%0 Journal Article
%T A Note on the Structure of Affine Subspaces of  <i>L</i><sup>2</sup>(R<i><sup>d</sup></i>)
%A Fengying Zhou
%A Xiaoyong Xu
%J Advances in Pure Mathematics
%P 62-70
%@ 2160-0384
%D 2015
%I Scientific Research Publishing
%R 10.4236/apm.2015.52008
%X <p style=\"text-align:justify;\">
	<span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\">This paper investigates the structure of general affine subspaces of <span style=\"white-space:nowrap;\"><span style=\"white-space:nowrap;\">&lt;i&gt;L&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt;(R&lt;i&gt;&lt;sup&gt;d&lt;/sup&gt;&lt;/i&gt;)</span></span> </span><span></span><span style=\"font-size:10.0pt;font-family:;\" \"=\"\"><span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\">. For a </span><i><span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\">d</span></i><span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\"> กม </span><i><span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\">d</span></i><span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\"> expansive matrix </span><i><span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\">A</span></i><span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\">, it shows that
every affine subspace can be decomposed as an orthogonal sum of spaces each of
which is generated by dilating some shift invariant space in this affine
subspace, and every non-zero and non-reducing affine subspace is the orthogonal
direct sum of a reducing subspace and a purely non-reducing subspace, and every
affine subspace is the orthogonal direct sum of at most three purely
non-reducing subspaces when |det</span><i><span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\">A</span></i><span style=\"font-size:12px;font-family:Verdana;line-height:1.5;\">| =
2.</span></span> 
</p>
%K Affine Subspace
%K Reducing Subspace
%K Shift Invariant Subspace
%K Orthogonal Sum
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=53586