毕 业 论 文

题 目：作 者：指导教师：职 称：院 系：专 业：班 级：日 期： 傅里叶级数及其应用 姜广辉 李博 讲师 理学院数学系 数学与应用数学 10级1班 2014年5月

傅里叶级数及其应用

摘 要：傅里叶级数是数学分析中的一个重要概念，具有较好的几何和代数性

质，伴随着科技的进步与发展，涉及了许多数学命题的讨论和应用，傅里叶级数的相关知识已经成为从事科学研究和工程设计等科技人员必备的数学基础.通过对傅里叶、拉格朗日、狄利克雷、黎曼等人在傅里叶级数方面的贡献，介绍了傅里叶级数起源和发展历程.同时文章以在图案设计和铁路客运量预测上的应用说明了傅里叶级数的价值.在图案设计设计方面，运用MATLAB软件，编写傅里叶级数的程序语言，通过自定义函数、编写画图函数程序、对图形多余部分处理、图形线条加粗等步骤，进而得到傅里叶级数的图形.通过对最基本的傅里叶级数的图形的组合、排列可以构成丰富的图案.在铁路客运量预测方面，基于傅里叶级数预测模型，以我国2004—2009年铁路客运量为数据基础，通过将时间序列划分为趋势性和季节性部分，分别采用最小二乘法和傅里叶级数预测法对两者进行拟合，应用MATLAB软件，求出预测模型，并进行预测.通过对预测结果的误差分析，表明：采用傅里叶级数预测法预测我国铁路客运量的效果较好.因此傅里叶级数在一定程度上受到了很多数学家的欢迎.

关键词：傅里叶级数；收敛性；MATLAB软件；图案设计；预测模型

Fourier series and its applications

Abstract：Fourier series is a mathematical analysis of an important concept，and

has good geometry and algebraic properties，along with the progress and development of technology，involving a lot of discussion and application of mathematical propositions，Fourier series of relevant knowledge has become a mathematical foundation for scientific research and engineering design and other technical personnel necessary. Through Fourier，Lagrange，Dirichlet, Riemann，who contribute in terms of Fourier series，Fourier series introduces the origin and development process，while the article in the graphic design and rail application passenger traffic forecast illustrates the value of the Fourier series. In the design of graphic design，the use of MATLAB software program written in the language of Fourier series，via a custom function，the preparation process of drawing functions，the excess part of the graphics processing，graphics，bold lines and other steps，then get the Fourier series pattern by the combination of the basic pattern of the Fourier series，the arrangement may constitute a rich patterns. Railway passenger traffic forecast，prediction model based on Fourier series to the railway passenger traffic volume of 2004-2009 data base，by the time series into trend and seasonal part，respectively，using the least squares method and fourier Fourier series prediction method for both fitting using MATLAB software，find the prediction model and predict the outcome of the prediction error by analysis showed that：Fourier series prediction method to predict the effect of China's railway passenger volume better. So to some extent，the Fourier series has been welcomed by many mathematicians.

Keywords：Fourier series；convergence；MATLAB software；graphic design；

prediction model

目 录

引 言 ............................................................... 1 1 傅里叶级数的起源 .................................................. 2

2 傅里叶级数的严密化 ................................................ 5

2.1 狄利克雷条件 ................................................... 5 2.2 黎曼引理 ....................................................... 5 2.3 吉布斯现象与一致收敛 ........................................... 6 2.4 连续傅里叶级数的收敛性 ......................................... 6

3 傅里叶级数的应用 .................................................. 8

3.1 傅里叶级数在图案设计上的应用 ................................... 8 3.2 傅立叶级数在铁路客运量预测上的应用 ........................... 14

3.2.1 傅里叶级数预测模型 ...................................... 14 3.2.2 实证分析 ................................................ 16

小 结 ............................................................... 19 致 谢 ............................................................... 20 参考文献 ............................................................. 21