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Vol.5 , No. 3, Publication Date: Sep. 3, 2018, Page: 66-84
 metaplectic representation of the semidirect product <img width="154" height="16" src="http://article.aascit.org/journal/921/9210811/image001.png" / between the Heisenberg group and the symplectic group. Subgroups <img width="68" height="16" src="http://article.aascit.org/journal/921/9210811/image002.png" /> with <img width="16" height="16" src="http://article.aascit.org/journal/921/9210811/image003.png" /> being a <img width="97" height="16" src="http://article.aascit.org/journal/921/9210811/image004.png" /> symmetric matrix and D a close subgroup of <img width="56" height="16" src="http://article.aascit.org/journal/921/9210811/image005.png" />, are the main concerned. They shall give a general setting for the reproducibility of such groups which include and assemble the ones for the single examples treated in Cordero et al.(2006) [3]. As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schrodinger representation of <img width="41" height="16" src="http://article.aascit.org/journal/921/9210811/image006.png" /> or the wavelet representation of <img width="60" height="16" src="http://article.aascit.org/journal/921/9210811/image007.png" /> with <img width="10" height="16" src="http://article.aascit.org/journal/921/9210811/image008.png" /> closed subgroup of <img width="71" height="16" src="http://article.aascit.org/journal/921/9210811/image009.png" /> by E. Cordero, A. Tabacco [11]. Finally, we shall provided new examples of reproducing groups of the type <img width="68" height="16" src="http://article.aascit.org/journal/921/9210811/image010.png" /> in dimension <img width="6" height="16" src="http://article.aascit.org/journal/921/9210811/image011.png" /> = 1.&picture=http://www.aascit.org/aascit/decorators/img/journal/921.jpg)
| [1] | Simon Joseph, Department of Mathematics, University of Juba, Juda, South Sudan. |
| [2] | Manal Juma, Department of Mathematics, Prince Sattam bin Abdulaziz University, Alkharj, Kingdom of Saudi Arabia. |
| [3] | Isra Mukhtar, Department of Mathematics, Shaqra University, Al-Riyadh, Kingdom of Saudi Arabia. |
Consider the (extended) metaplectic representation of the semidirect product 
with 
being a 
symmetric matrix and D a close subgroup of 
, are the main concerned. They shall give a general setting for the reproducibility of such groups which include and assemble the ones for the single examples treated in Cordero et al.(2006) [3]. As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schrodinger representation of 
or the wavelet representation of 
with 
closed subgroup of 
by E. Cordero, A. Tabacco [11]. Finally, we shall provided new examples of reproducing groups of the type 
in dimension 
= 1.
Keywords
Reproducing Formula, Metaplectic Representation, Wigner Distribution, Semidirect Product
Reference
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