再生核空间H^1[0,1]中的多尺度分析

Multiresolution analysis in reproducing kernel spaces H^1 [0,1]

小波分析在工程和技术的许多领域得到广泛应用,研究小波理论是必要的.人们所讨论的一维小波的构造都产生L2(R)的基.在某些应用中,我们感兴趣事的仅仅是实轴的一部分:如数值分析计算往往只在一个区间上有效;图像集中在一个短形框内;许多分析声音的系统将声音分成块等.所有这些都涉及到对支集在一个区间上的函数f的分解,比如说支集在[0,1]上.当然,令f在[0,1]以外为零,而用标准的小波基去分析它也是可以的,只是这将人为地在边界上造成跳跃.因此,研究适用于区间上的函数的小波是有意义的.这篇文章是首次在再生核空间H2[0,1]时论多尺度分析.本文利用积分算子建立了Hibert空间L2[0,1]与再生核空间H1[0.1]之间的同构映射,给出再生核空间H1[0,1]中的多尺度分析方法、小波逼近公式和采样公式.

Wavelet analysis is widely applied in many fields of engineering and technology. It is therefore necessary to further study the theory of wavelet. All the constructions of one-dimensional wavelets we have discussed so far lead to the bases for L2(R) . In many applications ,what one is interested in is only part of the real line the computations for numerical analysis generally work on an interval, images are concentrated on rectangles and many systems to analyze sound divide it in chunks. All these involve decompositions of functions f supported on an interval. Say[0,l ] of course, standard wavelet bases may be used to analyze/by setting the function equal to zero outside [0,1] ,but this causes an artificial'jump' at the edges. It is therefore useful to develop wavelets adapted to 'life on an interval'. Multiresolution anaysis in the reproducing kernel spaces H1 [0,1 ] is discussed here for the first time. An isomorphic mapping is built between Hibert space L2 [0,1 j and reproducing kernel space H1 [ 0,1 ] using integral operator while wavelet approximation expressions and sampling expressions in the H1 [0,1 j are discussed as well.