Frames generated by actions of locally compact groups

Let G be a second countable, locally compact group which is either compact or abelian, and let rho be a unitary representation of G on a separable Hilbert space Hrho. We examine frames of the form {rho(x)f j : x ∈ G, j ∈ I} for families {fj} {j ∈ I} in Hrho. In particular, we give necessary and sufficient conditions for the joint orbit of a family of vectors in mathcal{Hrho to form a continuous frame.;We pay special attention to this problem in the setting of shift invariance. In other words, we fix a larger second countable locally compact group Gamma ⊇ G containing G as a closed subgroup, and we let rho be the action of G on L2(Gamma) by left translation. In both the compact and the abelian settings, we introduce notions of Zak transforms on L2(Gamma) which simplify the analysis of group frames. Meanwhile, we run a parallel program that uses the Zak transform to classify closed subspaces of L2(Gamma) which are invariant under left translation by G. The two projects give compatible outcomes. This dissertation contains previously published material.