Vanishing moment conditions for wavelet atoms in higher dimensions
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- 作者：Hartmut Führ
- 关键词：Square ; integrable group representation ; Continuous wavelet transform ; Coorbit spaces ; Banach frames ; Irregular wavelet frames ; Vanishing moments ; Nonlinear approximation ; Shearlets ; Anisotropic wavelet systems ; 42C15 ; 42C40 ; 46E35
- 刊名：Advances in Computational Mathematics
- 出版年：2016
- 出版时间：February 2016
- 年：2016
- 卷：42
- 期：1
- 页码：127-153
- 全文大小：384 KB
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1. RWTH Aachen, Aachen, Germany - 刊物类别：Computer Science
- 刊物主题：Numeric Computing
Calculus of Variations and Optimal Control
Mathematics
Algebra
Theory of Computation - 出版者：Springer U.S.
- ISSN：1572-9044
文摘
We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in L2(ℝd), but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of GL(ℝd), the so-called dilation group.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Gröchenig, and they can be informally described as follows: Given a function ψ ∈ L2(ℝd) satisfying fairly mild decay, smoothness and vanishing moment conditions, any sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation. Keywords Square-integrable group representation Continuous wavelet transform Coorbit spaces Banach frames Irregular wavelet frames Vanishing moments Nonlinear approximation Shearlets Anisotropic wavelet systems