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On the Construction and Properties of Reducing Subspace Framelets
Author: ZhouFengYing
Tutor: LiYunZhang
School: Beijing University of Technology
Course: Probability Theory and Mathematical Statistics
Keywords: reducing subspace frame wavelet frame multiresolution analysis generalized multiresolution analysis generalized multiresolution structure
CLC: O174.2
Type: PhD thesis
Year: 2012
Downloads: 48
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Abstract
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 background and main results.In Chapter2, we introduce the notion of frame multiresolution analysis (FMRA), 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 reducing subspace must admit an FMRA. For an expansive matrix A withdet 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 sframe wavelets and FMRA frame wavelets.In Chapter3, we introduce the notion of generalized multiresolution analysis (GMRA) and develop GMRAbased 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 expansive matrix A withdet 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 framelike expansion for signals in reducing subspaces.In Chapter5, for an expansive matrix A withdet A=2, we investigate compactly 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 antisymmetry originated from refinable functions.In Chapter6, we investigate afne (quasiafne) dual frame wavelets in thesetting of reducing subspaces of L2(Rd). We establish a frame and dual framepreservation theorem between afne systems and quasiafne systems, and obtain a Fourierdomain characterization of afne (quasiafne) dual frame waveletswithout any decay assumptions. Furthermore, we also obtain a Fourierdomain characterization of afne Parseval frame wavelets.In Chapter7, for an expansive matrix A withdet A=2, we investigatedimension function characterization of Parseval frame wavelets (PFWs) in thesetting of reducing subspaces of L2(Rd). It is proved that all semiorthogonalPFWs (semiorthogonal MRA PFWs) are precisely the ones with their dimension functions being nonnegative integervalued (0or1). MRA PFWs are alsocharacterized.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Theory of functions > Fourier analysis ( classical harmonic analysis )
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