Minimal generator sets for finitely generated shift-invariant subspaces of L2(Rn)

Let S be a shift-invariant subspace of L2(Rn) defined by N generators and suppose that its length L, the minimal number of generators of S, is smaller than N. Then we show that at least one reduced family of generators can always be obtained by a linear combination of the original generators, without using translations. In fact, we prove that almost every such combination yields a new generator set. On the other hand, we construct an example where any rational linear combination fails.