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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.
##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems