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 $L^2\left({\mathbb{R}}^d\right)$. 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 $L^2\left({\mathbb{R}}^d\right)$.