K-frames, as a new generalization of frames, have important applications, especially in sampling theory, to help us to reconstruct elements from a range of a bounded linear operator K in a separable Hilbert space. In this paper, we focus on the reconstruction formulae to characterize all K-duals of a given K-frame. Also, we give several approaches for constructing K-frames.2010 AMS Mathematics subject classification. Primary 42C15, Secondary 41A58.

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Abstract. Let H be a locally compact group and K be an LCA group also let τ : H → Aut(K) be a continuous homomorphism and Gτ = H ⋉τ K be the semidirect product of H and K with respect to τ . In this article we define the Zak transform Z L on L 2 (Gτ ) with respect to a τ -invariant uniform lattice L of K and we also show that the Zak transform satisfies the Plancherel formula. As an application we show that how these techniques apply for the semidirect product group SL(2, Z) ⋉τ R 2 and also the Weyl-Heisenberg groups.