Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of L2(Rd)

Abstract

This article addresses frame multiresolution analyses (FMRAs) and FMRA frame wavelets in the setting of reducing subspaces of L2(Rd). For a general expansive matrix, we obtain a characterization and some conditions for a frame-scaling function to generate an FMRA, and we prove that an arbitrary reducing subspace must admit an FMRA. For an expansive matrix M with |det⁡M|=2, we establish a sufficient and necessary condition for FMRAs to admit a single FMRA frame wavelet, give an explicit construction of FMRA frame wavelets, and study the relation between s-frame wavelets and FMRA frame wavelets. These results are also new in the setting of L2(Rd).