Frame properties of generalized shift-invariant systems in discrete setting

In this paper, we present a simple characterization of those sequences {u(p) : p is an element of P} subset of l(2)(Z(d)) for which purely shift-invariant systems are normalized tight frames for l(2)(Z(d)). As an application, a characterization for the Gabor system to be a normalized tight frame can be directly derived. Further, we prove that if the Calderon condition holds, then such purely shift-invariant systems are still normalized tight frames for l(2)(Z(d)). At the end of the paper, a sufficient condition for those purely shift-invariant systems to be frames for l(2)(Z(d)) is established which is a variant of the classical wavelet systems to be frames for L-2(R-d).