Kernels of Gramian operators for frames in shift-invariant subspaces

For an invertible n×n matrix B and Φ a finite or countable subset of L2(Rn), consider the collection X={ϕ(·-Bk):ϕ∈Φ,k∈Zn} generating the closed subspace M of L2(Rn). Our main objects of interest in this paper are the kernel of the associated Gramian G(.) and dual Gramian operator-valued functions. We show in particular that the orthogonal complement of M in L2(Rn) can be generated by a Parseval frame obtained from a shift-invariant system having m generators where . Furthermore, this Parseval frame can be taken to be an orthonormal basis exactly when almost everywhere. Analogous results in terms of dim(Ker(G(.))) are also obtained concerning the existence of a collection of m sequences in the orthogonal complement of the range of analysis operator associated with the frame X whose shifts either form a Parseval frame or an orthonormal basis for that orthogonal complement.