The Characterization Of The Smoothness Of Refinable Functions

The purpose of this paper is to investigate the characterizations of smoothness of refinable functions.As we know,refinement equations play an important role in wavelet analysis and computer graphics. The researches to the convergence of subdivision schemes have obtained a lot of results. In practical applications, we always expect "better" wavelet(i.e. smooth wavelet).Then,by scaling and translating the wavelet, we can get the wavelet basis which play an important role in signal and image pro-cessing,etc.The more smooth refinement function is,the better wavelet is. This paper review several results of the smoothness with refinement functions detailedly. At the same time,there are some significative research proposals put forward as to investigate them for the future.[1]. In general L_p spaces,one can characterize the smoothness of refinement function with Lipschitz space. In particular, one can characterize the smoothness of refinement function in Sobolev spaces.[2]. We extend L_P(R) space to the multiple or multivariate space, then the expression of the refinement equation and the conditions of its convergence is different. Therefore, the conditions of characterization of smoothness of the refinable function is more complicated.