On The Construction And Properties Of Reducing Subspace Framelets

Framelet theory is one of the core issues of wavelet analysis. So far, the studyof framelets in L2(Rd)(especially in L2(R)) has seen significant achievements. Thestudy of subspace framelets has seen some progresses, but it is not systematic. Thisdissertation addresses the framelet theory in the setting of reducing subspaces ofL2(Rd).Let A be a d×d expansive matrix. A closed linear subspace X of L2(Rd) iscalled a reducing subspace if DX=X and TkX=X for each k∈Zd, whereThe concept of reducing subspace is a generalization of L2(Rd) and Hardyspace. The research on framelets in Hardy space can be dated back to the worksby Meyer in1990, Auscher in1992, Seip in1993, Volkmer in1995, etc.Our main work is as follows:Charper1is an introduction to this dissertation which includes the back-ground and main results.In Chapter2, we introduce the notion of frame multiresolution analysis (FM-RA), and investigate FMRA frame wavelets in the setting of reducing subspacesof L2(Rd). For a general expansive matrix, we obtain some sufcient conditions fora frame scaling function to generate an FMRA, and prove that an arbitrary reduc-ing subspace must admit an FMRA. For an expansive matrix A with|det A|=2,we establish a sufcient 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.In Chapter3, we introduce the notion of generalized multiresolution anal-ysis (GMRA) and develop GMRA-based construction procedures of Parsevalframelets in the setting of reducing subspaces of L2(Rd). For an expansive matrix,a unitary extension principle is established; in particular, for a general expan-sive matrix A with|det A|=2, an explicit construction of Parseval framelets isobtained.In Chapter4, we introduce the notion of generalized multiresolution structure(GMS) in the setting of reducing subspaces of L2(Rd). For a general expansivematrix, we obtain a necessary and sufcient condition for GMS, and prove theexistence of GMS in a reducing subspace. Using GMS, we obtain a pyramiddecomposition and a frame-like expansion for signals in reducing subspaces.In Chapter5, for an expansive matrix A with|det A|=2, we investigate com-pactly supported wavelets for L2(Rd). Starting with a pair of compactly supportedrefinable functions satisfying a mild condition, we obtain an explicit constructionof compactly supported Riesz basis wavelets for L2(Rd). This construction inheritsthe symmetry and anti-symmetry originated from refinable functions.In Chapter6, we investigate afne (quasi-afne) dual frame wavelets in thesetting of reducing subspaces of L2(Rd). We establish a frame and dual frame-preservation theorem between afne systems and quasi-afne systems, and ob-tain a Fourier-domain characterization of afne (quasi-afne) dual frame waveletswithout any decay assumptions. Furthermore, we also obtain a Fourier-domain characterization of afne Parseval frame wavelets.In Chapter7, for an expansive matrix A with|det A|=2, we investigatedimension function characterization of Parseval frame wavelets (PFWs) in thesetting of reducing subspaces of L2(Rd). It is proved that all semi-orthogonalPFWs (semi-orthogonal MRA PFWs) are precisely the ones with their dimen-sion functions being non-negative integer-valued (0or1). MRA PFWs are alsocharacterized.