Fourier Series .edu
Fourier Series
A Fourier series is an infinite series of the form
a+
X
n=1
bn cos(nx) +
X
cn sin(nx).
n=1
Virtually any periodic function that arises in applications can be represented as the
sum of a Fourier series. For example, consider the three functions whose graph are
shown below:
These are known, respectively, as the triangle wave (x), the sawtooth wave N(x),
and the square wave (x). Each of these functions can be expressed as the sum of
a Fourier series:
(x) = cos x +
cos 3x
cos 5x
cos 7x
cos 9x
+
+
+
+
2
2
2
3
5
7
92
N(x) = sin x +
sin 3x
sin 4x
sin 5x
sin 2x
+
+
+
+
2
3
4
5
(x) = sin x +
sin 3x
sin 5x
sin 7x
sin 9x
+
+
+
+
3
5
7
9
Fourier series are critically important to the study of differential equations, and they
have many applications throughout the sciences. In addition, Fourier series played an
important historical role in the development of analysis, and the desire to prove theorems about their convergence was a large part of the motivation for the development
of Lebesgue integration.
These notes develop Fourier series on the level of calculus. We will not be worrying
about convergence, and we will not be not be proving that any given function is
Fourier Series
2
n
cos nx
sin nx
2
cos2 x ? sin2 x
2 cos x sin x
3
cos3 x ? 3 cos x sin2 x
3 cos2 x sin x ? sin3 x
4
cos4 x ? 6 cos2 x sin2 x + sin4 x
4 cos3 x sin x ? 4 cos x sin3 x
5 cos5 x ? 10 cos3 x sin2 x + 5 cos x sin4 x 5 cos4 x sin x ? 10 cos2 x sin3 x + sin5 x
Table 1: Multiple-angle formulas.
actually equal to the sum of its Fourier series. We will revisit the theoretical aspects
of this topic later in the course after we have defined the Lebesgue integral and proven
Lebesgues dominated convergence theorem.
Trigonometric Polynomials
A trigonometric polynomial is a polynomial expression involving cos x and sin x:
cos5 x + 6 cos3 x sin2 x + 3 sin4 x + 2 cos2 x + 5
Because of the identity cos2 x + sin2 x = 1, most trigonometric polynomials can be
written in several different ways. For example, the above polynomial can be rewritten as
5 cos3 x sin2 x + 3 sin4 x + cos3 x ? 2 sin2 x + 7
Fourier Sums
A Fourier sum is a Fourier series with finitely many terms:
5 + 3 sin 2x + 4 cos 5x ? 3 sin 5x + 2 cos 8x.
Every Fourier sum is actually a trigonometric polynomial, and any trigonometric
polynomial can be expressed as a Fourier sum.
Converting a Fourier sum to a trigonometric polynomial is fairly straightforward: simply substitute the appropriate multiple-angle identity for each cos nx and
sin nx (see Table 1).
It is less obvious that every trigonometric polynomial can be expressed as a Fourier
Fourier Series
3
sum. This depends on the three product-to-sum formulas:
cos A cos B =
1
1
cos(A ? B) + cos(A + B)
2
2
sin A cos B =
1
1
sin(A ? B) + sin(A + B)
2
2
sin A sin B =
1
1
cos(A ? B) ? cos(A + B).
2
2
These identities allow us to transform any product of trigonometric functions into a
sum. By applying them repeatedly, we can remove all of the multiplications from a
trigonometric polynomial, resulting in a Fourier sum.
Alternatively, one can use these identities to derive power-reduction formulas
for cosj x sink x, the first few of which are listed below:
j=0
j=1
j=2
j=3
k=0
1
cos x
1 + cos 2x
2
3 cos x + cos 3x
4
k=1
sin x
sin 2x
2
sin x + sin 3x
4
2 sin 2x + sin 4x
8
k=2
1 ? cos 2x
2
cos x ? cos 3x
4
1 ? cos 4x
8
2 cos x ? cos 3x ? cos 5x
16
k=3
3 sin x ? sin 3x
4
2 sin 2x ? sin 4x
8
2 sin x + sin 3x ? sin 5x
16
3 sin 2x ? sin 6x
32
These formulas tell us how to convert each term of a trigonometric polynomial directly
into a Fourier sum.
Fourier Series
4
Orthogonality
There is a nice integral formula for finding the coefficients of any Fourier sum. This
is based on the orthogonality of the functions cos nx and sin nx:
Theorem 1 Orthogonality Relations
If j, k N, then:
Z
(
sin jx sin kx dx =
0
?
Z
cos jx cos kx dx =
?
and
Z
if j = k
otherwise,
sin jx cos kx dx = 0.
?
PROOF For n Z, observe that
Z
sin nx dx = 0
and
(
2
cos nx dx =
0
?
Z
?
if n = 0
otherwise.
If j, k N, we can use the product-to-sum identities to deduce that
(
Z
Z
cos (j ? k)x + cos (j + k)x
cos jx cos kx dx =
dx =
2
0
?
?
if j = k
otherwise,
and
Z
Z
sin jx sin kx dx =
?
and
?
Z
cos (j ? k)x ? cos (j + k)x
dx =
2
Z
sin jx cos kx dx =
?
?
(
0
if j = k
otherwise,
sin (j ? k)x + sin (j + k)x
dx = 0.
2
In general, the inner product of two functions f and g on an interval [a, b] is
Z b
hf, gi =
f (x) g(x) dx.
a
A collection F of nonzero functions on [a, b] is said to be orthogonal if hf, gi = 0
for all f, g F with f 6= g. According to the above theorem, the functions
{cos nx}nN {sin nx}nN
Fourier Series
5
are orthogonal on the interval [?, ]. Note that these functions are also orthogonal
to the constant function 1.
This definition of orthogonality is related to the notion of orthogonality in linear
algebra. Specifically, let C([a, b]) be the vector space of all real-valued continuous
functions on the interval [a, b]. Then the formula for hf, gi given above defines an
inner product on this vector space (analogous to the dot product on Rn ), under which
orthogonal functions are the same as orthogonal vectors.
Theorem 2 Fourier Coefficients
Let
N
X
f (x) = a +
bn cos nx +
n=1
Then
1
a =
2
N
X
cn sin nx.
n=1
Z
f (x) dx.
?
Furthermore, for all n {1, . . . , N },
Z
1
bn =
f (x) cos nx dx
and
?
cn
1
=
Z
f (x) sin nx dx.
?
PROOF The formula for a is fairly obvious. To derive the formula for the bs,
observe that
Z
f (x) cos kx dx
?
Z
= a
cos kx dx +
?
N
X
Z
bn
cos nx cos kx dx +
?
n=1
N
X
Z
cn
n=1
sin nx cos kx dx.
?
Applying the orthogonality relations reduces this to
Z
f (x) cos kx dx = bk
?
and the formula for bk follows. The derivation of the formula for ck is similar.
The formulas in the theorem above can be written as follows:
a =
hf, 1i
,
h1, 1i
bn =
hf, cos nxi
hcos nx, cos nxi
and
cn =
hf, sin nxi
.
hsin nx, sin nxi
................
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