Uncertainty principles in finitely generated shift-invariant spaces with additional invariance

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in L 2 ( R d ) . We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in L 2 ( R ) whose generator and its translates form a Riesz basis.