Characterizations of tight frame wavelets with special dilation matrices

We study all generalized low-pass filters and tight frame wavelet with special dilation matrix M (M-TFW), where M satisfy M^d = 2I_d and generates the checkerboard lattice. Firstly, we study the pseudo-scaling function, generalized low-pass filters and multiresolution analysis tight frame wavelets with dilation matrix M(MRA M-TFW), and give some important characterization about them. Then, we characterize all M-TFW by showing that they correspond precisely to those for which the dimension function is non-negative integer-valued.