傅里叶级数的其收敛性及其应用

傅里叶级数的收敛性及其应用

摘要

傅里叶级数是数学分析的一个重要组成部分.本文首先介绍了傅里叶级数的相关知识、以2π为周期函数的傅里叶级数展开式、以2l为周期函数的傅里叶级数展开形式.其次，通过狄利克雷积分和黎曼—勒贝格引理及局部化定理傅里叶

f t展开成傅里叶级数的收敛定理及其证明.级数的收敛定理分析了周期函数()

最后，给出了傅里叶级数一些简单应用，其原理主要是利用傅里叶级数均方误差证明了傅里叶级数部分和趋于无穷大时吉伯斯现象不存在以及利用傅里叶级数展开法研究了平顶高斯光束通过有光阑限制的近轴ABCD光学系统的传输特性问题.

关键词：傅里叶级数；收敛性；积分；周期函数

CONVERGENCE OF FOURIER SERIES AND

ITS APPLICATION

ABSTRACT

Fourier series is an important part in Mathematical Analysis. The first introduced the knowledge of Fourier series, toπ2for the periodic function of the Fourier series expansion, to l2for the periodic function of the Fourier series expansion. Second, analyzed periodic function()x f expand into Fourier series convergence theorem and its proof by Dirichlet integral and Riemann-Lebesgue Lemma and local theorem of Fourier series convergence theorem . Finally, some simple application of Fourier series, and its main principle is to use the mean square error of the Fourier series is proved, and tends to infinity, some of Gibbs phenomenon does not exist and the use of fourier Fourier series expansion of the flattened Gaussian beams through apertured paraxial optical system ABCD, the transmission characteristics of the problem.

Key words:Fourier series; Convergence; Integral; Periodic function

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