Wavelet frames for (not necessarily reducing) affine subspaces II: The structure of affine subspaces |
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Authors: | Qing Gu Deguang Han |
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Affiliation: | a Department of Mathematics, East China Normal University, Shanghai 200062, PR China b Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA |
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Abstract: | 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. |
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Keywords: | Affine subspaces Wavelet frames Translation and dilation reducing subspaces Shift-invariant subspaces |
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