MATH528

MATH528 Lesson18: Fourier series

Definition. The trigonometric basis of period $2 L$ is $𝒯_{L} = {\cos (π n x / L), \sin (π n x / L), n = 0, 1, 2, \dots} \ {0}$ .

Example. For $2 L = 2 π$ , the (canonical) trigonometric basis is ${1, \cos x, \sin x, \cos 2 x, \sin 2 x, \dots}$ .

We'll assume $L = π$ henceforth.

Definition. A trigonometric series is a linear combination of the trigonometric basis functions

$T = a_{0} + \sum_{n = 1}^{\infty} (a_{n} \cos n x + b_{n} \sin n x) .$

Many periodic functions $f (x) = f (x + 2 π)$ can be represented by Fourier series with coefficients:

a_{0} = \frac{1}{2 π} \int_{- π}^{π} f (x) d x,

a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) \cos (n x) d x,

b_{n} = \frac{1}{π} \int_{- π}^{π} f (x) \sin (n x) d x,

known as the Euler formulas.