Interpolation
*
Hector D. Ceniceros
1
Approximation Theory
Given
f
∈
C
[
a, b
], we would like to find a “good” approximation to it
by “simpler functions”, i.e.
functions in a given class (or family) Φ.
For
example,Φ =
P
n
=
{
all polynomials of degree
≤
n
}
.
A natural problem is that of finding the
best approximation
to
f
by func
tions in Φ. But how do we measure the accuracy of any approximation? that
is, what norm
1
do we use? We have several choices for norms of functions.
The most commonly used are:
1. The max or infinity norm:
k
f
k
∞
= sup
x
∈
[
a,b
]

f
(
x
)

.
2. The 2norm:
k
f
k
2
= (
R
b
a
f
2
(
x
)
dx
)
1
2
.
3. The pnorm:
k
f
k
p
= (
R
b
a
f
p
(
x
)
dx
)
1
p
.
Later, we will need to consider weighted norms: for some positive function
ω
(
x
) in [
a, b
] (it could be zero on a finite number of points) we define
k
f
k
ω,
2
=
Z
b
a
ω
(
x
)
f
2
(
x
)
dx
1
2
.
(1)
*
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1
A norm
k · k
is a real valued function on a vector space such that (1)
k
f
k
>
0
, f
6≡
0,
(2)
k
λf
k
=

λ
k
f
k
for any
λ
scalar, and (3)
k
f
+
g
k ≤ k
f
k
+
k
g
k
.
1
Then, by best approximation in Φ we mean a function
p
∈
Φ such that
k
f

p
k ≤ k
f

q
k
,
∀
q
∈
Φ
.
Computationally, it is often more efficient to seek not the best approximation
but one that is sufficiently accurate and fast converging to
f
. The central
building block for this approximation is the problem of interpolation.
2
Interpolation
Let us focus on the case of approximating a given function by a polynomial
of degree at most
n
. Then the
interpolation problem
can be stated as follows:
Given
n
+1 distinct points,
x
0
, x
1
, ..., x
n
called
nodes
and corresponding values
f
(
x
0
)
, f
(
x
1
)
, ..., f
(
x
n
), find a polynomial of degree at most
n
,
P
n
(
x
), which
satisfies (the interpolation property)
P
n
(
x
0
) =
f
(
x
0
)
P
n
(
x
1
) =
f
(
x
1
)
.
.
.
P
n
(
x
n
) =
f
(
x
n
)
.
Let us represent such polynomial as
P
n
(
x
) =
a
0
+
a
1
x
+
· · ·
+
a
n
x
n
. Then,
the interpolation property means
P
n
(
x
0
) =
f
(
x
0
)
, P
n
(
x
1
) =
f
(
x
1
)
,
· · ·
, P
n
(
x
n
) =
f
(
x
n
)
,
which implies
a
0
+
a
1
x
0
+
· · ·
+
a
n
x
n
0
=
f
(
x
0
)
a
0
+
a
1
x
1
+
· · ·
+
a
n
x
n
1
=
f
(
x
1
)
.
.
.
a
0
+
a
1
x
n
+
· · ·
+
a
n
x
n
n
=
f
(
x
n
)
.
This is a linear system of
n
+1 equations in
n
+1 unknowns (the polynomial
coefficients
a
0
, a
1
, . . . , a
n
). In matrix form:
1
x
0
x
2
0
· · ·
x
n
0
1
x
1
x
2
1
· · ·
x
n
1
.
.
.
1
x
n
x
2
n
· · ·
x
n
n
a
0
a
1
.
.
.
a
n
=
f
(
x
0
)
f
(
x
1
)
.
.
.
f
(
x
n
)
(2)
2
Does this linear system have a solution? Is this solution unique? The answer
is yes to both.
Here is a simple proof.
Take
f
≡
0, then
P
n
(
x
j
) = 0,for
j
= 0
,
1
, ..., n
but
P
n
is a polynomial of degree
≤
n
, it cannot have
n
+ 1
zeros unless
P
n
(
x
)
≡
0, which implies
a
0
=
a
1
=
· · ·
=
a
n
= 0.
That is,
the homogenous problem associated with (2) has only the trivial solution.