Fourier 分析
Student No.:50
Time:Mon/Thu 15:20-17:00(except for public holidays)
Instructor:Pin Yu,Shenghao Sun  
Place:Conference room 3, floor 2
Starting Date:2012-9-10
Ending Date:2012-12-27




第一部分:Archimedean情况 (约8周)

介绍欧式空间上的经典的Fourier分析。我们将学习L. Schwartz的分布理论,然后介绍能定义Fourier变换的最合适的空间:缓增分布空间。最后,我们学习一些Sobolev空间,并强调微局部的观点。



这部分讲解J. Tate的博士论文(1950年):"Fourier Analysis in Number Fields and Hecke's Zeta-Functions". 此文系统发展了局部域以及数域的adèles环上的Fourier分析,并用其重新证明了数域的zeta-函数的解析延拓和函数方程(由Hecke最早证明),而且把epsilon-因子表为局部因子的乘积。我们将从简要复习预备知识开始。


The course will be taught in CHINESE and it consists of two parts:

Part I Archimedean Case (8 weeks)

This is an introduction to classical Fourier analysis on Euclidean spaces. We start by studying the distribution theory of Laurent Schwartz, then introduce the tempered distribution space which is the ideal space to define Fourier transform. We shall end this part by studying certain Sobolev spaces and emphasizing the mircolocal point of view.

This part is designed specifically for third or fourth year undergraduate students.

Part II Non-Archimedean Case

This part is about J. Tate's thesis in 1950, "Fourier Analysis in Number Fields and Hecke's Zeta-Functions". It develops Fourier analysis on local fields and adèles of number fields systematically, and uses it to reprove the analytic continuation and functional equation of zeta functions of number fields (first proved by Hecke), and also gives an expression of the "epsilon-factor" as a product of local factors. We will start by briefly reviewing the background knowledge (see prerequisites below).


You can download the lecture notes written by Prof. Pin Yu from the MSC website or click here  

Please click here to download the lecture note written by Prof. Shenghao Sun.  







第一部分:除了微积分和线性代数,学生应熟悉Lebesgue dominant convergence定理,L^1和L^2空间。

第二部分:基本的代数数论(比如Lang, Algebraic Number Theory 2nd Ed. 的前三章),局部紧Abel群上的Fourier分析的基本知识。

Part I: Besides calculus and linear algebra, the students should be familiar with Lebesgue dominant convergence theorem, L^1 spaces and L^2 spaces.

Part II: Basic algebraic number theory (e.g. Lang, Algebraic Number Theory, 2nd Ed., Ch. 1, 2, 3 or equivalent), basics on Fourier analysis of locally compact abelian group.