频谱有限小波变换像空间的描述

发布时间：2018-07-29 14:06

【摘要】：在20世纪80年代,小波分析成为数学领域中的一个新的数学分支,也是继Fourier分析之后把数学工具成功应用在工程领域的一个重大突破。目前,已活跃在纯粹数学、应用数学和信号处理等领域。小波分析在进行信号处理时,不仅可以在时间和频率上同时做局部交换,而且从信号中能较有效地提取有用信息,因而小波分析理论比Fourier变换能更好的解决许多难题。基于小波分析的基础是连续小波变换,同时再生核Hilbert空间是连续小波变换的基础。因此本文对具体的频谱有限小波及其小波变换像空间进行了研究,具体研究内容如下:为了描述具有频谱有限且衰减速度较快的小波函数及其小波变换像空间的性质。首先针对由Shannon尺度函数构造的小波函数,借助小波分析理论给出这个小波变换及其像空间的性质。当固定尺度因子时,得到这个小波变换像空间中的再生核函数的两种具体的表达式,然后利用再生核空间理论描述这个小波变换像空间。此外,本文还研究了频谱有限的二进小波,它在时域具有对称性、衰减性、光滑性、在频域方面还具有紧支集、局部性等。本文首先给出二进小波变换的等距恒等式和估计式,其次给出二进小波变换像空间的再生核函数的具体解析表达式。这为一般的小波变换像空间的研究提供了依据。
[Abstract]:In the 1980s wavelet analysis became a new branch of mathematics and a major breakthrough in the application of mathematical tools to engineering after Fourier analysis. At present, has been active in pure mathematics, applied mathematics and signal processing and other fields. Wavelet analysis can not only do local exchange in time and frequency, but also extract useful information from signal effectively. So wavelet analysis theory can solve many problems better than Fourier transform. The basis of wavelet analysis is continuous wavelet transform, and the Hilbert space of reproducing kernel is the basis of continuous wavelet transform. Therefore, the specific spectral finite wavelet and its wavelet transform image space are studied in this paper. The main contents are as follows: in order to describe the wavelet function with limited spectrum and fast attenuation speed, the properties of wavelet transform image space are described. Firstly, for the wavelet function constructed by Shannon scaling function, the properties of the wavelet transform and its image space are given with the help of wavelet analysis theory. When the scale factor is fixed, two concrete expressions of the reproducing kernel function in the image space of the wavelet transform are obtained, and then the image space of the wavelet transform is described by the theory of the reproducing kernel space. In addition, we also study the dyadic wavelet with limited spectrum, which has symmetry, attenuation and smoothness in time domain and compact support set and locality in frequency domain. In this paper, we first give the isometric identity and estimator of dyadic wavelet transform, and then give the analytic expression of the reproducing kernel function in image space of dyadic wavelet transform. This provides the basis for the general research of wavelet transform image space.
【学位授予单位】：哈尔滨理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.2

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