Some properties of the Fourier transform on R^n
- It is unitary
- It exchanges convolution and multiplication
- It exchanges translation and modulation
- It exchanges regularity and decay
- The kernel is unimodular. In particular, it is L^1 -> L^infty bounded.
- Uncertainty principle in the bulk: If f has support in a convex body, then F[f] is locally constant on each dual convex body
- Uncertainty principle in the tails: if f has compact support, then F[f] has full support. The proof uses complex analysis.