Riesz Sequences and Frames of Exponentials associated with non-full rank lattices.

Abstract

Let Rd be a measurable set of nite positive measure (not necessarily bounded). Let (cj)kj =1 be a given collection of vectors in Rd, and let H be the dual lattice of a full rank lattice K Rd. For 2 Rd, let e denote the exponential e (x) := e2 ih ;xi; x 2 Rd: It is known that, the collection E( ) := fe : 2 g; where = f(cj + h) 2 Rd : h 2 H; j 2 f1; :::; kgg; forms Riesz basis on Rd if the domain is a k-tile domain and if, in addition, it satis es an extra arithmetic property, called the admissibility condition. The theory of shift invariant spaces generated by the full rank lattice K plays an important role to analyze and solve the above problem. The main goal of this thesis is to study a variant of the problem above where the dual lattice H is replaced by a non-full rank lattice in Rd. In particular, given an at most countable index set J and a collection of vectors (cj)j2J Rd, we examine the existence of Riesz sequences, frames and Riesz bases of the form E( ) := fe : 2 g; where = f(cj +h) 2 Rd : h 2 H; j 2 Jg; on Rd as above, and H, a non-full rank lattice in Rd. Our results are obtained using an extention of the theory of shift invariant subspaces of L2(Rd), where the shifts are now generated by a non-full rank lattice in Rd.