Construction Of Symmetric Complex Wavelets And Tight Framelets With Dilation 3

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.