Properties of oblique dual frames in shift-invariant systems

Gramian analysis is used to study properties of a shift-invariant system X = { ϕ ( ⋅ − B k ) :   ϕ ∈ Φ , k ∈ Z n } , where B is an invertible n × n matrix and Φ a finite or countable subset of L 2 ( R n ) under the assumption that the system forms a frame for the closed subspace M of L 2 ( R n ) . In particular, the relationship between various features of such system, such as being a frame for the whole space L 2 ( R n ) , being a Riesz sequence and having a unique shift-generated dual of type I or II is discussed in details. Several interesting examples are presented.