John Lindsay Orr

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.