Methods from abstract harmonic analysis are used to derive a new formulation of the wavelet dimension function and its natural generalizations to higher dimensions. By means of this abstract description, necessary and sufficient conditions are derived for a multiwavelet in N dimensions, relative to an arbitrary expansive integral matrix A, to be a multiwavelet that arises from a multiresolution analysis (MRA), i.e., is an MRA wavelet. Even in the classical case, it is shown that this abstract approach gives new results.
2000 Academic Press 1 , ..., n a ``multiplicity function,'' which, in the classical case, agrees with, and gives an abstract description of, the very powerful wavelet dimension function D introduced by Auscher. Using this abstract description, we derive some necessary, some sufficient, and some both necessary and sufficient conditions for a multiwavelet to be a multiresolution analysis (MRA) wavelet. In applications of wavelets to computational problems, the MRA wavelets appear to be the ones of value thus far, the non-MRA wavelets being regarded by some as merely pathological examples. Though these non-MRA wavelets exist in abundance, and in fact constitute a theoretically fascinating and complex collection of wavelets, clearly worthy