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Fourier Series

7
...
Then, for any f, g E C[a, b] we have that the product j g is also continuous
on [a, b] and hence integrable on [a, b]
...

The inner product ( , ) is defined on

C[a, b] by
b

(j, g)

=

l j x g x dx
( ) ( )

The three properties of an inner product are satisfied because
b

b

(1) (f, f) J f(x)f(x) d x 2: 0 for all f E C[a, b] and (j, f) J f(x)f(x) dx 0 if
and only if f (x)
0 for all x E [a, b]
...

One interesting consequence is that the norm of a function f with respect to this

(

inner product is

11!11

=

b

l

) 1/2

j2(x) dx

Intuitively, this is quite satisfactory as a measure of how far the function is from the
zero function
...


Fourier Series
Let CP2rr denote the space of continuous real-valued functions of a real variable that
are periodic with period 2n
...
Examples

cos x, sin x, cos 2x, sin 3x, etc
...


However, its "fundamentai (smallest) period" is n
...
5
...


y

7r

2
Figure 7
...
5

A continuous periodic function
...
, cos nx, sin nx,
...

We formulate the questions and ideas as follows
...
)
(i) For any n, the set of functions { l, cos x, sin x, cos 2x, sin 2x,
...

This subspace will be denoted CP2rr,n·

(ii) Given an arbitrary function f in CP2rr, how well can it be approximated by a
function in CP2rr,n? We expect from our experience with distance and subspaces
that the closest approximation to f in CP2rr,n is projcP:z
...
The coefficients
for Fourier's representation off by a linear combination of { 1, cos x, sin x,
...
}, called Fourier coefficients, are found by considering this
projection
...
Since the distance
fromf to the n-th approximation prokP:z
...
n /II, to test if the ap­
proximation improves, we must examine whether II perpCP:z
...

Let us consider these statements in more detail
...
, cosnx, sinnx} is orthogonal
...
nf in CP2rr,n to an

...
2 and 7
...

That is, we use the projection formula, given an orthogonal basis {v1,
...

vk
proJs 1 1iJ v 1 +
+
2
···
llvkll
=

There is a standard way to label the coefficients of this linear combination:
prokp2"
...
Thus, we have

rr

(f, 1)

1
ao = li = ;

rr

f(x) dx

(f, cos mx) 1
= am =
7r
11 cos mxll2
(f, sin mx)
1
bm -2
7r
II sin mxll
_

_

(iii) Is projcPin,, fequal to fin the limit

rr
I-rr
rr f( )
...
The
question being asked is a question about the convergence of series-and in fact, about
series of functions
...
(The short answer is "yes, the series converges to f provided
that f is continuous
...
) Questions about convergence are important in physical and
engineering applications
...

Solution: We have

rr lxl dx
I-rr
1 rr
I-rr
rr lxl 2x dx
I-rrrr
1
lxl
3x dx
I-rrrr
1
lxl x dx
I-rr
1 rr
lxl
I
rr 3x dx
I
l

ao = a,= rr

a1= -

cos

7r

7r

=

0
4

cos

bi = -

sin

b1 = -

sin2xdx = 0

7r

= --

9rr

=0

- rr

I

b3 = -

�

rr and

4

a3 = 7r

Hence, prokPin
...
5
...


�

lxlcos xdx = --

1

7r

x

= rr

7r

7r

�

lxl sin

= 0

-rr

- ; cos x -

t,; cos 3x
...
5
...
1 fx
( )

-

Y

=

( )
projCP:u,,3 fx
7r

t

Graphs of projCP,,,, f and prokP,, , f compared to the graph of f(x)
...
J f= ; sinx

-

1

7r
1

7r
1

7r
1
7r
1
7r
1
7r
1
rr
-

x

lf
Jlf
Jlf
Jlf
Jlf
Jlf
Jlf
J

if

-

7r � x �

/2

if -rr/2 < x � rr/2
ifrr/2 < x � rr

fdx = 0

-;r

fcosxdx = 0

-;r

fcos 2xdx = 0

-;r

fcos 3xdx = 0

-;r

-;r

fsinxdx =

4

7r

fsin 2xdx = 0

-;r

-;r

fsin 3xdx = -

4

9rr

-

tr sin 3x
...
5
...


y

1r

2

-

--

-�
Figure 7
...
7

Graphs of prokPi
...
1 f(x)
Y projCPin,J f(x)
=

f and projCPi
...


...
5
Computer Problems
Cl Use a computer to calculate projCPin,,, f for
n =3,7, and 11 for each of the following functions
...

(a)

f(x) =x2,

-;r

�

x

�

f(x) =

1

-

x

n �

if

-

ifO

:5

7r

n :5
<

x

x
�

:5
7r

o

7r

PROBLEMS 7 5

...
1)

2 Why is it easier to determine coordinates with re­
spect to an orthonormal basis than with respect to an
arbitrary basis? What are some special features of
the change of coordinates matrix from an orthonor­
mal basis to the standard basis? What is an orthogo­
nal matrix? (Section 7
...
(Section 7
...
2)

5 Outline how to use the ideas of orthogonality to find
the best-fitting line for a given set of data points
{( t; y; ) Ii=1,
...
(Section 7
...

Give an example of an inner product on M(2, 3)
...
4)

Chapter Quiz
El

Determine whether the following sets are orthog­

a vector in §, use the orthonormality of 'B to deter­

onal, and which are orthonormal
...


{ t -> -i}
{ � �}
{}, � -�}

decide
...


' Vs

(b) Prove that if P and R are n x n orthogonal ma­
trices, then so is PR
...


Consider the orthonormal set
-1
1
1
1

(b) Determine the point in S closest to x

1
-2

...
Let S be the sub-

1

0

space of JR
...
Given that x

=

2
5
1 is

ES

Determine whether each of the following functions
( , ) defines an inner product on M(2, 2)
...

(a) (A, B)

=

(b) (A,B)

=

det(AB)

a11b11+2a12b12+2a21b21 + a22b22

3

-2

Further Problems
Fl

(lsometries of JR
...
Suppose that

(a) A linear mapping is an isometry of JR
...
Prove that an

isometry preserves dot products and angles as
well as lengths
...
Let

v and

w

be vectors such that {u, v,

orthonormal

[u

v

basis

for

w]
...
(Hint: See
Problem 3
...
l
...
)

(c) Explain why an isometry of JR
...
Based on Problem 7
...
D3 (b), these
must be±1
...
3

1
021

and

012
A*

w}

let

is an

P

]

where the right-hand side is a partitioned ma­
trix, with OiJ being the i x

j zero matrix, and

with A• being a 2 x 2 orthogonal matrix
...


Note that an analogous form can be obtained
for pT AP in the case where one eigenvalue
is 1
-

(e) Use


...
F5

to

analyze

the

A*

of

part (d) and explain why every isometry of IR
...


IR
...
11 is called an involu­

F2 A linear mapping L :
tion if L o L
Id
...
Prove that any two of the

c

S2

c

·

·

·

c

Si

c

·

·

·

c

The i-th approximation is then projs;
the approximations improve as

V

v
...

(a) A is the matrix of an involution
...

(c) A is an isometry
...
Prove that (S + T)
...
l n T
...

+

=

F4 A problem of finding a sequence of approximations
to some vector (or function) v in a possibly infinite­
dimensional inner product space V can often be

llv

-

proj
...
,

vii

FS QR-factorization
...
Prove that A can be written as the
product of an orthogonal matrix Q and a upper­
triangular matrix R : A
QR
...
)

Note that this QR-factorization is important in a
numerical procedure for determining eigenvalues
of symmetric matrices

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