MAU34206 Harmonic analysis
| Module Code | MAU34206 |
|---|---|
| Module Title | Harmonic analysis |
| Semester taught | Semester 2 |
| ECTS Credits | 5 |
| Module Lecturer | Prof. Dmitri Zaitsev |
| Module Prerequisites | MAU22200 Advanced analysis |
Assessment Details
- This module is examined in a 2-hour examination at the end of Semester 2.
- Students are assessed based on the exam alone.
- Re-assessment, if needed, consists of 100% exam.
Contact Hours
11 weeks of teaching with 3 lectures per week.
Learning Outcomes
On successful completion of this module, students will be able to
- Give appropriate definitions, theorems and proofs related to the topics introduced in this module including topics in Fourier analysis on intervals and the real line, convergence of Fourier series and integrals, Fourier analysis on locally compact abelian groups, dual group, Pontryagin duality, Plancherel theorem and discrete groups.
- Solve problems requiring manipulation or application of one or more of the concepts and results introduced in this module.
- Formulate mathematical arguments in appropriately precise terms related to the concepts and results introduced in this module.
- Apply their knowledge in mathematical domains where harmonic analytic techniques are relevant.
Module Content
- Introduction: Review of Fourier series for periodic functions, the unit circle as either the domain or the target for characters of LCA (locally compact abelian) groups, dual group, Fourier transform in concrete and abstract settings, Haar integral on LCA groups.
- Convergence results: Fourier series for differentiable functions, Dirichlet and Fejer kernels, Abel convergence.
- Abstract Fourier theory: Pontryagin duality, Haar integrals on certain kinds of LCA groups, Plancherel theorem.
- Additional topics: Fast Fourier transform, wavelets.