| The concept of multiwavelets was first initiated by Goodman, Lee and Tang. The main contribution for multiwavelets is that they not only maintain the nicer localize features of the time domain and the frequency domain, but also overcome the drawback of wavelets, i.e., they can simultaneously possess desired properties such as symmetry, orthogonality, smooth property, higher approximation order and shorter support and so on, which are impossible for any real-valued wavelet. Due to the advantages in theory and the potential in practical application, multiwavelets have received considerable attentions from the wavelet researchers. The constructions of multiwavelets with desired properties are the hotspots in multiwavelets research. The constructions of multiwavelets are mainly indicate the constructions of the multiscaling functions and the multiwavelet functions.This thesis presents my contribution to the constructions of multiwavelets with desired properties by using two methods. It is organized as follows:In chapter 1, we will mainly give the basic concepts, related notions and results which will be used in chapter 2 and chapter 3.In chapter 2, we pay our considerable attentions to the case of M-band multiwavelets. We present the concept of M-band two-scale similarity transform(MTST), and the relevant algorithm construction of transform matrix T(ω) is obtained. Further, we prove that under the conditions on T(ω) we given, by using the corresponding MTST, not only the approximation order of the primal M-band multiscaling function is raised, but also the symmetry and compact support can be preserved. Finally, an example will be given.In chapter 3, we mostly discuss a class of symmetric-antisymmetric orthonormal multiwavelets about the case of 2-band. We give a easy method to construct a class of symmetric-antisymmetric orthogonal multiwavelets with multiplicity 2Υ, which will show that any length-2N multiwavelets can be constructed from a length-(2N+l) and vice verse. Finally, examples are also given. |
