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Construction of Symmetric Complex Wavelets and Tight Framelets with Dilation 3

Author: ShenYanFeng
Tutor: YangShouZhi
School: Shantou University
Course: Applied Mathematics
Keywords: Scaling function Orthogonality Symmetricity Smoothness Wavelet frame
CLC: O174.2
Type: Master's thesis
Year: 2011
Downloads: 49
Quote: 0
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Abstract


Except Haar wavelet there are no other wavelets which are compactly sup-ported, orthogonal and symmetric. While multi-band wavelets, multiwaveletsand high-dimension wavelets may have all these properties. Researchers arefocusing on these wavelets these years. We will concentrate in studying themethods of construction of wavelets and wavelet frames with dilation 3 in thisthesis. This thesis is divided into four chapters, The main contents of which areformed as follows:In chapter 1, we introduce the history of wavelet analysis and the currentresearch in the world. Then outline the construction of this thesis.In chapter 2 give some basic concepts and results in wavelet analysis, whichwill be used in this thesis.In chapter 3 discuss the construction of compactly supported orthogonalsymmetric complex wavelet function with dilation 3. At first we construct scal-ing function with dilation 3 satisfies all the property mentioned above. Then bythe method of reducing the order of polynomial vector progressively to constructa paraunitary matrix, we present an algorithm to obtain the wavelet symbols（?）1（ξ）, （?）2（ξ）. What is more, we prove that the corresponding symmetric complexwavelets, whose translated and scaled system form a basis of L2（R）.Pseudo-splines were first introduced in [30] by Daubechies to constructtight framelets, many works about pseudo-splines have been done. Inchapter 4 introduce the concept of 3 band pseudo-splines, present a method toconstruct tight framelets with better regularity from orthogonal scaling function（Haar Type）. Furthermore, based on 3 band pseudo-splines, we also propose akind of compactly supported symmetric complex tight framelet.

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