High-dimensional Space Wavelet Frame Theory And Its Structure

Wavelet analysis is the emergence of a new branch of mathematics in recent years, From the date of its birth, it has been an important focus of mathematicians and experts in other fields of research. Wavelet frame theory is an important part of the wavelet analysis, it is a major research tool and has played a very important role in the development of wavelet analysis.In recent years, people gained many important achievements in the framework theory. In the one-dimensional space, the study from the construction of compactly supported orthogonal wavelets to the construction of semiorthogonal wavelets, biorthogonal vavelets and Riesz wavelet, wavelet frame. Research in high-dimensional space ranging from two-dimensional, three-dimensional space for constant2extended to high-dimensional space for the matrix case.This thesis introduces the concept of high-dimensional wavelet frames, Bessel sequence and its properties, and then derives the properties into frames, and discusses the special nature of the high-dimensional space frame and the wavelet frame, makes a further review and improvement for the theoretical of wavelet frame for high-dimension.The thesis also describes the FMRA theory in L2(Rd), and gives the conditions for the existence of the A-dilation FMRA wavelet, and discusses wavelet tight frames in high-dimensional, using the unitary expansion principle, gives its construction method, enriching and developing the theory of the structure of wavelet tight frames. And it is of great practical value on the research of signal denoising,image fusion and so on.This thesis consists of five parts:Chapter I:Introduction. Briefly describes the emergence and development of the wavelet analysis theory and the generation of the framework theory and the current study of it.Chapter II:frame conclusions in high-dimensional space. Firstly listed the definition of framework and affine department.combined with the conclusions of the relevant literature, it gives the relationship of affine department, sequence, framework and parseval framework, and made a strict proof of them, spreads the relevant theory.ChapterⅢ:FMRA and FMRA wavelets on L2(Rd). Firstly introduces the concept of the FMRA, FMRA is a promotion of MRA to the framework.then investigate the existence of the A-dilation FMRA wavelet, gives the existence conditions of the A-dilation FMRA wavelet.Chapter IV:the definition and the structure of wavelet tight frames in L2(Rd). It has two parts, the first part describes the definition of wavelet frame and the frame operator in high-dimensional, and studies its properties; the second part using the knowledge of FMRA in chapter three and the unitary expansion principle, gives a construction of A-dilation wavelet tight frames in two-dimensional space.Chapter VI:summary.