On the relations between multiresolution analyses and wavelets

Abstract

This thesis is devoted to a study of the relations between multiresolution analyses and wavelets in generalized sense. First, we characterize the Riesz wavelets which are associated with multiresolution analyses (MRAs) and characterize the Riesz wavelets whose duals are also Riesz wavelets. The characterizations show that if a Riesz wavelet is associated with an MRA, then it has a dual Riesz wavelet. We then improve Wang`s characterization for a pair of biorthogonal wavelets to be associated with biorthogonal MRAs by showing that one of the two conditions in his characterization is redundant. We generalize the above results to frame multiresolution analyses (FMRAs) and frame wavelets. We characterize semi-orthogonal frame wavelets by generalizing the characterization of orthonormal wavelets. We then characterize those semi-orthogonal frame wavelets that are associated with FMRAs. Moreover, we introduce the concepts of quasi-biorthogonal FMRAs and quasi-biorthogonal frame wavelets which are natural generalizations of biorthogonal MRAs and biorthogonal wavelets, respectively. Necessary and sufficient conditions for quasi-biorthogonal FMRAs to admit quasi-biorthogonal wavelet frames are given and we characterize the pair of quasi-biorthogonal frame wavelets that are associated with quasi-biorthogonal FMRAs. On the other hand, we also study the relationship between MRAs and FMRAs. We characterize a scaling function of an FMRA and also characterize the spectrum of the `central space` of an FMRA, which determines the structure of the FMRA. We prove that an FMRA is always contained in an MRA, and then we characterize those MRAs that contain FMRAs in terms of the unique low-pass filters of the MRAs and the spectrums of the central spaces of the FMRAs to be contained. This characterization shows, in particular, that if the low-pass filter of an MRA is almost everywhere zero-free, as is the case of the MRAs of Daubechies, then the MRA contains no FMRAs other than i...