| Wavelet analysis is a new mathematical branch which just appeared recently.It has attracted a great deal of interest and effort of mathematical workers andexperts and scholars of other fields since it appeared. Construction of waveletbases is an important content. In this thesis, we construct a kind of waveletswhose Fourier transform are supported in an interval In. Furthermore, thewavelet V0 subspaces generated by the wavelets are invariant under thetranslate operators gn={Tm/2n:m∈Z},n∈N. We study the construction ofwavelet sets of R; Because frame wavelet are more elastic than wavelets,weconstruct frame wavelet and study the relation between frame wavelet sets andwavelet sets. This thesis consists of four sections:In section one, the background and the development of wavelet analysis areintroduced. Especially, we introduce the importance of the wavelet basesConstruction in both theory and application, the development of wavelet basesconstruction. At last, the structure and main results of this thesis areintroduced.In section two, we construct a kind of wavelets whose Fourier transform aresupported compactly. Given a wavelet, its translations under a certain scalegenerate subsets of L2(R),which are called wavelet subspaces.These wavelet subspaces generate a generalized multiresolution analysis.We obtain a necessary and sufficient condition for a functionψto be a wavelet,whose Fourier transform are supported in a defined interval In.Furthermore, the wavelet V0 subspaces generated by the wavelets are invariantunder the translate operators gn={Tm/2n:m∈Z},n∈N. In section three, the construction of wavelet sets are studied. InGeneral a set can be judged as a wavelet set if the groups {2nW:n∈Z} and{W+2kπ:k∈Z} are partitions of R. In this section, we define a map between aset W and a wavelet set E,thus the set can be judged as a wavelet set if themap is a bijection. Generalizing the translation setΓ={2kπ:k∈Z} and dilation set D={2n:n∈Z}, we drawa conclusion:ifΩis adomain of R, (Ω,Γ) is a spectral pair,Ωis a multiplicative D-tile of R,thenψ=(?)Ωis a (D,Γ) wavelet; Conversely, ifψ=(?)Ωis a (D,Γ) wavelet, and0∈F,then (Ω,Γ) is a spectral pair,Ωis a multiplicative D-tile of R. Inthis chapter, we construct a wavelet according to the equation (?)=h1(?)w1+h2(?)w2.In section four, frame wavelet sets are studied. Firstly, we introduce somerelated notions such as frame of Hilbert space, tight frame, normalized tightframe, frame wavelet, normalized tight frame wavelet, frame wavelet set, tightframe wavelet set, normalized tight frame wavelet set etc. Secondly, we constructa frame wavelet which supports in a wavelet set E using a necessary conditionof frame wavelet. When we change the support, we can obtain a series of framewavelets. Basing on the closed relation between wavelet sets and normalized tightframe wavelet set, We obtain a necessary and sufficient condition for a normalizedtight frame wavelet set to beawavelet set.
|
PDF Full Text Request