B$ in $\mathbb\{C\}^n$ if and only if $f \times \mu _r (z) = 0$ for $r &gt; B+|z|$ for all $z \in \mathbb\{C\}^n.$ This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When $n = 1$ we show that the two conditions $f \times \mu _r (z) = \mu _r \times f (z) = 0$ for $r &gt; B+|z|$ imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter result does not generalize to higher dimensions.},
affiliation = {Indian Institute of Science Department of Mathematics Bangalore 560 012 (India)},
author = {Narayanan, E. K., Thangavelu, S.},
journal = {Annales de l’institut Fourier},
keywords = {Spectral Paley-Wiener theorem; twisted spherical means; special Hermite operator; Laguerre functions; support theorem; spherical harmonics; spectral decomposition; Paley-Wiener theorem; Heisenberg group; spherical mean},
language = {eng},
number = {2},
pages = {459-473},
publisher = {Association des Annales de l’institut Fourier},
title = {A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $\mathbb\{C\}^n$},
url = {http://eudml.org/doc/10153},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Narayanan, E. K.
AU - Thangavelu, S.
TI - A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $\mathbb{C}^n$
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 2
SP - 459
EP - 473
AB - We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on $\mathbb{C}^n.$ If $f(z) e^{\frac{1}{4}|z|^2}$ is a Schwartz class function we show that $f$ is supported in a ball of radius $B$ in $\mathbb{C}^n$ if and only if $f \times \mu _r (z) = 0$ for $r &gt; B+|z|$ for all $z \in \mathbb{C}^n.$ This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When $n = 1$ we show that the two conditions $f \times \mu _r (z) = \mu _r \times f (z) = 0$ for $r &gt; B+|z|$ imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter result does not generalize to higher dimensions.
LA - eng
KW - Spectral Paley-Wiener theorem; twisted spherical means; special Hermite operator; Laguerre functions; support theorem; spherical harmonics; spectral decomposition; Paley-Wiener theorem; Heisenberg group; spherical mean
UR - http://eudml.org/doc/10153
ER -

References

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  9. G. Sajith, S. Thangavelu, On the injectivity of twisted spherical means on ℂ n , Israel J. Math. 122 (2), 79-92 Zbl0986.43001MR1826492
  10. R. S. Strichartz, Harmonic analysis as spectral theory of Laplacians, J. Funct. Anal. 87 (1989), 51-148 Zbl0694.43008MR1025883
  11. G. Szego, Orthogonal polynomials, 23 (1967), Amer. Math. Soc., Providence, R. I. 
  12. S. Thangavelu, Lectures on Hermite and Laguerre expansions, 42 (1993), Princeton University Press, Princeton, NJ Zbl0791.41030MR1215939
  13. S. Thangavelu, Harmonic analysis on the Heisenberg group, 159 (1998), Birkhäuser Boston, Boston, MA Zbl0892.43001MR1633042
  14. S. Thangavelu, An introduction to the uncertainty principle, 217 (2004), Birkhäuser Boston, Boston, MA Zbl1188.43010MR2008480

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