IJPAM: Volume 93, No. 5 (2014)

NON COMMUTATIVE FOURIER TRANSFORM AND
PLANCHEREL THEOREM FOR THE AFFINE GROUP

Kahar El-Hussein$^1$, Badahi Ould Mohamed$^2$
$^1$Department of Mathematics
Faculty of Science
Al Furat University
Dear El Zore, SYRIA
$^{1,2}$Department of Mathematics
Faculty of Arts Science at Al Qurayat
Al-Jouf University, KINGDOM OF SAUDI ARABIA


Abstract. Let $G=SL(2,\mathbb{R})$ be the $2\times 2$ connected real semisimple Lie group. Let $AF=\mathbb{R}^{2}\rtimes GL(2,$ $\mathbb{R})$ be the affine group, which is the semidirect product of the two groups $\mathbb{R}^{2}$ with $GL(2,$ $\mathbb{R})$, whivh plays an important role in technology. The purpose of this paper is to define the Fourier transform in order to obtain the Plancherel formula for the group $SL(2,$ $\mathbb{R})$, and then we establish the Plancherel theorem for the group $P=\mathbb{R}^{2}\rtimes SL(2, $ $\mathbb{R})$. To this end a Plancherel theorem for the affine group $AF=\mathbb{R}^{2}\rtimes GL(2,$ $\mathbb{R})$ will be obtained

Received: March 18, 2014

AMS Subject Classification: 43A30, 35D05

Key Words and Phrases: Iwasawa decomposition, affine group $\mathbb{R}^{2}\rtimes GL(2,\mathbb{R})$, Fourier transform, Plancherel theorem

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DOI: 10.12732/ijpam.v93i5.9 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 93
Issue: 5
Pages: 699 - 714

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).