# A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on ${\u2102}^{n}$

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

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topNarayanan, 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 $r > B+|z|$ 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 $r > B+|z|$ 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 $r > B+|z|$ 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 $r > B+|z|$ 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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