We generalize classical Fourier analysis over the real line or the circle to any locally compact abelian (LCA) group. In this setting, the Fourier transforms of certain functions on a LCA group are functions on the dual of this group. We prove the Pontryagin duality theorem -- that every LCA group is the dual group of its dual group -- and show that it leads to a duality between compact and discrete abelian groups.
This talk should be accessible to everyone -- elementary analysis and algebra recommended.