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

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