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On constructions of shearlets using incomplete quasi-beta functions in \(L^2(\mathbb{R}^3)\). (Chinese. English summary) Zbl 1363.42071
Summary: The structure of two sort of shearlet groups \(\mathbb{N}_{(d, s)}\) and \(\mathbb{H}_d (d = 1, 2, 3)\) is first investigated in this paper, and the isomorphic and automorphic groups for \(\mathbb{N}_{(d, s)}\) and \(\mathbb{H}_d\) are discussed. Next, by methods of group product, higher order groups are constructed. Henceforth, the corresponding relationships among unitary operators in \(L^2(\mathbb{R}^3)\) and associated shearlet groups and higher order groups are constructed, and some functions \(f\in L^2(\mathbb{R}^3)\) satisfied with the Parseval identity are developed underlying the spirit of group representation. Finally, concrete examples of shearlets in \(L^2(\mathbb{R}^3)\) are constructed by utilizing two kinds of incomplete quasi-beta functions.
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems