Existence of representation frames based on wave packet groups

Year 2020, Volume: 49 Issue: 5, 1825 - 1842, 06.10.2020

Ali Akbar Arefijamaal , Atefe Razghandi

https://doi.org/10.15672/hujms.540946

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

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