Some Research On Vector-valued Wavelet

Wavelet analysis.is nowadays a new area which rapidly developed in applied mathematics and engineering mathematics. It is a great creation after Fourier analy-sis and solves many problems which can not be settled in Fourier analysis. Moreover, it provides us with a powerful tool for theory science and applied science, and has promoting effects in studying non-linear problems, numerical calculation,network and information security and so on.Vector-valued wavelets have become a hot spot which is studied by many wavelet experts recently at home and abroad, and the concept of which were firstly given by Xia and Suter in 1996. Vector-valued wavelets are a kind of wide-sense multi-wavelets, and can own desirable characters including orthogonality, compact support and symmetry features. Furthermore, multi-wavelets can be generated by the component function of vector-valued wavelets, and they have wide application nowadays, but vector-valued wavelets have a much larger space for applying.Compared with the multi-wavelets, vector-valued wavelets have obvious advan-tages. For example, with the development of the multi-channel technology, vector-valued wavelets can supply with a powerful tool and have important value in dealing with multi-channel sigal. As we know, before practicing discrete vector-valued trans-formation we need not, which can largely reduce our work. Besides, the application field is related to signal processing, image processing, numerical information, air and see wave analysis and so on. Because the study for vector-valued wavelets has dual value of theory and application, it has become a concerned hot spot internal and external. However, the development of vector-valued wavelets theory still belongs to a primary stage and the study in various aspects have not been ripe. So many problems need us solve.In this paper, the Mallat algorithm of bivariate compactly supported orthogonal vector-valued wavelets with dilation factor a and the construction of bivariate vector-valued wavelets are mainly investigated, and the corresponding result is. obtained, which push forward the development of vector-valued wavelets theory. this paper is composed of four chapters.Chapter one gives a concise introduction of the emergence and development of wavelets analysis theory and a current situation of vector-valued wavelets. Chapter two is about the existence of vector-valued wavelets. Firstly, the space of vector-valued function and corresponding preparing knowledge are given. Then, vector-valued multi-resolution analysis is builded, and vector-valued wavelets are defined by it. The corresponding characters are also studied. Lastly, the results about vector-valued scaling function and the existence of vector-valued wavelets are deduced.Chapter three mainly studies the Mallat algorithm of vector-valued wavelets ac-cording to chapter two. The decomposition and reconstruction algorithm of vector-valued signal is given by the approximation of vector-valued signal in space of scaling functions and the othogonality of scaling functions and wavelet functions.Chapter four further investigates the construction of bivariate compactly sup-ported orthogonal vector-valued wavelets. Fistly, the definitions of bivariate vector-valued scaling functions and wavelets are given by introducing bivariate vector-valued multi-resolution analysis and the corresponding characters are studied. Then, an effective algorithm for constructing bivariate compactly supported orthogonal vector-valued wavelets is obtained. Finally, a numerical example is given.