Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups
Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups
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DOI：
10.1016/j.acha.2016.03.003
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发表时间：
2014-07
期刊：
影响因子：
--
通讯作者：
Hartmut Fuhr;R. R. Tousi-R.
中科院分区：
文献类型：
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作者：
Hartmut Fuhr;R. R. Tousi-R.
我们考虑与半直积群 G= R d⋊ H 的平方可积、不可约拟正则表示相关的坐标理论。最近已经建立了这种非常一般设置的坐标空间的存在，以及用于分析可在坐标方案中使用的矢量和原子的具体消失矩准则。这些标准取决于对扩张群的对偶作用的相当技术性的假设，本文的主要目的之一就是大大简化这些假设。然后，我们继续验证大类膨胀群的假设，特别是任意维度的所有阿贝尔膨胀群，以及称为广义剪切波膨胀群的类，包含并扩展二维及更高维度中使用的剪切波膨胀群的所有已知示例。我们解释了如何从相同维度的某些交换结合代数系统地构造这些群，并给出了第三维和第四维剪切群的完整列表，直到共轭。在后一种情况下，发现了三个以前未知的群体。结果，对于所有涉及的群，都建立了由紧支撑小波组成的巴纳赫框架的存在，并且在整个坐标空间范围内同时收敛。
We consider the coorbit theory associated to a square-integrable, irreducible quasi-regular representation of a semidirect product group G= R d⋊ H. The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups in arbitrary dimensions, as well as a class called generalized shearlet dilation groups, containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups are found. As a result, the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved.