A Class Of Totally Positive Symmetric Smooth Refinable Function And Framelets

Wavelet analysis is a rapid development area in the modern mathematics.The produce and application are very important. Wavelet theory is the momen-tous breakthrough of the traditional Fourier transform. It begot a big attentionof the many international science organization sodality. It was been the currentadvanced subject. The development was very fast. It became appropriate char-acter model of the sound,picture and signal. It especially fit disposal signaland picture with break in a short time. So it has a very abroad application.Based on deep research to the statuses of the wavelet theory at home andabroad, we do some research on the refinable function and tight wavelet frame.The following are our main results:Firstly, the construction of a class of totally positive refinable function withdilation factor M is investigated, and its properties,such as accuracy, smoothnessand symmetry, are discussed. We provided an explicit way for constructiontotally positive, symmetry and smoothness refinable function , and also provethat many properties of this class of refinable are similar to the cardinal B-spline. Hence its smoothness can be lifted by using the method of convolution.Secondly, we introduce the concept and basic features of the wavelet frame .In this paper, we quoted the theory of the existing tight frame and the methodof construction compact wavelet frame, and then we give a new method onthe construction of the wavelet frame with dilation factor M .We resolve theproblem about the existing tight wavelet frame of the total positivity refinablefunction .At last, we give some examples on the tight wavelet frame of the totalpositivity refinable function .