Harmonic Analysis

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• 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.
• 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 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.

• 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

Teaching and learning methods

Lectures, tutorials, private study.

Resources & Reading list

Textbooks

Anton Deitmar and Siegried Echterhoff. Principles of Harmonic Analysis. Springer.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat Information

Repeat type: Internal & External