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).

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.

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.

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.