Chapter 2
Function Approximation
We saw in the introductory chapter that one key step in the construction of
a numerical method to approximate a definite integral is the approximation
of the integrand by a simpler function, which we can integrate exactly.
The problem of function approximation is central to many numerical
methods: given a continuous function
f
in an interval [
a, b
], we would like to
find a good approximation to it by simpler functions, such as polynomials,
trigonometric polynomials, wavelets, rational functions, etc.
We are going
to measure the accuracy of an approximation using norms and ask whether
or not there is a best approximation out of functions from a given family of
simpler functions. These are the main topics of this introductory chapter to
Approximation Theory.
2.1
Norms
A norm on a vector space
V
over a field
K
=
R
(or
C
) is a mapping
k
·
k
:
V
!
[0
,
1
)
,
which satisfy the following properties:
(i)
k
x
k ≥
0
8
x
2
V
and
k
x
k
= 0 i
↵
x
= 0
.
(ii)
k
x
+
y
k k
x
k
+
k
y
k 8
x, y
2
V
.
(iii)
k
λ
x
k
=

λ

k
x
k 8
x
2
V,
λ
2
K
.
17
18
CHAPTER 2.
FUNCTION APPROXIMATION
If we relax (i) to just
k
x
k ≥
0, we obtain a
seminorm
.
We recall first some of the most important examples of norms in the finite
dimensional case
V
=
R
n
(or
V
=
C
n
):
k
x
k
1
=

x
1

+
. . .
+

x
n

,
(2.1)
k
x
k
2
=
p

x
1

2
+
. . .
+

x
n

2
,
(2.2)
k
x
k
1
= max
{
x
1

, . . . ,

x
n
}
.
(2.3)
These are all special cases of the
l
p
norm:
k
x
k
p
= (

x
1

p
+
. . .
+

x
n

p
)
1
/p
,
1
p
1
.
(2.4)
If we have weights
w
i
>
0 for
i
= 1
, . . . , n
we can also define a weighted
p
norm by
k
x
k
w,p
= (
w
1

x
1

p
+
. . .
+
w
n

x
n

p
)
1
/p
,
1
p
1
.
(2.5)
All norms in a finite dimensional space
V
are equivalent, in the sense that
there are two constants
c
and
C
such that
k
x
k
↵
C
k
x
k
β
,
(2.6)
k
x
k
β
c
k
x
k
↵
,
(2.7)
for all
x
2
V
and for any two norms
k
·
k
↵
and
k
·
k
β
defined in
V
.
If
V
is a space of functions defined on a interval [
a, b
], for example
C
[
a, b
],
the corresponding norms to (2.1)(2.4) are given by
k
u
k
1
=
Z
b
a

u
(
x
)

dx,
(2.8)
k
u
k
2
=
✓Z
b
a

u
(
x
)

2
dx
◆
1
/
2
,
(2.9)
k
u
k
1
= sup
x
2
[
a,b
]

u
(
x
)

,
(2.10)
k
u
k
p
=
✓Z
b
a

u
(
x
)

p
dx
◆
1
/p
,
1
p
1
(2.11)
and are called the
L
1
,
L
2
,
L
1
, and
L
p
norms, respectively. Similarly to (2.5)
we can defined a weighted
L
p
norm by
k
u
k
p
=
✓Z
b
a
w
(
x
)

u
(
x
)

p
dx
◆
1
/p
,
1
p
1
,
(2.12)
2.2.
UNIFORM POLYNOMIAL APPROXIMATION
19
where
w
is a given positive weight function defined in [
a, b
]. If
w
(
x
)
≥
0, we
get a seminorm.
Lemma 1.
Let
k
·
k
be a norm on a vector space
V
then

k
x
k  k
y
k

k
x

y
k
.
(2.13)
This lemma implies that a norm is a continuous function (on
V
to
R
).
Proof.
k
x
k
=
k
x

y
+
y
k k
x

y
k
+
k
y
k
which gives that
k
x
k  k
y
k k
x

y
k
.
(2.14)
By reversing the roles of
x
and
y
we also get
k
y
k  k
x
k k
x

y
k
.
(2.15)
2.2
Uniform Polynomial Approximation
There is a fundamental result in approximation theory, which states that
any continuous function can be approximated uniformly, i.e. using the norm
k
·
k
1
, with arbitrary accuracy by a polynomial. This is the celebrated Weier
strass Approximation Theorem. We are going to present a constructive proof
due to Sergei Bernstein, which uses a class of polynomials that have found
widespread applications in computer graphics and animation. Historically,