Fourier Series

Definition 1 (Periodic functions)
A function f(t) is said to have a period T or to be periodic with period T if for all t, f(t+T)=f(t), where T is a positive constant. The least value of T>0 is called the principal period or the fundamental period or simply the period of f(t).

Definition 2 (Periodic expansion)
Let a function f be declared on the interval [0,T). The periodic expansion

of f is defined by the formula

Definition 3 (Piecewise continuous functions)
A function f defined on I=[a,b] is said to be piecewise continuous on I if and only if

(i): there is a subdivision such that f is continuous on each subinterval and
(ii): at each of the subdivision points both one-sided limits of f exist.

Theorem 1
Let f be continuous on

. Suppose that the series

(1)

converges uniformly to f for all

. Then

(2)

Definition 4 (Fourier coefficients, Fourier series)
The numbers a_n and b_n are called the Fourier coefficients of f. When a_n and b_n are given by (2), the trigonometric series (1) is called the Fourier series of the function f.

Remark 1
If f is any integrable function then the coefficients a_n and b_n may be computed. However, there is no assurance that the Fourier series will converge to f if f is an arbitrary integrable function. In general, we write

to indicate that the series on the right may or may not converge to f at some points.

Remark 2 (Complex Notation for Fourier series)
Using Euler's identities,

where i is the imaginary unit such that i²=-1, the Fourier series of f(x) can be written in complex form as

(3)

where

(4)

and

Example 4
Let f(x) be defined in the interval [0,T] and determined outside of this interval by its periodic extension, i.e. assume that f(x) has the period T. The Fourier series corresponding to f(x) (with