Characterizations of tight frame wavelets associated with frame multiresolution analysis

Abstract

The orthonormal multiresolution analysis with a single scaling function was introduced in order to construct an orthonormal wavelet basis. The conditions for an orthonormal wavelet to be associated with a multiresolution analysis were known. In contrast to a multiresolution analysis, a frame multiresolution analysis with a single scaling function may or may not have a singly generated wavelet. It was shown that there always exist at most two frame wavelets derived from a frame multiresolution analysis. In this thesis, we introduce the concepts of semi-orthogonal tight frame wavelets, which are generalizations of orthonormal wavelets, and give some examples derived from a frame multiresolution analysis. We find the necessary and sufficient conditions for which two wavelets are semi-orthogonal tight frame wavelets. These conditions include the orthonormal case. Finally, we characterize the semi-orthogonal tight frame wavelets which are associated with a frame multiresolution analysis.