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).

Example 1
The function has periods , since all equal .

Example 2
Let . If f(x) has the period then has the period T. (substitute )

Example 3
If f has the period T then

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

Theorem 1
Let f be continuous on . Suppose that the series


converges uniformly to f for all . Then


Definition 4 (Fourier coefficients, Fourier series)
The numbers an and bn are called the Fourier coefficients of f. When an and bn 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 an and bn 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





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 ) is


where the Fourier coefficients an and bn are



Example 5
Let an and bn be the Fourier coefficients of f, see above. The phase angle form of the Fourier series of f is

with amplitudes

and phase angles

See the applet to gain intuition as to the effects of the amplitude and the phases on the function behavior.

Example 6
We compute the Fourier series of the function f given by

Since f is an odd function, so is , and therefore


For the coefficient bn is given by

It follows

The Gibbs Phenomenon

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