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Let N be a step two connected and simply connected non commutative nilpotent Lie group which is square-integrable modulo the center. Let Z be the center of N . Assume that N = P ⋊ M such that P , and M are simply connected, connected abelian Lie groups, M acts non-trivially on P by automorphisms and dim P/Z = dim M . We study bandlimited subspaces of L 2 (N ) which admit Parseval frames generated by discrete translates of a single function. We also find characteristics of bandlimited subspaces of L 2 (N ) which do not admit a single Parseval frame. We also provide some conditions under which continuous wavelets transforms related to the left regular representation admit discretization, by some discrete set Γ ⊂ N . Finally, we show some explicit examples in the last section.

Let G = N H be a Lie group where N, H are closed connected subgroups of G, and N is an exponential solvable Lie group which is normal in G. Suppose furthermore that N admits a unitary character χ λ corresponding to a linear functional λ of its Lie algebra. We assume that the map h → Ad h −1 * λ defines an immersion at the identity of H. Fixing a Haar measure on H, we consider the unitary representation π of G obtained by inducing χ λ . This representation which is realized as acting in L 2 (H, dµ H ) is generally not irreducible, and we do not assume that it satisfies any integrability condition. One of our main results establishes the existence of a countable set Γ ⊂ G and a function f ∈ L 2 (H, dµ H ) which is compactly supported and bounded such that {π (γ) f : γ ∈ Γ} is a frame. Additionally, we prove that f can be constructed to be continuous. In fact, f can be taken to be as smooth as desired. Our findings extend the work started in [28] to the more general case where H is any connected Lie group. We also solve a problem left open in [28]. Precisely, we prove that in the case where H is an exponential solvable group, there exist a continuous (or smooth) function f and a countable set Γ such that {π (γ) f : γ ∈ Γ} is a Parseval frame. Since the concept of well-localized frames is central to time-frequency analysis, wavelet, shearlet and generalized shearlet theories, our results are relevant to these topics and our approach leads to new constructions which bear potential for applications. Moreover, our work sets itself apart from other discretization schemes in many ways. (1) We give an explicit construction of Hilbert frames and Parseval frames generated by bounded and compactly supported windows. (2) We provide a systematic method that can be exploited to compute frame bounds for our constructions. (3) We make no assumption on the irreducibility or integrability of the representations of interest.

In this paper, we study the Plancherel measure of a class of non-connected nilpotent groups which is of special interest in Gabor theory. Let G be a time-frequency group. That is Even in the case where G is not type I, we are able to obtain a decomposition of the left regular representation of G into a direct integral decomposition of irreducible representations when d = 1. Some interesting applications to Gabor theory are given as well. For example, when B is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of G.