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CHARACTERISATION OF THE FOURIER TRANSFORM ON COMPACT GROUPS

  • N. SHRAVAN KUMAR (a1) and S. SIVANANTHAN (a2)
Abstract

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 $\ell ^{2}\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$ .

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shravankumar@maths.iitd.ac.in
References
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[1]Alesker, S., Artstein-Avidan, S. and Milman, V., ‘A characterization of the Fourier transform and related topics’, C. R. Math. Acad. Sci. Paris 346 (2008), 625628.
[2]Alesker, S., Artstein-Avidan, S. and Milman, V., ‘A characterization of the Fourier transform and related topics’, in: Linear and Complex Analysis: Dedicated to V. P. Havin on the Occasion of his 75th Birthday, Advances in the Mathematical Sciences, American Mathematical Society Translations Series 2, 226 (American Mathematical Society, Providence, RI, 2009), 1126.
[3]Folland, G. B., A Course in Abstract Harmonic Analysis (CRC Press, Boca Raton, FL, 1995).
[4]Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis, Vol. II: Structure and Analysis of Compact Groups. Analysis on Locally Compact Abelian Groups, Grundlehren der Mathematischen Wissenschaften, 152 (Springer, Berlin, 1970).
[5]Jaming, P., ‘A characterization of Fourier transforms’, Colloq. Math. 118 (2010), 569580.
[6]Lakshmi Lavanya, R. and Thangavelu, S., ‘A characterization of the Fourier transform on the Heisenberg group’, Ann. Funct. Anal. 3 (2012), 109120.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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