We define a new function-valued inner product on L2(G), called j-bracket product, where G is a locally compact abelian group and j is a topological isomorphism on G. We investigate the notion of j -orthogonality, Bessel's Inequality and j -orthonormal bases with respect to this inner product on L2(G).

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Let G be a locally compact abelian group with a uniform lattice sub-group. In this paper, we verify extension of shift-invariant systems in L2 (G) to tight frames. We show that any shift-invariant Bessel sequence with an at most countable number of generators in L2 (G) can be extended to a tight frame for its closed linear span by adding another shift-invariant system with at most the same number of generators in L2 (G). Also, we yield an extension of the given Bessel sequence to a pair of dual frame sequences.

Let L be the category of all locally compact abelian (LCA) groups. In this paper, the groups G in L are determined such that every extension 0®X®Y®G®0 with divisible, s-compact X in L splits. We also determine the discrete or compactly generated LCA groups H such that every pure extension 0®H®Y®X®0 splits for each divisible group X in L.

This article presents a systematic study for abstract harmonic analysis aspects of wave-packet transforms over locally compact abelian (LCA) groups. Let HH be a locally compact group, let KK be an LCA group, and let q:H®Aut(K) q:H®Aut(K) be a continuous homomorphism. We introduce the abstract notion of the wave-packet group generated by q, and we study basic properties of wave-packet groups. Then we study theoretical aspects of wave-packet transforms. Finally, we will illustrate application of these techniques in the case of some well-known examples.

We investigate shift invariant subspaces of L2 (G), where G is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift invariant subspaces of L2 (G) in terms of range functions. Finally, we investigate shift preserving operators on locally compact abelian groups. We show that there is a one-to-one correspondence between shift preserving operators and range operators on L2 (G) where Gis a locally compact abelian group.

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Let à be the category of totally disconnected, locally compact abelian groups. In this paper, we determine the discrete or compact pure injective groups in Ã. Also, we determine the compact pure projective groups in Ã.