3-4新数学分支简介

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（3） One disadvantage of Fourier series is that its building blocks, sines and cosines, are periodic waves that continue forever. While this approach may be appropriate for filtering or compressing signals that have time-independent wavelike features, other signals may have more localized features for which sines and cosines do not model very well.
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In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions. Any set of functions that form a complete orthogonal system have a corresponding generalized Fourier series analogous to the Fourier series. For example, using orthogonality of the roots of a Bessel function of the first kind gives a so-called Fourier-Bessel series.
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傅里叶级数与傅里叶变换的内容自19世纪以来已经非 常丰富了，关于此论题已发表和出版了大量的研究论
文和书籍（大学程度和研究生程度）。相比而言，小
波的出现却是近几年的事。尽管可追溯到几十年前，
但只是在最近的二十年里，小波才成为信号分析和其
他应用领域中非常流行的工具。在一定程度上，这应 当归功于Ingrid Daubechies女士①在构造紧支撑正交小 波方面的杰出工作。
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(4) One of the difficult tasks in the construction of a wavelet is to make sure that its translates and rescalings satisfy analogous orthogonality relationships, so that efficient algorithms for the computation of the wavelet coefficients of a given signal can be found.
新数学分支简介
4.3 Wavelets and Fourier analysis
1. 新单词与术语 wavelet （小波） Fourier analysis （傅里叶分析） vibrate （震动） filter out （过滤掉） high –frequency wiggle （高频抽动） eliminate （排除） data compression （数据压缩） digit bits （数字位） tolerance error （容许误差） time-independent wavelike feature （时间独立的波状特征） scaling （换算、定标） rescaling （换算） translate （平移） truncate （截断） 1
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2. 例句分析
（1） All we attempt here is an elementary presentation of the relations that have to be known in order to follow, without difficulty and with a critical mind, the scientific literature dealing with the concept of time in physics in a fractal environment.
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（2） The basic goal of Fourier series is to take a signal, which will be considered as a function of the time variable t, and decompose it into its various frequency components. 傅里叶级数的基本目标是把一个时间t 的函数的信号分 解成各种不同的频率成分。
整数维）与我们度量我们所在的空间，测量其边界，
测定其内容的方式有密切的关系。首先，我们将看到
维数是一个物理的概念。
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皮亚诺（Peano）曲线
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A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. Examples of successive approximations to common functions using Fourier series are illustrated above.
在这里我们只向大家简单介绍一些关系，为了使你能
没有困难地、审慎地读懂一些处理分形环境下物理现
象中的时间概念的文献，这些关系是必须知道的。
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（2） In this study the reader will find that the key element of fractal geometry, the non-integral dimension, is intimately associated with the way in which we evaluate our space, measure its boundaries and “weigh” its contents, the meaning of the concept of dimension will be seen to be, above all, physical. 在这篇文章里，读者将看到分形几何的关键元素（非
2. 例句分析 （ 1 ） Fourier series and the Fourier transform have been around since nineteenth century and many research articles and books have been written about these topics. By contrast, the development of wavelets has been much more recent. While its origins go back many decades, the subject of wavelets has become a popular tool in signal analysis and other areas of applications only within the last two decades or so partly as a result of Ingrid Daubechies’s celebrated work on the construction of compactly supported, orthonormal wavelets.
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As an example, consider the graph given in Figure 3-46. This may represent a sound signal with two isolated noisy pops that need to be filtered out. Since these pops are isolated, sines and cosines do not model this signal very well. A different set of building blocks, called wavelets, is designed to model these type of signals. In a rough sense, a wavelet looks like a wave that travels for one or more periods and is nonzero only over a finite interval instead of propagating forever the way sines and cosines do.