The University of Southampton
Courses

# MATH6155 Harmonic Analysis

## Module Overview

Harmonic analysis extends key ideas of Fourier analysis from Euclidean spaces to general topological groups. A fundamental goal is understanding algebras of functions on a group in terms of elementary functions. These correspond t the idea representing signals in terms of standing waves. Harmonic analysis is now a key part of modern mathematics with important applications in physics and engineering.

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Explain the concept of Haar measure and identify Haar measures for the group of the integers, the reals under addition and multiplication, the torus, and the ax+b group.
• Define Banach algebra and list main examples.
• State the Gelfand-Naimark theorem and use it to identify the C* algebra of the groups Rn and Zn.
• Explain the concept of Pontryagin duality and the connection with the Fourier series and Fourier transform.
• Use the Pontryagin duality to identify duals of examples of locally compact abelian groups.

### Syllabus

• Locally compact topological groups • The Haar measure • Convolution algebra of a group • Banach algebras • Gelfand map and Gelfand-Naimark theorem • Dual groups • C*-algebra of a locally compact abelian group • Plancherel theorem • Pontryagin duality • Applications to wavelet theory, if time permits

### Learning and Teaching

#### Teaching and learning methods

Lectures, tutorials, private study.

TypeHours
Lecture24
Preparation for scheduled sessions24
Total study time150

Anton Deitmar and Siegried Echterhoff. Principles of Harmonic Analysis.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 100%

#### Referral

MethodPercentage contribution
Written exam 100%

#### Repeat Information

Repeat type: Internal & External