AB-wavelet Frames in L2(Rn)

In order to provide a unified treatment for the continuum and digital realm of multivariate data, Guo, Labate, Weiss and Wilson [Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 78-87] introduced the notion of AB-wavelets in the context of multiscale analysis. We continue and extend their work by studying the frame properties of AB-wavelet systems {D(A)D(B)Tk psi(l) (k is an element of Z(n);1 <= l <= L)} in L-2 (R-n). More precisely, we establish four theorems giving sufficient conditions under which the AB-wavelet system constitutes a frame for L-2(R-n). The proposed conditions are stated in terms of the Fourier transforms of the generating functions.