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Math 240, Spring 2023, Dr. Honigs
Lecture Notes, April 3
Today’s Lecture: The projection formula, projecting on an orthogonal basis
(§4.2), Fourier series
Last time we saw:
• The dot product definition
• v · w = ||v|| ||w|| cos(θ)
• The formula for projecting v onto w is
Example
v·w
w·w w.
Where does the projection formula come from?
Projecting onto an orthogonal basis.
Fourier series
570
Inner Product Spaces
Example 10.5.1
Find the fifth Fourier approximation to the function f (x) defined on [−π , π ] as follows:
y
π + x if − π ≤ x < 0
f (x) =
π − x if 0 ≤ x ≤ π
π
−π
0
π
x
Solution. The graph of y = f (x) appears in the top diagram.
The Fourier coefficients are computed as follows. The details
of the integrations (usually by parts) are omitted.
y
4
a0 =
3
1
2π
2
ak =
1
-4 -3 -2 -1 0 1 2 3 4
f5 (x)
x
y
1
π
−π
Z π
−π
Z π
−π
f (x)dx =
π
2
f (x) cos(kx)dx =
2
[1 − cos(kπ )] =
π k2
f (x) sin(kx)dx = 0
0
4
π k2
if k is even
if k is odd
for all k = 1, 2, . . .
Hence the fifth Fourier approximation is
o
n
f5 (x) = π2 + π4 cos x + 312 cos(3x) + 512 cos(5x)
4
3
2
1
-4 -3 -2 -1 0 1 2 3 4
f13 (x)
bk =
1
π
Z π
x
This is plotted in the middle diagram and is already a reasonable
approximation to f (x). By comparison, f13 (x) is also plotted
in the bottom diagram.
We say that a function f is an even function if f (x) = f (−x) holds for all x; f is called an odd function
if f (−x) = − f (x) holds for all x. Examples of even functions are constant functions, the even powers x2 ,
x4 , . . . , and cos(kx); these functions are characterized by the fact that the graph of y = f (x) is symmetric
about the y axis. Examples of odd functions are the odd powers x, x3 , . . . , and sin(kx) where k > 0, and
the graph of y = f (x) is symmetric about the origin if f is odd. The usefulness of these functions stems
from the fact that
Rπ
f (x)dx = 0 R
if f is odd
R−ππ
π
f
(x)dx
=
2
f
(x)dx
if
f is even
−π
0
These facts often simplify the computations of the Fourier coefficients. For example:
1. The Fourier sine coefficients bk all vanish if f is even.
2. The Fourier cosine coefficients ak all vanish if f is odd.
This is because f (x) sin(kx) is odd in the first case and f (x) cos(kx) is odd in the second case.
The functions 1, cos(kx), and sin(kx) that occur in the Fourier approximation for f (x) are all easy to
generate as an electrical voltage (when x is time). By summing these signals (with the amplitudes given
by the Fourier coefficients), it is possible to produce an electrical signal with (the approximation to) f (x)
as the voltage. Hence these Fourier approximations play a fundamental role in electronics.