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CHARACTERISATION OF THE FOURIER TRANSFORM ON COMPACT GROUPS

Part of: Compact groups Abstract harmonic analysis

Published online by Cambridge University Press:  13 November 2015

N. SHRAVAN KUMAR  and
S. SIVANANTHAN
N. SHRAVAN KUMAR*
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, Delhi-110016, India email shravankumar@maths.iitd.ac.in
S. SIVANANTHAN
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, Delhi-110016, India email siva@maths.iitd.ac.in
*
✉shravankumar@maths.iitd.ac.in
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Abstract

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Let $G$ be a compact group. The aim of this note is to show that the only continuous *-homomorphism from $L^{1}(G)$ to $\ell ^{\infty }\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$ that transforms a convolution product into a pointwise product is, essentially, a Fourier transform. A similar result is also deduced for maps from $L^{2}(G)$ to $\ell ^{2}\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$.

Keywords

compact groupsgroup Fourier transform

MSC classification

Primary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
Secondary: 22C05: Compact groups 43A32: Other transforms and operators of Fourier type 43A77: Analysis on general compact groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 3 , June 2016 , pp. 467 - 472
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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