Characterization of Generators for Multiresolution Analyses with Composite Dilations

Abstract

This paper introduces multiresolution analyses with composite dilations (AB-MRAs) and addresses frame multiresolution analyses with composite dilations in the setting of reducing subspaces of $L^2\left({?}^n\right)$ (AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspace $L^2{\left(S\right)}^{\vee }$. For a general expansive matrix, we obtain the characterizations for a scaling function to generate an AB-RMRA, and the main theorems generalize the classical results. Finally, some examples are provided to illustrate the general theory.