1. MA5238 Fourier Analysis Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sgmatwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1 Lecture 1. Tuesday 12 Jan 2010

2. Administrative MA5238Module Title FOURIER ANALYSIS SemesterSemester 2, 2009/2010 Modular Credits4 Teaching Staff ASSOC PROF Lawton, Wayne Michael matwml@nus.edu.sgmatwml@nus.edu.sg Weblinks http://www.math.nus.edu.sg/~matwml/courses/ my personal website which contains mountains of materialshttp://www.math.nus.edu.sg/~matwml/courses/ AIMS & OBJECTIVES This module is designed for graduate students in mathematics. It covers the following major topics: Fourier series, Fourier transform on R^n, distributions and generalised functions, Sobolev spaces and their applications to partial differential equations. Introduction to singular integrals. PREREQUISITES MA5205 Graduate Analysis I and {MA3266 Introduction to Fourier Analysis or MA3266S Intr. FA version S} SCHEDULE Final Examination 26-04-2010 AM LECTURE Class [SL1] TUESDAY From 1000 hrs to 1200 hrs in S16-0306, Week(s): EVERY WEEK. FRIDAY From 1000 hrs to 1200 hrs in S16-0306, Week(s): EVERY WEEK. SYNOPSIS required textbook A Guide to Distribution Theory and Fourier Transforms by Robert S. Strichartz SYLLABUS PRACTICAL WORK none ASSESSMENT Test 1 25% Test 2 25% Final Examination 50% PRE-CLUSIONS NIL WORKLOAD 3-0-0-3-4

3. Textbook A Guide to Distribution Theory and Fourier Transforms by Robert S Strichartz, World Scientific, Singapore, 2003. Available in the Science COOP Bookstore at a significantly reduced student price The use of this textbook is compulsory because you are expected to read ALL of it and work out solutions to most of the problems located at the ends of each of the 8 chapters. If time permits we will supplement the material in this textbook with additional material covering singular integral operators and selected topics in harmonic analysis.

4. Contents of Textbook 1.What are Distributions? 2.The Calculus of Distributions 3.Fourier Transforms 4.Fourier Transforms of Tempered Distributions 5.Solving Partial Differential Equations 6.The Structure of Distributions 7.Fourier Analysis 8.Sobolev Theory and Microlocal Analysis My aim to is cover all of the material in about 9 weeks, to spend 1 week for tests, and cover supplementary topics in the remaining weeks

5. Functions as Operators Lets forget about details for now – those WILL come associate to a function the operator what’s this ? Question What properties does enjoy ?

6. Very Useful Notation Corollary Proof Wow, that’s deep !

7. Integration by Parts Definition Lemma Proof to be worked out in class

8. Heaviside Function Definition Theorem Proof to be worked out in class Definition Notation Definition Lemma

9. Assignment 1 Read Preface and Chapter 1 Do Problems 1-14 and prepare to solve on the board in class for Friday