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.