Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

version of 26 March 2000

There are three distinct versions of the Fourier
series in common use, the "full"
Fourier series, the Fourier sine series, and the Fourier cosine series.
They are based on slightly different choices of an orthogonal basis set, whether
(2.6) for the Fourier sine series,
(2.7) for the Fourier cosine series, or
(2.8) for the full Fourier series. Notice that while the
full set (2.8) contains both sines and cosines, it does not contain as many frequencies
as either of the other sets, since the trig functions depend on **2**
n x/L rather
than all integer multiples of
x/L. We will also discuss an
"exponential" Fourier series,
which uses complex functions of the form

exp(2 i x/L),

but this is actually the same series as the full Fourier series except that it has been written in a convenient way using exponential functions.

The three different sets do an equally good job of approximating functions in the r.m.s. sense, but their relationship to symmetries is different. For this reason, different sets of Fourier functions will arise in our study of partial differential equations, depending on which kind of boundary conditions are physically appropriate for a given problem.

The full Fourier series consists of functions which are *periodic* , and are
particularly good for representing or approximating periodic functions, that is,
functions for which f(x + L) = f(x). Examples include sounds built on a basic
frequency with overtones or a physical function depending on an angular variable
such as longitude. The Fourier sine series, in contrast, is well adapted to functions
which are zero at x=0 and x=L, since all the functions sin(n \pi x/L) have this property.
A related but less obvious property that begs for the use of sine functions is an
odd symmetry: f(-x) = - f(x). If the function you wish to approximate is an odd function,
then the sines are a good candidate for a basis set. Notice that the sines are not only odd
under reflection about 0, but also about the other end of the interval, x=L.
Fourier cosine series are better suited for even functions.
A Fourier sine series represents a function as

The formula for the coefficients b_{n} is the same as in
equation 2.14, except that the Fourier sine series
uses all integer multiples of
x/L, where L=b-a, not just the even multiples:

**The powerful theorem behind the Fourier
sine series III.3a**.

If f(x) is square-integrable on an interval [a,b], then

a) All of the coefficients b_{n} of (3.a2) are well defined.

b) All of the coefficients b_{n} are uniquely
determined by f(x) and depend linearly on f(x).

c) The series

converges to f(x) in the r.m.s. sense. In other words,

We shall express this by writing

(Remember, however, that this series converges in a mean sense, and not
necessarily at any given point.) It is also true that it converges
a.e.

d)

and the right side is guaranteed to converge.

e) If g is a second square-integrable function, with Fourier sine coefficients b-tilde, then

This is known as the *
Parseval formula* for the Fourier sine series.

Conversely, given a square-summable sequence b_{n}, i.e., real or
complex numbers such that

is finite, they determine a square integrable function f(x) uniquely a.e. such that all the statements of this theorem hold.

In addition, if f(x) is piecewise continuous, then Theorem III.6 likewise holds, except at the end points of the interval. If 0 < x < L, the Fourier sine series will converge to the average of the right and left limiting values of f at x. When x=0 or L, however, the Fourier sine series will always converge to 0, since all the sine functions are 0 at those two points.

Compare with the Fourier cosine series

Compare with the full Fourier series

Compare with the exponential Fourier series

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