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

FOURIER 项目：基于浏览器的程序，用于增强跨学科本科傅立叶分析课程

• 批准号: 9950738
• 负责人: David Eubanks
• 金额: $ 13.57万
• 依托单位: Coker College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9950738&HistoricalAwards=false
• 关键词: Project; FOURIER; Browser; based; Program

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.

基本傅立叶分析是科学家和工程师教育的重要组成部分。理论与应用之间的相互作用使其成为数学学生教育的理想顶峰课程，特别是那些希望学习数学为我们的文化做出贡献的一些方式的潜在教师。传统的介绍这一机构的数学集中在使用傅立叶级数来解决边界值问题和一些相关的世纪定理，处理问题的收敛。分布和FFT的发展使得有可能为本科生提供一个全面的现代傅立叶分析理论和应用的基本介绍。本文介绍了大卫W. Kammler提供了这些思想的精彩组织，并包含了各种各样的应用程序，肯定会吸引许多学科的学生。计算机辅助可视化（和声化）对于傅立叶分析的现代应用至关重要。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来使课程更有效。静态插图包含在每一个小学数学教科书。我们设想使用互联网（或光盘）提供动态的多媒体插图。文中提出了振动弦运动的公式，学生只需点击几下鼠标就可以看到计算机模拟。教授描述了一个信号的小波分解，并点击几下鼠标，让学生听到这种处理对一个简单的音乐主题的影响。(You可以在pascal.math.stthomas.edu/fourier/demos上体验原型。美国国家科学基金会的一部分资金（NSF-USE 89503，NSF-USE 9054179和NSF-USE 9156064）提供了开发FOURIER的资源，FOURIER是一个软件包，旨在为学生提供这种即时体验。FOURIER被十几所大学的教师用来提高学生对傅立叶分析的理解。这个软件包是由PI和大卫Kammler共同开发的。这项工作始于1990年，软件设计用于DOS环境。该项目的合作伙伴目前正在使用该程序向圣托马斯大学的本科生教授傅立叶分析。在对该软件进行了近十年的实验后，我们现在已经准备好进行自上而下的重新编码，这将导致一个在Netscape或Internet Explorer等Web浏览器下运行的商业产品。这个想法是，程序将是独立于平台的，并且可以通过互联网、本地网络、CD或硬盘访问。这一提议是为了这一目的而向NSF提出的资金请求。将保留FOURIER的基本设计特征。PI将负责大部分编程。合作PI将带头设计几个教程和模块，与FOURIER一起打包，以CD或万维网的形式分发。参加过傅立叶分析课程的圣托马斯大学的本科生将协助co-PI进行这项奋进。大卫卡姆勒已同意担任该项目的顾问。此外，路易斯安那州立大学和阿克伦大学的教师将参加该项目的测试阶段。