【作者】 徐振民；

【导师】 高德智；

【作者基本信息】 山东科技大学 ， 应用数学， 2007， 硕士

【摘要】 小波分析是近年来迅速发展起来的的一门应用数学学科，系统的研究开始于20世纪80年代初期。它从产生到现在虽然仅仅几十年的时间，但它在信号传输、图像处理、数字水印、偏微分方程、电磁场数值计算等众多领域都有广泛的应用。本文主要从两个方面研究小波分析：一、从框架方面研究了近Riesz基的稳定性与摄动；二、从高维空间方面研究了框架多分辨分析的性质和小波。在第一章中，首先介绍了小波分析的发展历史及发展前景，然后介绍了本文的主要工作。在第二章中，先介绍了框架的定义与性质，接着描述了Riesz基、近Riesz基性质与它们各自之间的等价关系，最后重点研究了近Riesz基的稳定性与摄动问题，得出了关于近Riesz基摄动新的充要条件。在第三章中，在介绍了高维空间多分辨分析的定义与性质后，得出了高维空间框架多分辨分析的一些新的性质以及一个集合是框架多分辨分析的谱充要条件。在第四章中，我们首先介绍了高维空间的框架小波的定义与性质，然后介绍了小波集相关定义，得出了一个集合是相应小波集的等价条件。在第五章中，利用二维空间整数的分类，得出了的一个重要结论，在此基础上，得出二维框架多分辨分析存在一个Parseval框架小波的充要条件，并给出Parseval框架小波的表示形式。更多还原

【Abstract】 Wavelet analyses is one of the disciplines of applied mathematics whichdeveloped rapidly in recent years, Research began in the early 80s of the 20thcentury. Ithas a wide range of applications such as signal processing, imageprocessing, digital watermarking, partial differential equations, numericalcalculation of electromagnetic fields and so on, though the wavelet analysesfrom produces to present merely several dozens years.Two topics of the dissertation research about the wavelet analyses include:First, we studied the stability and the perturbation of the near-Riesz base fromframe; Second, we studied the properties of frame multiresolution analyses andwavelet in high-dimension spaces. In chapter 1, first introduced the developmenthistory and the prospects for development of the wavelet analyzes, then introducethe mainly conclusion of this article. In chapter 2, we first introduced theframe definition and properties, and then describes the properties of the Rieszbase and the near-Riesz base and their respective equivalence relations betweenthemseleves, finally we studied the stability and perturbation of the near-Rieszbase on emphasis, then obtained a new necessary and sufficient condition of theperturbation of the near-Riesz base. In chapter 3, first introduced thedefinition of frame multiresolution analyses in high-dimension space, thenobtained some properties and a necessary and sufficient condition of a set isa spectrum of frame multiresolution analyses. In chapter 4, we first introducedthe definition and properties of frame wavelet in high-dimension spaces, thenintroduced the definition of wavelet set, finally obtained a necessary andsufficient condition for a set is a wavelet set in a high-dimension spaces. Inchapter 5, we obtained a important conclusion though integer classification intwo-dimension space, use it, we obtaioned a necessary and sufficient conditionof multiresolutions analyses has the single Parseval frame wavelet in two- dimension space, and we give the express of Parseval frame wavelet.更多还原