Principles of Fourier Analysis, 2nd Edition

8

Heuristic Derivation of the Fourier Series Formulas

Suppose we have a “reasonable” (whatever that means) periodic function f with period p. If Fourier’s bold conjecture is true, then this function can be expressed as a (possibly infinite) linear combination of sines and cosines. Let us naively accept Fourier’s bold conjecture as true and see if we can derive precise formulas for this linear combination. That is, we will assume there are ω’s and corresponding constants A_ω’s and B_ω’s such that

$f\left(t\right)={\displaystyle \sum _{\omega =?}^{?}{A}_{\omega }\cos \left(2\pi \omega t\right)}+{\displaystyle \sum _{\omega =?}^{?}{B}_{\omega }\sin \left(2\pi \omega t\right)}\text{for}\text{all}t\text{in}ℝ.$

(8.1)

Then we will derive (without too much concern for rigor) formulas for the ω’s and corresponding A_ω’s and B_ω’s. Later, we’ll investigate the validity of our naively derived formulas. ...