Based on a guest lecture at Instituto Superior Técnico (University of Lisbon), I use calculus and abstract algebra to derive the Fourier Tranform using the Fourier Series.
The lecture notes are based on Tom Coates' lecture on toric varieties. A few references:
M. Audin, Toric actions on symplectic manifolds
W. Fulton, Toric varieties
Cox-Schenck-Little, Toric varieties