# Fourier Notes

This page collects some useful results from Fourier Analysis.

## Fourier Series

Definition 1. Let be a Lebesgue integrable function on and define the *Fourier coefficients* by

Theorem 2. Suppose that is continuous on and is absolutely summable. Then
where the series converges uniformly on .

Theorem 3. **(Parseval's Theorem)** Let . Then its Fourier coefficients are in and
Thus the map is unitary.

## Fourier Transforms

Definition 4. Let and define the *Fourier transform*
Likewise for integrable define the *inverse Fourier transform*

Theorem 5. **(Fourier Inversion Theorem)** If and also then
for almost every .

Corollary 6. If, in addition, is continuous, then for all .

Theorem 7. **(Plancherel Theorem)** If then and

Thus the map is an isometry from into ,
which extends to a unitary . This unitary is called the *Plancherel transform*.