Quantifying invariance properties of shift-invariant spaces

The norm of the commutator of a projection onto a closed subspace and an operator S can be understood as a quantitative measure of the lack of invariance of the space under S  . In this paper we study principal shift-invariant spaces V(φ)V(φ) and their invariance properties under arbitrary translations. It is known that no principal shift-invariant space with an integrable orthonormal basis generator can be fully translation-invariant. In this paper we present several quantitative versions of this statement.