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HARDY’S THEOREM FOR GABOR TRANSFORM

Part of: Lie groups Abstract harmonic analysis Locally compact groups and their algebras

Published online by Cambridge University Press:  15 August 2018

ASHISH BANSAL Open the ORCID record for ASHISH BANSAL [Opens in a new window] ,
AJAY KUMAR Open the ORCID record for AJAY KUMAR [Opens in a new window]  and
JYOTI SHARMA
ASHISH BANSAL
Affiliation:
Department of Mathematics, Keshav Mahavidyalaya (University of Delhi), H-4-5 Zone, Pitampura, Delhi-110034, India email abansal@keshav.du.ac.in
AJAY KUMAR*
Affiliation:
Department of Mathematics, University of Delhi, Delhi-110007, India email akumar@maths.du.ac.in
JYOTI SHARMA
Affiliation:
Department of Mathematics, University of Delhi, Delhi-110007, India email jsharma3698@gmail.com
*
✉akumar@maths.du.ac.in
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Abstract

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Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form $\mathbb{R}^{n}\times K$, where $K$ is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group $G$ which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form $G\times D$, where $D$ is a discrete group.

Keywords

Hardy’s theoremFourier transformcontinuous Gabor transformnilpotent Lie group

MSC classification

Primary: 43A32: Other transforms and operators of Fourier type
Secondary: 22D99: None of the above, but in this section 22E25: Nilpotent and solvable Lie groups

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 106 , Issue 2 , April 2019 , pp. 143 - 159
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The last author is supported by UGC under joint UGC-CSIR Junior Research Fellowship (Ref. No. 21/12/2014(ii)EU-V).

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