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.

In this paper we extend the notion of approximate dual to fusion frames and present some approaches to obtain alternate dual and approximate alternate dual fusion frames. We also study the stability of alternate dual and approximate alternate dual fusion frames.

Let H be a locally compact group and K be a locally compact abelian group. Also let G = H × τ K denote the semidirect product group of H and K, respectively. Then the unitary representation (U,is called the quasi regular representation. The properties of this representation in the case K = (R n , +), have been studied by many authors under some specific assumptions. In this paper we aim to consider a general case and extend some of these properties when K is an arbitrary locally compact abelian group. In particular we wish to show that the two conditions (i) δ H ≡ 1, and (ii) the stabilizers H ω are compact for a.e. ω ∈ K; both are necessary for square integrability of U . Furthermore, we shall consider some sufficient conditions for the square integrability of U . Also, for the square integrability of subrepresentations of U , we will introduce a concrete form of the Duflo-Moore operator.

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