A Note on the Structure of Affine Subspaces of <i>L</i>²(R<i>^d</i>)

Abstract

This paper investigates the structure of general affine subspaces of <i>L</i>²(R<i>^d</i>) . For a d × d expansive matrix A , 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 A | = 2.