Let \({\mathbb {G}}= {\mathbb {K}}\ltimes {\mathbb {H}}\) be a semi-direct product, as considered in Sect.  2.2. Here, we are interested in operators \(L:L^2({\mathbb {H}})\rightarrow L^2({\mathbb {G}})\), which we call lift operators for obvious reasons. Observe that, via the isomorphism \(\sigma ^*:B_2({\mathbb {H}})\rightarrow L^2({\mathbb {H}}^\flat )\) any lift \(L:L^2({\mathbb {H}}^\flat )\rightarrow L^2({\mathbb {G}}^\flat )\) induces a lift \(L':B_2({\mathbb {H}})\rightarrow B_2({\mathbb {G}})\) of Besicovitch almost periodic functions. We are mainly interested in identifying the action of the quasi-regular representation on \(f\in L^2({\mathbb {H}})\) by analyzing the Fourier transform of the lift \(Lf\in L^2({\mathbb {G}})\). Thus, the first, and more natural, requirement on the lift operation is to intertwine the quasi-regular representation acting on \(L^2({\mathbb {H}})\) with the left regular representations on \(L^2({\mathbb {G}})\). 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 \(\widehat{Lf}(T^\lambda )\). 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 \(\widehat{Lf}(T^\lambda )\) for these lifts.


Left-invariant lift Cyclic lift Wavelet transforms 

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Laboratoire des Signaux et des SystèmesCéntraleSupelecGif sur YvetteFrance
  2. 2.LSISUniversité de ToulonLa Garde CedexFrance

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