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Frame properties of generalized shift-invariant systems in discrete setting

In this paper, we present a simple characterization of those sequences for which purely shift-invariant systems are normalized tight frames for . 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 Calderón condition holds, then such purely shift-invariant systems are still normalized tight frames for . At the end of the paper, a sufficient condition for those purely shift-invariant systems to be frames for is established which is a variant of the classical wavelet systems to be frames for .