Project FOURIER : A Browser-based Program for Enhancing an Interdisciplinary Undergraduate Fourier Analysis Course

初等傅立叶分析是科学家和工程师教育的重要组成部分。理论和应用之间的相互作用使它成为数学学生教育的理想顶点课程，特别是那些想要学习数学对我们文化贡献的一些方式的潜在教师。这一数学体系的传统介绍侧重于使用傅立叶级数来解决边值问题和一些19世纪处理收敛问题的相关定理。随着分布和FFT的发展，我们有可能为本科生介绍傅里叶分析的理论和应用。David W. Kammler的新书《傅里叶分析第一课程》（Prentice-Hall）对这些思想进行了精彩的组织，并包含了各种各样的应用，肯定会吸引许多学科的学生。计算机辅助可视化（和超声）对于傅里叶分析的现代应用是必不可少的。学生可以使用Maple、Mathematical或Matlab来绘制各种函数，并为某些应用程序准备动画或声音文件。然而，准备这样的代码需要时间，我们希望通过让学生能够立即体验最常用的可视化和声音化，使课程更有效。静态插图包括在每一个初等数学文本。我们设想使用互联网（或CDROM）来提供动态多媒体插图。本文给出了振动弦运动的公式，学生只需点击几下鼠标就能看到计算机模拟结果。教授描述了信号的小波分解，并通过几次鼠标点击让学生听到这种处理对简单音乐主题的影响。(你可以在pascal.math.stthomas.edu/fourier/demos体验一个原型。html)。美国国家科学基金会资助的一部分（NSF- use 89503， NSF- use 9054179和NSF- use 9156064）提供了开发FOURIER的资源，这是一个旨在为学生产生这种即时体验的软件包。十几所大学的教师使用傅里叶来提高学生对傅里叶分析的理解。这个软件包是由PI和David Kammler共同开发的。这项工作开始于1990年，软件被设计为在DOS环境下工作。该提案的共同负责人目前正在使用该程序向圣托马斯大学（University of St. Thomas）的本科生教授傅里叶分析。在对该软件进行了近十年的试验之后，我们现在已经准备好进行从上到下的重新编码，这将导致在Netscape或Internet Explorer等网络浏览器下运行的商业可用产品。这个想法是，程序将是独立于平台的，并且可以通过Internet、本地网络、CD或硬盘访问。本提案是为此目的请求国家科学基金会资助。傅里叶的基本设计特征将被保留。PI将负责大部分的编程工作。联合项目负责人将带头设计几个教程和模块，与傅里叶一起打包在CD或万维网上分发。圣托马斯大学上过傅里叶分析课程的本科生将协助副pi完成这项工作。大卫·卡姆勒已同意担任该项目的顾问。此外，来自路易斯安那州立大学和阿克伦大学的教师将参与该项目的测试阶段。

Elementary Fourier analysis is an essential component in the education of scientists and engineers. The interplay between theory and applications makes it an ideal capstone course in the education of mathematics students, especially potential teachers who want to learn some ways that mathematics has contributed to our culture. Traditional presentations of this body of mathematics focus on the use of Fourier series to solve boundary value problems and some of the related 19th century theorems that deal with issues of convergence. The development of distributions and the FFT have made it possible to present a thoroughly modern elementary introduction to the theory and applications of Fourier analysis for undergraduate students. A new text, A First Course in Fourier Analysis (Prentice-Hall) by David W. Kammler, provides a wonderful organization of these ideas and contains a wide variety of applications sure to appeal to students across many disciplines.Computer assisted visualization (and sonification) is essential for the modern applications of Fourier analysis. Students can use Maple, Mathematical, or Matlab to plot various functions and to prepare animations or sound files for certain applications. It takes time to prepare such code, however, and we would like to make the course more efficient by giving students the ability to experience the most commonly used visualizations and sonifications instantly. Static illustrations are included in every elementary mathematics text. We envision the use of the Internet (or a CDROM) to provide dynamic multimedia illustrations. The text develops the formula for the motion of a vibrating string and with a few mouse clicks the student sees a computer simulation. The professor describes a wavelet decomposition of a signal and with a few mouse clicks lets the students hear the effect of such processing on a simple musical theme. (You can experience a prototype at pascal.math.stthomas.edu/fourier/demos. html).A portion of NSF funding (NSF-USE 89503, NSF-USE 9054179, and NSF-USE 9156064) provided resources to develop FOURIER, a software package designed to produce such instant experiences for students. FOURIER is used by faculty at a dozen universities to enhance student understanding of Fourier analysis. This software package was developed jointly by the PI and David Kammler. This work was initiated in 1990 and the software was designed to work in a DOS environment. The co-PI on the proposal is currently using the program to teach Fourier analysis to undergraduates at the University of St. Thomas.After experimenting with the software for nearly a decade we are now ready for a top-to bottom recoding that will result in a commercially available product that runs under a web browser such as Netscape or Internet Explorer. The idea is that the program will then be platform-independent as well as accessible via the Internet, on a local network, or on a CD or hard disk. This proposal is a request for NSF funding for this purpose. The basic design features of FOURIER will be preserved. The PI will be responsible for the bulk of the programming. The co-PI will take the lead in designing several tutorials and modules to be packaged with FOURIER for distribution on CD or the World Wide Web. Undergraduate students at the University of St. Thomas who have taken the Fourier Analysis course, will assist the co-PI in this endeavor. David Kammler has agreed to serve as a consultant on the project. In addition, faculty from Louisiana State University and the University of Akron will take part in the testing phase of the project.