Reducing subspace frame multiresolution analysis and frame wavelets

Abstract

Multiresolution analysis (<strong>MRA</strong>) and frame multiresolution analysis (<strong>FMRA</strong>) in $L^2(\mathbb R)$ play a significant role in the construction of wavelets and frame wavelets for $L^2(\mathbb R)$. In this paper, the notions of <strong>MRA</strong> and <strong>FMRA</strong> in a reducing subspace of $L^2(\mathbb R)$ are introduced, from which the construction of wavelets and frame wavelets for this subspace is obtained. Many examples are also provided to illustrate the general theory. In particular, most of them are about the space with its frequency lying in $[0,\infty)$, which is closely related to various Paley-Wiener Theorems.