- Hacettepe Journal of Mathematics and Statistics
- Vol: 49 Issue: 5
- Existence of representation frames based on wave packet groups
Existence of representation frames based on wave packet groups
Authors : Ali Akbar Arefijamaal, Atefe Razghandi
Pages : 1825-1842
Doi:10.15672/hujms.540946
View : 9 | Download : 4
Publication Date : 2020-10-06
Article Type : Research
Abstract :Let $H$ be a locally compact group, $K$ a locally compact abelian group with dual group $\hat{K}$. In this article, we consider the wave packet group $G_{\Theta}$ which is the semidirect product of locally compact groups $H$ and $K\times \hat{K}$, where $\Theta$ is a continuous homomorphism from $H$ into $Aut(K\times\hat{K})$. We review the quasi regular representation on $G_{\Theta}$ and extend the continuous Zak transform to $L^{2}(G_{\Theta})$. Moreover, we state a continuous frame based on $G_{\Theta}$ to reconstruct the element of $L^{2}\left(K\times \hat{K}\right)$. These results are extended to more general wave packet groups. Finally, we establish some methods to find dual of such continuous frames in the form of original frames.Keywords : semidirect product groups, quasi regular representation, wave packet groups