Short Courses in Discrete and Continuous Fourier Analysis

1989年和1990年的夏天，一门关于傅里叶分析的创新短期课程在SIU的自然接触会议中心为18名大学教师开设。课程讨论离散和连续函数的傅里叶表示；算子恒等式有助于深入理解这一学科的规则和对称性；从基于推导的高斯插值到现代变体的快速傅里叶变换，如Bracewell最近获得专利的快速哈特利变换；广泛应用于数学中的广义函数，即分布的一个初等新理论及其应用并从数学、物理和工程科学中选择这些思想的应用。学员将使用个人电脑（配备特别设计的用户友好软件）加深对讲座中发展的规则和算子身份的理解，体验快速运算，即0(N log N)，而不是缓慢运算，即O(N2)， N位数运算，并使用快速变换方法解决实际问题。短期课程的课堂讲稿、习题集和计算机光盘为学员在本国院校引入相应的跨学科本科傅立叶分析课程或丰富和现代化现有的数值分析、傅立叶级数、常微分方程和偏微分方程等课程做了准备。已经在课程中出现了。数学科学部也为该项目提供支持。大学将承担该项目的费用，个人参与者的家庭机构也承担参与成本。

An innovative short course on Fourier analysis is being given to groups of 18 college faculty at SIU's Touch of Nature Conference Center during the summers of 1989 and 1990. Lectures are discussing Fourier's representation of discrete and continuous functions; the operator identities that facilitate a deep understanding of the rules and symmetries of this discipline; the fast Fourier transform from Gauss` interpolation bases on derivation to modern variants such as Bracewell's recently patented fast Hartley transform; an elementary new theory of generalized functions, i.e., distributions, that are widely used in mathematics and its applications; and selected applications of these ideas from the mathematical, physical, and engineering sciences. Participants are using personal computers (with especially designed user friendly software) to deepen their understanding of the rules and operator identities developed in the lectures, to experience fast, i.e., 0(N log N), as opposed to slow, i.e., O(N2), N-digit arithmetic, and to solve practical problems using fast transform methods. The lecture notes, exercise sets, and computer discs from the short courses are preparing the participants to introduce corresponding interdisciplinary undergraduate Fourier analysis courses in their home institutions or to enrich and modernize existing courses in numerical analysis, Fourier series, ordinary and partial differentials equations, etc., already in place in the curriculum. The Division of Mathematical Sciences is also providing support for this project. The University is contributing to the cost of the program, with the individual participant's home institutions also bearing a cost of the participation.