MATH6155 Harmonic Analysis
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
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.
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.
• 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.
|Preparation for scheduled sessions||24|
|Completion of assessment task||66|
|Wider reading or practice||36|
|Total study time||300|
Resources & Reading list
Anton Deitmar and Siegried Echterhoff. Principles of Harmonic Analysis.
Repeat type: Internal & External