分析学教程

 

Princeton Lectures in Analysis,Elias M. Stein and Rami Shakarchi

Elias M. Stein (调和分析大师,1999年Wolf奖获得者)和 Rami Shakarchi在2000年开始,给Princeton的本科生教分析课。他们的目的是,用统一的、联系的观点来把现代分析的“核心”内容教给本科生们,力图使本科生的分析学课程能接上现代数学研究的脉络。每学期一门课,共四门,顺序是:
I. Fourier series and Integrals (包括Finite Fourier analysis,不包括wavelet analysis).
II. Complex analysis.
III. Real Analysis:Measure theory,Lebesgue integration, and Hilbert Spaces.
IV. A selection of further topics, including functional analysis, distributions, and elements of probability theory.
前三个部分已经分别出书,
[1]Elias M. Stein; Rami Shakarchi,  Fourier analysis. An introduction. Princeton Lectures in Analysis, 1. Princeton University Press, Princeton, NJ, 2003. xvi+311 pp. ISBN: 0-691-11384-X
[2]Elias M. Stein; Rami Shakarchi,  Complex analysis. Princeton Lectures in Analysis, II. Princeton University Press, Princeton, NJ, 2003. xviii+379 pp. ISBN: 0-691-11385-8
[3]Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ, 2005. xx+402 pp. ISBN: 0-691-11386-6
其中[1]已经可以由国内买到,世图书号:7-5062-7287-3 ,29Y。
 
可以看出,这些课程仅仅假定读者读过大一微积分和线性代数,所以可看作是大二到大三共四个学期的必修课程,每学期一门。
 
非常值得注意的是,作者把Fourier analysis作为学完大一微积分后的第一门高级分析课。我本人是极为赞同这种做法的,一者,现代数学中Fourier分析无处不在,既在纯数学,如数论的各个方面都有深入的应用,又在应用数学中是绝对的基础工具。二者,Fourier分析不光有用,其本身的内容,可以说,就能够把数学中的几大主要思想都体现出来,如对称观,分解观,求和法,表示论方法。这样,学生们先学这门课,对数学就能有鲜活的了解,既知道它的用处,又能够“连续”地欣赏到数学中的各种大思想、大美妙。接着,是学同样具有深刻应用和理论优美性于一体的Complex analysis。学完这两门课,学生已经有了相当多的例子和感觉,既懂得其用又懂得其妙。这样,再学后面比较抽象的Real analysis 和functional analysis时,就自然得多、动机充分得多。
 
下面是[1]的书评
1.  世图书评(我改了几个字):
内容简介:
本书由在国际上享有盛誉普林斯顿大学教授Stein撰写而成,是一部傅立叶分析的入门教材,理论与实践并重,为了便于非数专业的学生学习,全书内容简明、易懂.全书分为三部分,第一部分介绍傅立叶级数的基本理论及其在等周不等式和等分布中的应用;第二部分研究傅立叶变换及其在经典偏微分方程及Radom变换中的应用;第三部分研究有限阿贝尔群上的傅立叶分析。书中各章均有练习题(Exercise)及思考题(Problem)。
目次:傅立叶积积分的起源;傅立叶级数和基本性质;傅立叶级数的收敛性;傅立叶积分的应用;R上的傅立叶变换;R^d上的傅立叶变换;有限傅立叶分析;狄利克雷素数定理。
读者对象:理工科高年级本科生、研究生及数学工作者。
 
 作者简介:
Stein在国际上享有盛誉,现任美国普林斯顿大学数学系教授。
他是当代分析,特别是调和分析和分析领域领袖人物之一。古典调和分析最困难问题之一是推广到多维。他是多维欧氏调和分析的创造者之一,为此他发展了许多先进工具如奇异积分、Radon变换、极大函数等。他还发展了多个实变元的Hardy空间理论,推广了1971年F. John和L. Nirenberg的重要发现:即Hardy空间与BMO空间的对偶。在群上的调和分析方面也有贡献,例如同R.Kunze一起发现所谓Kunze-Stein现象。除此之外,他对多复变问题也做出了突出成绩。
除了研究工作之外,他的许多书成为影响学科发展的重要参考文献。为此,他荣获1984年美国数学会在论述方面的Steele奖。
由于他的成就,他在1974年被选为美国国家科学院院士,1982年被选为美国文理学院院士,1993年获得瑞士科学院颁发的Schock奖。1999年获得世界性Wolf数学奖。
 
2.AMS Review : 作者们的三本书
 
第一本书评:
E. M. Stein is certainly one of the great avatars and developers of Fourier analysis in modern times. R. Shakarchi is a recent student of Charles Fefferman, so obviously is very well trained in the discipline. We are fortunate indeed that two such prominent exponents of one of the central parts of analysis have taken the time to write an instructional book of this kind. For this is not an entree to singular integrals, nor to pseudodifferential operators or paradifferential operators or wave front sets. It is instead a basic introduction to very classical topics of Fourier analysis. This includes Fourier series, the Fourier transform, and Fourier analysis on (some) abelian groups. There is even some material on the fast Fourier transform, and some nice Fourier-analytic ideas from number theory.
The book is written in a lively and inviting style. It is accessible to bright upper-division undergraduates and first-year graduate students. It does not assume acquaintance with either the Lebesgue integral or with functional analysis. So this book will be a delightful adventure for math majors as well as for mathematically oriented engineers, economists, physicists, and others who use analytical reasoning.
 
This work is clearly a labor of love. It has all the accoutrements that one expects of a good text: (i) a detailed preface, explaining whom the book is for and what are the pre-requisites, (ii) a solid and complete index, and (iii) a table of notation. The book is loaded with beautiful graphics and many crisp, clean examples (the kind for which Stein is justifiably renowned). The authors go the extra distance in providing appendices on the Riemann integral and on the sorts of multiple integrals that are used in modern Fourier analysis. Bravo!
 
A very important feature of this book is the exercise sets. They are broad and rich and deep. In fact a very unusual feature is that there are: (a) collections of queries that are labeled "Exercises"---these range from the straightforward to the tricky, but are tied fairly closely to the text; and (b) collections of queries that are labeled "Problems". These are more exploratory, and frequently go well beyond the direct subject matter that has been covered.
 
This is a great book from which to teach and to learn. It is based on a semester-long course that Stein taught at Princeton. It is probably too ambitious for a semester-long course at most schools, but it will be easy for the interested instructor to craft a useful and stimulating educational experience, based on this book, for the students at his/her institution.
 
I look forward to reading and reviewing the next three books in this series (by the same authors). One of the exciting features of this collection is that it establishes many non-obvious connections among different parts of analysis (real analysis, complex analysis, Fourier analysis, and probability). It will be instructive for student and mentor alike. This first volume is a terrific beginning, and promises to stand as a classic for many years to come.
 
Reviewed by Steven George Krantz
 

 
所以,我真得很希望,我们系可以试试这样的教法。有大师级数学家写的书做后盾,再加上本系这么多高水平的老师认真教,学生们应该是可以更快、更好地去明白数学的实质和美。20060426
 
 
相关讨论见新浪博客