Some Research About Mra Wavelet

This thesis mainly studies MRA (Multiresolution Analysis) wavelets. As the decomposition and reconstruction property that an orthonormal wavelet must satisfy, it makes the construction of a wavelet difficult. Especially orthonormal wavelets that have the most desirable feature, it does impose more restrictions. Since the theory of Multiresolution Analysis was introduced by Mallat, people gradually take more interest in studying wavelet on this road.By learning wavelets and some related courses, and absorbing the basis of previous studies and scholars, we mainly study the conditions that MRA wavelets exist in n-dimension space and its reproducing subspace. In addition, we make some study on MRA TFW (MRA tight frame wavelet). The thesis is divided into four chapters, the main contents of each chapter are as follows:Chapter one talks about the generation and development of wavelet analysis, focuses on the status development and the study necessity of the MRA wavelet.Chapter two is about the definitions of MRA wavelet and dimension function. First introduce the concept of multiresolution analysis, orthonormal wavelet, and then according to the reference, we combined the two concepts and give the definition of MRA wavelet. Then we give the definition of dimension function, further deduced the nature of the function and give a definite illustration of its name. Finally, we link MRA wavelet and dimension function, and draw the relationship between the two.Chapter three mainly characterizes subspace MRA wavelets. First we introduce the concept of reducing subspace and several related properties. Then we study the conditions for the existence of MRA wavelets in n-dimensional space. Based on the above studies, we use the conclusions in the subspace,get the similiar result and give a detailed derivation. Finally, a specific example is given to strengthen the methodological guidance of the certification process.Chapter four mainly studies on the characterization on MRA tight frame wavelet. On the basis of the previous chapter, and further integration of the references, we extend the results to tight frame wavelets. Which explores the properties of the TFW, and obtains the characterization on the existent conditions of MRA wavelets with dimension function.