【作者】 闫立军；

【导师】 冉启文；

【作者基本信息】 哈尔滨工业大学 ， 应用数学， 2006， 硕士

【摘要】 随着小波变换理论研究的不断深入和实际应用的日益广泛,小波分析的各种优势在不断的明确。MRA(Multiresolution analysis)作为一种优美的数学思想,越来越多的为广大科学家、工程师所接受,但是随着小波应用到各个不同领域,展现出来的问题也越来越多,其中一个突出的问题就是要求我们找出更多的具有不同性质的小波。这篇文章主旨是建立任意尺度a > 1的小波紧框架的完全特征,特别是正交小波的特征,这里的小波是由L~2 := L~2(R)中的一族函数生成的。Weiss及其同伴们做了一些关于整数尺度伸缩小波的工作,这里是对他们工作的一般化。作为一个应用,我们将给出由一个L~2中的函数生成的具有任意尺度伸缩因子a > 1的正交小波的例子,并且这个小波还有很好的时频定位性。作为另外一个应用,我们将说明当a > 1且a满足对于任意的j∈Z~+有a~j∈Q~c时,具有良好时频局限性的a尺度正交小波是不存在的。这就回答了Daubechies在《小波十讲》里面提出的关于所有无理尺度伸缩因子的问题。经典的MRA小波可能是现在最重要的一类小波。许多著名的例子都属于这一类,它们已经被应用到很多领域。这样,一个问题就自然凸现出来了:一个小波是MRA小波的充分必要条件是什么。与这个问题的答案比较接近的包括维函数的应用。这里我们证明了具有整数尺度伸缩因子的紧框架小波维函数几乎处处取整数值。更多还原

【Abstract】 With the deeper study of the theory of wavelets and their more extensive applications, many advantages have emerged gradually. MRA (Multiresolution analysis), as a kind of graceful mathematics thought, is accepted by more and more scientists and engineers. However, accompanying with the application to different fields, there also comes many problems, one of which is asking for more wavelets with different characters.The objective of this paper is to establish a complete characterization of tight frames, and particularly of orthonormal wavelets, for an arbitrary dilation factor a >1, that are generated by a family of finitely many functions in L~2 : = L~2(R) This is a generalization of the fundamental work of Weiss and his colleagues who considered only integer dilations. As an application, we give an example of tight frames generated by one single function for an arbitrary dilation a >1 that possess“good”time-frequency localization. As another application, we also show that there does not exist an orthonormal wavelet with good time-frequency localization when the dilation factor a >1 is irrational such that a jremains irrational for any positive integer j . This answers a question in Daubechies’Ten Lectures book for almost all irrational dilation factors. Other applications include a generalization of the notion of s-elementary wavelets of Dai and Larson to s-elementary wavelet families with arbitrary dilation factors a >1.The“classical”MRA wavelets are probably the most important class of orthonormal wavelets. Many of the better known examples as well as those often used in applications belong to this class. Thus, it was a natural question to find necessary and sufficient conditions for an orthonormal wavelet to be an MRA wavelet. An interesting approach to this involves the dimension function. Here, we prove that the dimension functions of the tight frame wavelets, for an integer dilation factor, are integer valued.更多还原