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Wavelet frames for (not necessarily reducing) affine subspaces II: The structure of affine subspaces

机译：仿射子空间的小波帧II：仿射子空间的结构

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This is a continuation of the investigation into the theory of wavelet frames for general affine subspaces. The main focus of this paper is on the structural properties of affine subspaces. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces, while every reducing subspace (with respect to the dilation and translation operators) is the orthogonal direct sum of two purely non-reducing ones. This result is obtained through considering the basic question as to when the orthogonal complement of an affine subspace in another one is still affine. Motivated by the fundamental question as to whether every affine subspace is singly-generated, and by a recent result that every singly generated purely non-reducing subspace admits a singly generated wavelet frame, we prove that every affine subspace can be decomposed into the direct sum of a singly generated affine subspace and some space of "small size". As a consequence we establish a connection between the above mentioned two questions.
机译：这是对一般仿射子空间小波框架理论研究的延续。本文的主要重点是仿射子空间的结构性质。我们证明，每个仿射子空间都是至多三个纯非归约子空间的正交正和，而每个归约子空间（相对于扩张和平移算子）都是两个纯非归约子空间的正交正和。通过考虑关于另一个仿射子空间中的正交补数何时仍是仿射的基本问题来获得此结果。受关于每个仿射子空间是否单独生成的基本问题的启发，以及最近的结果（即每个单独生成的纯非约化子空间都接受一个单独生成的小波框架），我们证明了每个仿射子空间都可以分解为直接和单个生成的仿射子空间和一些“小尺寸”空间。结果，我们在上述两个问题之间建立了联系。

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