Lifts

Abstract

Let be a semi-direct product, as considered in Sect. 2.2. Here, we are interested in operators, which we call lift operators for obvious reasons. Observe that, via the isomorphism induces a lift of Besicovitch almost periodic functions. We are mainly interested in identifying the action of the quasi-regular representation on by analyzing the Fourier transform of the lift. Thus, the first, and more natural, requirement on the lift operation is to intertwine the quasi-regular representation acting on with the left regular representations on. We call these type of lifts left-invariant. We show that, under some mild regularity assumptions on L, left-invariant lifts coincide with wavelet transforms, as defined in Sect. 2.2.3. These kind of lifts have been extensively studied in, e.g., [33], and related works. Unfortunately, left-invariant lifts have a huge drawback for our purposes: they never have an invertible non-commutative Fourier transform. The second part of this chapter is then devoted to the generalization of the concept of cyclic lift, introduced in [73] exactly to overcome the above problem. In this general context, we will present a cyclic lift as a combination of an almost-left-invariant lift and a centering operation, as defined in Definition 2.2. As a consequence, we obtain a precise characterization of the invertibility of for these lifts.