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On the Characterization of Nonuniform Wavelet Sets on Positive Half Line

The concepts of wavelet and multiresolution analysis has been extended to many different setups. One can replace the dilation factor 2 by an integer and one needs to construct wavelets to generate the whole space . In general, in higher dimensions, it can be replaced by a dilation matrix , in which case the number of wavelets required is . But in all these cases, the translation set is always a group. In the two papers [5, 6], Gabardo and Nashed considered a generalization of Mallat's [13] celebrated theory of MRA, in which the translation set acting on the scaling function associated with the MRA to generate the subspace is no longer a group, but is the union of and a translate of . More precisely, this set is of the form , where is an integer, , is an odd integer relatively prime to . They