Some Research On A - And Narrow Wave Episode

Wavelet analysis is a rapidly developed new interdisciplinary in applied mathematic-s and engineering mathematics. Due to the great applied value, the method of its construction is one of the core contents. It was found that the construction of the wavelet sets in the frequency domain is a new construction method of the wavelet, and then the research turns to the construction of the wavelet set. Starting from the most simple Shannon wavelet set in dimension one, gradually to the high dimension of the wavelet set, it is obtained that the generalized form of the Shannon wavelet set——matrix dilation wavelet set, after using the inverse Fourier transform we can get a weak smoothness and compact support of single orthogonal wavelet. However, there are certain limitations for a single wavelet in application areas, the deficiency of linear phase when it processes multi-channel color image, so the concept of multiwavelet has been proposed. Multiwavelet overcomes this defect and it has a shorter support, so the construction of the multiwavelet becomes a new research direction. It is very naturally that people expect to use the construction method of a single wavelet to construct the multiwavelet, then the concept of multiwavelet set and the construction of multiwavelet set are discussed.Based on the research of the single wavelet, single and multiwavelet set, this thesis focuses on the real expansive matrix A-dilation wavelet set and A-dilation multiwavelet set for the separable closed subspace of X and L2(Rd), some conclusions have been obtained. In the thesis it is composed of four parts:The Chapter1is an introduction which summarizes the emergence, development of wavelet analysis and wavelet set. In Chapter2, the sufficient and necessary conditions of A-dilation single wavelet set and single subspace wavelet set are obtained and proved by the related knowledge of the set and the operator theory. In Chapter3, based on the research of A-dilation single wavelet set, under the premise of MRA, the concept of the A-dilation multiwavelet set is introduced, the characterization of A-dilation single wavelet set is a tile, which is enlarged to A-dilation multiwavelet set. A sufficient and necessary condition of a positive measurable set family{Wn}n=1q-1to be A-dilation multiwav-elet set for L2(Rd) is obtained, and it is proved by the related knowledge. In Chapter4, By analogizing the conclusions of Chapter3, the concept of the A-dilation subspace multiwavelet set is introduced, A sufficient and necessary condition of a positive measurable set family{Wn}n=1L to be A-dilation subspace multiwavelet set for the separabl-e closed subspace X is obtained.