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

#### Module Aims

This module brings together methods learned in algebra, group theory and analysis courses to introduce the students to harmonic analysis. Harmonic analysis extends key ideas of Fourier analysis from Euclidean spaces to general topological groups. A fundamental goal is understand algebras of functions on a group in terms of elementary functions. These correspond to the idea representing signals in terms of standing waves.

#### 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
Preparation for scheduled sessions24
Teaching24
Independent Study126
Lecture24
Total study time300

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