Affine frames, quasi-affine frames, and their duals

The notion of quasi-affine frame was recently introduced by Ron and Shen 9] in order to achieve shift-invariance of the discrete wavelet transform. In this paper, we establish a duality-preservation theorem for quasi-affine frames. Furthermore, the preservation of frame bounds when changing an affine frame to a quasi-affine frame is shown to hold without the decay assumptions in 9]. Our consideration leads naturally to the study of certain sesquilinear operators which are defined by two affine systems. The translation-invariance of such operators is characterized in terms of certain intrinsic properties of the two affine systems. This revised version was published online in August 2006 with corrections to the Cover Date.