论文标题:关于细分函数光滑性的特征刻画
The Characterization of the Smoothness of Refinable Functions
论文作者
论文导师 李松,论文学位 硕士,论文专业 应用数学
论文单位 浙江大学,点击次数 81,论文页数 42页File Size1039K
2006-05-01论文网 http://www.lw23.com/lunwen_58426707/
refinement equation;subdivision schemes;joint spectral radius;convergence rate;L_p-solution;smoothness;Lipschitz space;cascade operator;smoothness
本文考虑细分函数在不同空间的光滑性的特征刻画。我们知道细分方程在小波分析和计算几何中总起着重要作用。在以往的研究中对细分格式的收敛性研究已经取得了很多进展,在实际应用中我们总想找到更“好”的小波:即光滑的小波,然后通过伸缩及平移构造小波基用于图像处理、信号处理等。细分函数越光滑,相应地小波函数也就越光滑。本文将详细介绍有关细分函数光滑性的若干重要结果,同时也提出一些有意义的研究课题,有待于本人在今后的时间里去探讨。 [1].对于一般的L_p空间,在细分方程收敛的前提下,对细分函数光滑性的刻画用一般的Lipschitz空间即可;特别地,当p=2时,可以在Sobolev空间中去刻画其光滑性。 [2].将L_p(R)空间扩张到多重及多元多重空间,则细分方程的表示形式以及收敛的条件有了变化,相应地,对细分函数光滑性的刻画所需要的条件也变得复杂起来。
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.
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