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

    Authors

    Q. Gu;D. G. Han

    Comments

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    Abbreviated Journal Title

    J. Funct. Anal.

    Keywords

    Affine subspaces; Wavelet frames; Translation and dilation reducing; subspaces; Shift-invariant subspaces; INVARIANT SUBSPACES; Mathematics

    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. (C) 2010 Elsevier Inc. All rights reserved.

    Journal Title

    Journal of Functional Analysis

    Volume

    260

    Issue/Number

    6

    Publication Date

    1-1-2011

    Document Type

    Article

    Language

    English

    First Page

    1615

    Last Page

    1636

    WOS Identifier

    WOS:000286559700002

    ISSN

    0022-1236

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