An Abstract Interpretation of the Wavelet Dimension Function Using Group Representations

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.