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;
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,
Abstract Fourier theory:
Pontryagin duality, Haar integrals on (certain kinds of locally compact abelian) groups,
Possible additional topics (if time permits):
Fast Fourier transform, Wavelets.
This module will be examined in a 2 hour examination in Trinity term. The examination will contribute 100% to the final grade. Re-assessments if required will consist of 100% exam.