Theory Of Multiframelets And Sampling Theorem In Sobolev Spaces

In this paper, I commit myself to multiframelets and multiwavelet samplingtheorem in Sobolev spaces ????(???) , and the framelets and multiwavelets in thespecial Sobolev spaces ??2(???) , ??≥1 .The history and studying stative of wavelet analysis are introduced con-cisely in Chapter 1.In Chapter 2, we give some conceptions and notations used frequently inthis paper.In Chapter 3, we mainly study the ?? -multiframelets in (????(???),?????(???))with ??∈?+ and ?? being an isotropic dilation matrix. The Bessel property ofmultiframelets in (????(???),?????(???)) is systematically studied. It is found thatthe multiframelets in ????(???) does not necessarily have vanishing moments.Moreover, the conditions for the Bessel property of multiframelets in ?????(???)take great di?erences from those of framelets. Precisely, the property is not onlyrelated to frame functions themselves but also to re?nable function vectors. Af-ter constructing a class of re?nable function vectors satisfying Bessel property,we give the algorithm for dual ?? -multiframelets in (????(???),?????(???)) .The traditional wavelet (multiwavelet) sampling theorem only works forthe signals in wavelet (multiwavelet) subspaces. Suppose ??∈??2(?) is con-tinuous but does not belong to any wavelet (multiwavelet) subspaces, or it isvery di?cult to check whether it belongs to some subspace, then the traditionalsampling theorem is invalid. Based on Chapter 3, we construct a special classof dual multiframelets in Chapter 4, from which we then establish the multi-wavelet sampling theorem in ????(?) with ?? > 1/2 . The continuous signals in??2(?) can be precisely reconstructed by this sampling theorem.In Chapter 5, we give a fast algorithm for raising the approximation orderof re?nable function vectors, and give the parametrization of orthogonal andsymmetric multiwavelets. (I) Two-scale Similarity Transform (TST) is an im- portant method for raising approximation order of re?nable function vectors.However, implementing TST each time, the approximation order is raised byonly one order. In this chapter, a general algorithm is given for constructingappropriate TST transform matrices. The algorithm can raise the approxima-tion order to any desired integer each time if an appropriate transform matrixis chosen. Furthermore, it can preserve the symmetry of the re?nable vec-tor functions. (II) Symmetry and orthogonality are two important properties ofmultiwavelets. A class of paraunitary symmetric matrices is constructed. Basedon the paraunitary symmetric matrices, parametrization of symmetric and or-thogonal multiwavelets is given. Appropriately selecting some parameters, wecan obtain symmetric and orthogonal multiwavelets with some additionally de-sirable properties such as Armlets.In Chapter 6, we give the constructing algorithm for framelets in ??2(???) .(I) The nonseparable dual framelets in ??2(???) are constructed explicitly. Itis very easy to construct multivariate wavelets and framelets via the tensorproduct. However, the separable wavelets and framelets have some drawbacksin applications. Especially, they impose an unnecessary product structure onthe plane, which is arti?cial for natural images. Based on the dual ?? -frameletsin ??2(???1) and the dual ?? -framelets in ??2(???2) , the dualΩ-framelets in[]??2(???) are constructed, where andare matrices of integers. Moreover, a scheme for improving the regularity offramelets as well as an algorithm for symmetric framelets are given. (II) Anexplicit algorithm for symmetric and orthogonal framelets in ??2(???) is given.From symmetric and orthogonal framelets in ??2(???) , based on Han's projectionmethod, we shall construct symmetric and orthogonal framelets in ??2(???) ,...