CYCLIC SUBSPACES FOR UNITARY REPRESENTATIONS OF LCA GROUPS: GENERALIZED ZAK TRANSFORM EUGENIO HERN ´ ANDEZ, HRVOJE ˇ SIKI ´ C, GUIDO WEISS, AND EDWARD WILSON 1. Introduction Cyclic subspaces generated by integer translations of a single function ψ ∈ L 2 (R) have attracted attention as building blocks for more general translation invariant subspaces of L 2 (R). Also, they play a key role in the theory of Multiresolution Analysis (MRA) where the scaling spaces V 0 and the zero resolution wavelet space W 0 are both principal invariant subspaces of L 2 (R). For ψ ∈ L 2 (R), let <ψ> be the closure in L 2 (R) of the finite linear combinations of the integer translations T k ψ(x)= ψ(x - k). Properties of the collection {T k ψ : k ∈ Z} in <ψ> can be studied using the “periodization” function P ψ (ξ )= X l∈Z | b ψ(ξ + l)| 2 . For example, {T k ψ : k ∈ Z} is an orthonormal basis for <ψ> if and only if P ψ (ξ )=1 a.e. (see [9], Proposition 1.11, Chapter 1.) More properties of the collection {T k ψ : k ∈ Z} and its relation to the function P ψ (ξ ) is the subject of [10]. The present paper includes the results of [10] and many more. Now consider integer translations T k ψ(x)= ψ(x - k),x ∈ R and integer modula- tions M l ψ(x)= e 2πikx ψ(x) of a single function ψ ∈ L 2 (R). We obtain the collection {T k M l ψ : k,l ∈ Z} in L 2 (R), known as Gabor system, that can be studied using the Zak transform Zψ(x, ξ )= X l∈Z ψ(x + l)e 2πilξ , as the analog of P ψ . For example, {T k M l ψ : k,l ∈ Z} is an orthonormal basis of L 2 (R) if and only if |Zψ(x, ξ )| = 1 a.e. (see [8], Theorem 4.3.3 and the references given there.) We show that these results are particular cases of a more general theory involving representations of locally compact abelian groups. Suppose that G is a locally compact Date : September 2, 2008. 2000 Mathematics Subject Classification. 42C40, 43A65,43A70. Key words and phrases. Cyclic subspaces, invariant subspaces, unitary representations, locally compact abelian groups, translations, dilations, Gabor systems, Zak transform. The research of E. Hern´ andez is supported by grants MTM2007-60952 of Spain and SIMUMAT S-0505/ESP-0158 of the Madrid Community Region. The research of H. ˇ Siki´ c, G. Weiss and E. Wilson is supported by the US-Croatian grant NSF-INT-0245238. The research of H. ˇ Siki´ c is also supported by the MZOS grant 037-0372790-2799 of the Republic of Croatia. 1