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[Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {G}}= {\mathbb {K}}\ltimes {\mathbb {H}}$$\end{document} be a semi-direct product, as considered in Sect. 2.2. Here, we are interested in operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L:L^2({\mathbb {H}})\rightarrow L^2({\mathbb {G}})$$\end{document}, which we call lift operators for obvious reasons. Observe that, via the isomorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*:B_2({\mathbb {H}})\rightarrow L^2({\mathbb {H}}^\flat )$$\end{document} any lift \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L:L^2({\mathbb {H}}^\flat )\rightarrow L^2({\mathbb {G}}^\flat )$$\end{document} induces a lift \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L':B_2({\mathbb {H}})\rightarrow B_2({\mathbb {G}})$$\end{document} of Besicovitch almost periodic functions. We are mainly interested in identifying the action of the quasi-regular representation on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in L^2({\mathbb {H}})$$\end{document} by analyzing the Fourier transform of the lift \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Lf\in L^2({\mathbb {G}})$$\end{document}. Thus, the first, and more natural, requirement on the lift operation is to intertwine the quasi-regular representation acting on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {H}})$$\end{document} with the left regular representations on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {G}})$$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{Lf}(T^\lambda )$$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{Lf}(T^\lambda )$$\end{document} for these lifts.]
Published: Jun 12, 2018
Keywords: Left-invariant lift; Cyclic lift; Wavelet transforms
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