Wavelet frames for (not necessarily reducing) affine subspaces

An affine subspace is a closed linear subspace of L2(R) generated by an affine system for some subset Φ⊆L2(R). Among affine subspaces, those that are reducing with respect to translation and dilation operators are well understood. The existence of singly generated wavelet frames for each reducing subspace has long been established, yet most affine subspaces are not reducing. This naturally leads to the question of whether every affine subspace admits a singly generated Parseval wavelet frame. We show that if an affine subspace is singly generated (i.e., if Φ={ψ}), then it admits a Parseval wavelet frame with at most two generators. We provide some sufficient conditions under which a singly generated affine subspace admits a singly generated Parseval wavelet frame. In particular, this is true whenever and is a Bessel sequence.