Characterization of generators for multiresolution analyses with composite dilations (Q535987)

Summary: 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 \(L^2 (\mathbb R^n)\) (AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspace \(L^2(S)^{\vee}\). 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.