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MA5238 Fourier Analysis. Lecture 1. Tuesday 12 Jan 2010. Wayne Lawton Department of Mathematics S17-08-17, 65162749 [email protected] http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1. Administrative. MA5238Module Title FOURIER ANALYSIS

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Ma5238 fourier analysis

MA5238 Fourier Analysis

Lecture 1. Tuesday 12 Jan 2010

Wayne Lawton

Department of Mathematics

S17-08-17, 65162749 [email protected]

http://www.math.nus.edu.sg/~matwml/

http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1


Administrative
Administrative

MA5238Module TitleFOURIER ANALYSIS

SemesterSemester 2, 2009/2010

Modular Credits4

Teaching StaffASSOC PROF Lawton, Wayne Michael [email protected]

Weblinks http://www.math.nus.edu.sg/~matwml/courses/ my personal website which contains mountains of materials

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


Textbook

A Guide to Distribution Theory and Fourier Transforms by Robert S Strichartz, World Scientific, Singapore, 2003.

Textbook

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.


Contents of textbook

  • What are Distributions? Robert S Strichartz, World Scientific, Singapore, 2003.

  • The Calculus of Distributions

  • Fourier Transforms

  • Fourier Transforms of Tempered Distributions

  • Solving Partial Differential Equations

  • The Structure of Distributions

  • Fourier Analysis

  • Sobolev Theory and Microlocal Analysis

Contents of Textbook

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


Functions as operators

Lets forget about details for now – those WILL come Robert S Strichartz, World Scientific, Singapore, 2003.

Functions as Operators

associate to a function

the operator

what’s this ?

enjoy ?

Question What properties does


Very useful notation

Very Useful Notation Robert S Strichartz, World Scientific, Singapore, 2003.

Corollary

Proof Wow, that’s deep !


Integration by parts

Integration by Parts Robert S Strichartz, World Scientific, Singapore, 2003.

Definition

Lemma

Proof to be worked out in class


Heaviside function

Heaviside Function Robert S Strichartz, World Scientific, Singapore, 2003.

Definition

Theorem

Proof to be worked out in class

Definition

Notation

Definition

Lemma


Assignment 1

Assignment 1 Robert S Strichartz, World Scientific, Singapore, 2003.

Read Preface and Chapter 1

Do Problems 1-14 and prepare to

solve on the board in class for Friday


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