A Characterization of Affine Dual Frames in L²($\text{R}$ⁿ)

Abstract

We give a characterization of all (quasi)affine frames in L²($\text{R}$ⁿ) which have a (quasi)affine dual in terms of the two simple equations in the Fourier transform domain. In particular, if the dual frame is the same as the original system, i.e., it is a tight frame, we obtain the well-known characterization of wavelets. Although these equations have already been proven under some special conditions we show that these characterizations are valid without any decay assumptions on the generators of the affine system.