# Module MA342A: Harmonic Analysis I

**Credit weighting (ECTS)**- 5 credits
**Semester/term taught**- Michaelmas term 2015-16
**Contact Hours**- 11 weeks, 3 lectures including tutorials per week
**Lecturer**- Prof Richard Timoney
**Learning Outcomes**- On successful completion of this module, students will be able to:
- Give the appropriate definitions, theorems and proofs concerning the syllabus topics, 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, discrete groups;
- Solve problems requiring manipulation or application of one or more of the concepts and results studied;
- Formulate mathematical arguments in appropriately precise terms for the subject matter;
- Apply their knowledge in mathematical domains where harmionic analytic techniques are relevant;

**Module Content**-
- Introduction: Review of Fourier series for periodic functions, the unit circle as a domain and as a target for characters of general 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 locally compact abelian) groups, Plancherel theorem.
- Possible additional topics (if time permits): Fast Fourier transform, Wavelets.

For further information refer to the module web pages.

**Module Prerequisite**- MA2223-Metric Spaces and MA2224-Lebesque Integral are required
**Assessment Detail**-
This module will be examined in a 2 hour
**examination**in Trinity term.**Continuous assessment**will contribute 15% to the final grade for the module at the annual examination session.