Existence of representation frames based on wave packet groups

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 $Autinsert ignore into journalissuearticles values(K\times\hat{K});$. We review the quasi regular representation on $G_{\Theta}$ and extend the continuous Zak transform to $L^{2}insert ignore into journalissuearticles values(G_{\Theta});$. Moreover, we state a continuous frame based on $G_{\Theta}$ to reconstruct the element of $L^{2}\leftinsert ignore into journalissuearticles values(K\times \hat{K}\right);$. 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.