Research On Some Problems Of Sampling Theorem And Parseval Frame Wavelet

Since 1980's, wavelet analysis has been a popular field in scientific research. Its application almost involves all the branches in natural science and engineering technology. Nowadays, it has been a powerful tool for exploring and solving many complicated problems in natural science and engineering computation. Wavelet analysis is applied widely in signal processing and digital communications, and the sampling theorem plays a crucial role in signal processing and digital communications too: it tells us how to convert an analog sinal into a sequence of numbers. The sampling theorem in the wavelet subspaces is a popular field in modern stage. As the generalization of orthogonal bases, Parseval frame wavelet almost preserve the nice properties of orthogonal bases except the orthogonality. So far, frame theory is successfully used by many fields such as signal processing, mapping processing, data compress and sampling theorem. Thus, it is important to study Parseval frame wavelet.The dissertation is composed of two parts. In part one, we discuss the construction and property of Parseval frame wavelets and Parseval frame scaling functions. In part two, we investigate the lattic sampling theorem in subspace of L2 ( R ) and sampling theorem associated to frame in multiwavelet subspace. The important and significant results obtained in this dissertation can be sumarized as follows:1.We outline the development histories of wavelet analysis and some current research situations of wavelet theory and the sampling theorem in the wavelet subspaces. Main results of the thesis will be presented.2.Notice that all the waveltes constructed by general multiresolution analysis(GMRA) have the property of sime-orthogonal, so we firstly study the property of semi-orthogonal frame wavelet, and get the necessary conditions for the semi-orthogonal frame wavelet to hold, and prove the necessary and sufficent conditions for the semi-orthogonal frame wavelet to hold. Then, we discover that all the GMRA Parseval wavelets are equivalent to the affine system {T kψ: k∈Zd} being the Parseval frame of a closed subspace W0 such that improving the theoretical basis of GMRA. Finally, by the minimal vector-filter, we depict an important property of frame wavelets.3.We consider the Parseval frame scaling function set and MSF Parseval frame wavelets, and get several significant results. At first, we show the necessary and sufficent conditions for a measurable set S to be the Parseval frame scaling function set. Then, by discussing the property of the measurable set S, we provide a method of construction of Parseval frame scaling function set, and prove that all the Parseval frame scaling function sets are obtained by the construction method. At last, by discovering the relation between the MSF Parseval frame wavelet and the Parseval frame scaling function set, we get two important results: Firstly, each MSF Parseval frame wavelet with the additional property arise from the Parseval frame scaling function set. Secondly, for the dimention function of MSF Parseval frame waveletψsatisfies Dψ(ξ) =χ? (ξ)≤1, there must exist a Parseval frame scaling function set s such thatψ? =χBS \S. By the MSF Parseval frame wavelet defined by its Fourier transformψ? =χW, the study of the property of the wavelet is reduced to the study of the property of the set W. So, we not only provide a new construction method of the MSF Parseval frame wavelet, but also have a better understanding of wavelets in general.4.We generalize the results in [101] to the Parseval frame scaling function, and derive the necessary and sufficent conditions for the measurable set G to be the support G of the Fourier transform of the Parseval frame scaling functions. In the special case, by studying the relations among the sets G,τ( G),12G and G \12G , we discover the difference between the MRA scaling function and the FMRA scaling function. And, by considering the support of the Fourier transform of the band-limited scaling function, we get an important property of the band-limited scaling function.5.We discuss the lattice sampling theorem based on frame in subspace of L2 ( R ). Through investigating the property of the translation operator and the dilation operator, we prove a necessary and sufficient condition for a > 0 lattice sampling theorem to hold in subspace of L2 ( R ), and obtain the formula of lattice sampling function in frequency domain.6 . We study the uniform noninteger sampling theorem and irregular sampling in multiwavelet subspace. Firstly, by Zak transform, we provide a construction method for the uniform noninteger sampling function with the additional property, and establish the uniform noninteger sampling theorem based on frame in multiwavelet subspace. Then, by irregular sampling be also useful in practice, we consider the irregular sampling theorem based on frame, and obtain the necessary condition for the irregular sampling theorem to hold in multiwavelet subspace. At last, from our obtained results, we discuss the perturbations of uniform noninteger sampling, and establish the algorithm for perturbations of uniform noninteger sampling in multiwavelet subspace.Finally , sums up the work and research results, brings forward future research considerations and objects.