【作者】 贾彩燕；

【导师】 高协平；

【作者基本信息】 湘潭大学 ， 计算数学， 2001， 硕士

【摘要】 本文首先利用Zak变换将Walter取样定理推广至多小波子空间，给出了一类多小波子空间的取样定理，使得现有的一大类具有给定特性（正交性、紧支性，对称性，高的逼近阶、平衡性等）的多小波子空间的任一函数可以利用我们的结果实现精确重构。并且，目前结果较好I.W.Selesnick的具有插值特性的多小波子空间的取样定理只是我们定理的特例。特别值得指出的是：I.W.Selesnick所构造的一类具有插值特性、紧支、正交、平衡、具有一定逼近阶的多尺度函数和多小波不能同时拥有对称特性（这不利于数字信号处理和图像压缩），而我们的实例表明，对于诸如CHM等具有较好特性的多小波（可以具有对称性而不具有插值特性），仍可用我们的定理进行精确重构，实现精确的A/D和D/A。同时，我们发现具有插值特性的多小波子空间的取样定理其插值（综合）函数即为尺度（分析）函数，不但取样形式简单，而且可以实现精确的A/D和D/A。因此，我们给出了广义的插值特性的概念，讨论了广义的插值特性和正交性、紧支性、对称性之间的关系，给出了广义插值正交和广义插值对称的充要条件。由于利用Singular软件求Grobner基而后获得尺度函数和小波基，当问题规模稍微大一点时就有相当大的时间复杂度，因此我们借助于Hopfield反馈型神经网络解非线性系统构造了具有给定特性的广义插值多小波，不但极大地减少了时间复杂度，而且获得了令人满意的结果。更多还原

【Abstract】 A general sampling theorem for multiwavelet subspaces is introduced by using Zak transform.At present..the sampling theorem for tnultiwavelet subspaces given hy [22] reconstructing a signal perfectly and completing A/D and D/A exactly~ While his sampling theorem in mnltiresolution spaces with scaling functions as interpolant.s is just the special case of our results.Moreover. his multiwavelets have required the solution of large system of nonlinear equation and can抰 have symmetry property except for orthogonalitv.short-compact support.high appromation order.halanced propertv,etc. which is disadvantaged to signal processing,especially,irnage compressing.hi this paper. the emphasis of our theorems are on a large family of multiwavelets with good properties including symmetry(GHM抯 mnlt.iwavelet. is a case and completes t.he exact. A,/D and D/A,t.oo). In addition.we devote to a study of general cardinal rnultiwavelet svstem~based on a signal in the mult.iresolutiou space V0Q~) can be perfectly reconstructed by integer and semiinteger uniform shift sampling in the case of the multiscaLing function with general cardinal propert.y.In this case.t.he exact. A/D and D,/A can be compelet. So we investigate the relationship among interpolating propertv.orthogonalitv.shortcompact. and symmetry and mainly present and prove the necessity and sufficiencv of the conditions hot.h of general cardinal orthogonal multiwavelets and general cardinal symmetrical multiwavelets.The examples of the general cardinal multiwavelet,s with some good properties are given by Hopfield feedback neural networks instead of t.he software Sin gii.Iar.It. not. only significantly get.s over the time梒ost. by sin 9uI(Jr carrying out the G b bnFr 1)aSis cornpiit.atioiis.hnt. also obtain the satisfied solutions when the proper initial values are chosen.更多还原