nodes are not equally spaced and tend to cluster toward the end points of
the interval.
3.2
Connection to Best Uniform Approxima
tion
Given a continuous function
f
in [
a, b
], its best uniform approximation
p
*
n
in
P
n
is characterized by an error,
e
n
=
f

p
*
n
, which equioscillates, as defined
in (2.40), at least
n
+ 2 times. Therefore
e
n
has a minimum of
n
+ 1 zeros
and consequently, there exists
x
0
, . . . , x
n
such that
p
*
n
(
x
0
) =
f
(
x
0
)
,
p
*
n
(
x
1
) =
f
(
x
1
)
,
.
.
.
p
*
n
(
x
n
) =
f
(
x
n
)
,
(3.15)
In other words,
p
*
n
is the polynomial of degree at most
n
that interpolates
the function
f
at
n
+ 1 zeros of
e
n
. Of course, we do not construct
p
*
n
by
finding these particular
n
+1 interpolation nodes. A more practical question
is: given (
x
0
, f
(
x
0
))
,
(
x
1
, f
(
x
1
))
, . . . ,
(
x
n
, f
(
x
n
)), where
x
0
, . . . , x
n
are distinct
interpolation nodes in [
a, b
], how close is
p
n
, the interpolating polynomial of
degree at most
n
of
f
at the given nodes, to the best uniform approximation
p
*
n
of
f
in
P
n
?
To obtain a bound for
k
p
n

p
*
n
k
∞
we note that
p
n

p
*
n
is a polynomial of
degree at most
n
which interpolates
f

p
*
n
. Therefore, we can use Lagrange
formula to represent it
p
n
(
x
)

p
*
n
(
x
) =
n
X
j
=0
l
j
(
x
)(
f
(
x
j
)

p
*
n
(
x
j
))
.
(3.16)
It then follows that
k
p
n

p
*
n
k
∞
≤
Λ
n
k
f

p
*
n
k
∞
,
(3.17)
42
CHAPTER 3.
INTERPOLATION
where
Λ
n
= max
a
≤
x
≤
b
n
X
j
=0

l
j
(
x
)

(3.18)
is called the
Lebesgue Constant
and depends only on the interpolation nodes,
not on
f
. On the other hand, we have that
k
f

p
n
k
∞
=
k
f

p
*
n

p
n
+
p
*
n
k
∞
≤ k
f

p
*
n
k
∞
+
k
p
n

p
*
n
k
∞
.
(3.19)
Using (3.17) we obtain
k
f

p
n
k
∞
≤
(1 + Λ
n
)
k
f

p
*
n
k
∞
.
(3.20)
This inequality connects the interpolation error
k
f

p
n
k
∞
with the best
approximation error
k
f

p
*
n
k
∞
. What happens to these errors as we increase
n
? To make it more concrete, suppose we have a triangular array of nodes
as follows:
x
(0)
0
x
(1)
0
x
(1)
1
x
(2)
0
x
(2)
1
x
(2)
2
.
.
.
x
(
n
)
0
x
(
n
)
1
. . .
x
(
n
)
n
.
.
.
(3.21)
where
a
≤
x
(
n
)
0
< x
(
n
)
1
<
· · ·
< x
(
n
)
n
≤
b
for
n
= 0
,
1
, . . .
.
Let
p
n
be the
interpolating polynomial of degree at most
n
of
f
at the nodes corresponding
to the
n
+ 1 row of (3.21).
By the Weierstrass Approximation Theorem (
p
*
n
is a better approxima
tion or at least as good as that provided by the Bernstein polynomial),
k
f

p
*
n
k
∞
→
0
as
n
→ ∞
.
(3.22)
However, it can be proved that
Λ
n
>
2
π
2
log
n

1
(3.23)
3.3.
BARYCENTRIC FORMULA
43
and hence the Lebesgue constant is not bounded in
n
. Therefore, we cannot
conclude from (3.20) and (3.22) that
k
f

p
n
k
∞
as
n
→ ∞
, i.e.
that the
interpolating polynomial, as we add more and more nodes, converges uni
formly to
f
. That depends on the regularity of
f
and on the distribution of
the nodes. In fact, if we are given the triangular array of interpolation nodes
(3.21) in advance, it is possible to construct a continuous function
f
such
that
p
n
will not converge uniformly to
f
as
n
→ ∞
.
3.3
Barycentric Formula
The Lagrange form of the interpolating polynomial is not convenient for com
putations.
If we want to increase the degree of the polynomial we cannot
reuse the work done in getting and evaluating a lower degree one.
How
ever, we can obtain a very efficient formula by rewriting the interpolating
polynomial in the following way. Let