Document Type: Research Paper

**Authors**

Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O.Box 397, Sabzevar, Iran.

**Abstract**

In this paper we consider (extended) metaplectic representation of the semidirect product $G_{\mathbb{J}}=\mathbb{R}^{2d}\times\mathbb{J}$ where $\mathbb{J}$ is a closed subgroup of $Sp(d,\mathbb{R})$, the symplectic group. We will investigate continuous representation frame on $G_{\mathbb{J}}$. We also discuss the existence of duals for such frames and give several characterization for them. Finally, we rewrite the dual conditions, by using the Wigner distribution and obtain more reconstruction formulas.

**Keywords**

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