A generalization of the notion of multiresolution analysis, based on the theory of spectral pairs, is considered. In contrast to the standard setting, the associated subspace V 0 of L 2 (R) has, as an orthonormal basis, a collection of translates of the scaling function , of the form [,(x&*)] * # 4 where 4=[0, rÂN]+2Z, N 1 is an integer, and r is an odd integer with 1 r 2N&1 such that r and N are relatively prime and Z is the set of all integers. Furthermore, the corresponding dilation factor is 2N, the case where N=1 corresponding to the usual definition of a multiresolution analysis with dilation factor 2. A necessary and sufficient condition for the existence of associated wavelets, which is always satisfied when N=1 or 2, is obtained and is shown to always hold if the Fourier transform of , is a constant multiple of the characteristic function of a set. Academic Press

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Given an invertible n × n matrix B and a finite or countable subset of L 2 (R n ), we consider the collection X = {φ(· − Bk) : φ ∈ , k ∈ Z n } generating the closed subspace M of L 2 (R n ). If that collection forms a frame for M, one can introduce two different types of shift-generated (SG) dual frames for X, called type I and type II SG-duals, respectively. The main distinction between them is that a SG-dual of type I is required to be contained in the space M generated by the original frame while, for a type II SG-dual, one imposes that the range of the frame transform associated with the dual be contained in the range of the frame transform associated with the original frame. We characterize the uniqueness of both types of duals using the Gramian and dual Gramian operators which were introduced in an article by Ron and Shen and are known to play an important role in the theory of shift-invariant spaces.

In this work, the Balian-Low theorem is extended to Gabor ͑also called Weyl-Heisenberg͒ frames for subspaces and, more particularly, its relationship with the unique Gabor dual property for subspace Gabor frames is pointed out. To achieve this goal, the subspace Gabor frames which have a unique Gabor dual of type I ͑resp. type II͒ are defined and characterized in terms of the Zak transform for the rational parameter case. This characterization is then used to prove the Balian-Low theorem for subspace Gabor frames. Along the same line, the same characterization is used to prove a duality theorem for the unique Gabor dual property which is an analogue of the Ron and Shen duality theorem.

The well-known density theorem for one-dimensional Gabor systems of the form {e 2πimbx g(x − na)} m,n∈Z , where g ∈ L 2 (R), states that a necessary and sufficient condition for the existence of such a system whose linear span is dense in L 2 (R), or which forms a frame for L 2 (R), is that the density condition a b ≤ 1 is satisfied. The main goal of this paper is to study the analogous problem for Gabor systems for which the window function g vanishes outside a periodic set S ⊂ R which is a Z-shift invariant. We obtain measure-theoretic conditions that are necessary and sufficient for the existence of a window g such that the linear span of the corresponding Gabor system is dense in L 2 (S). Moreover, we show that if this density condition holds, there exists, in fact, a measurable set E ⊂ R with the property that the Gabor system associated with the same parameters a, b and the window g = χ E , forms a tight frame for L 2 (S).