. In fact it diverged quite dramatically toward the end points of the
interval. On the other hand, we noticed fast and uniform convergence of
p
n
to
f
when the nodes were given by
x
j
= cos
✓
j
⇡
n
◆
,
j
= 0
, . . . , n.
(3.48)
44
CHAPTER 3.
INTERPOLATION
These points are called
Chebyshev
nodes or
GaussLobatto
nodes and are
the preferred choice in practice when high order, very accurate polynomial
interpolation is needed.
It is then natural to ask: Given any
f
2
C
[
a, b
], can we guarantee that if
we choose the Chebyshev nodes
k
f

p
n
k
1
!
0? The answer is no. Bernstein
and Faber proved in 1914 that given any distribution of points, organized in
a triangular array as
x
(0)
0
x
(1)
0
,
x
(1)
1
x
(2)
0
,
x
(2)
1
,
x
(2)
2
.
.
.
(3.49)
it is possible to construct a continuous function
f
for which its interpolating
polynomial
p
n
(corresponding to the nodes on the
n
th row of (3.49)) will not
converge uniformly to
f
as
n
! 1
. However, if
f
is slightly smoother, for
example
f
2
C
1
[
a, b
], then for the Chebyshev array of nodes
k
f

p
n
k
1
!
0.
In one of the homework examples
f
(
x
) =
e

x
2
and we noticed convergence
of
p
n
even with the equidistributed nodes.
What is so special about this
function? The function
f
(
z
) =
e

z
2
,
(3.50)
z
=
x
+
iy
is analytic in the entire complex plane. Using complex variables
analysis it can be shown that if
f
is analytic in a su
ffi
ciently large region in
the complex plane containing [
a, b
] then
k
f

p
n
k
1
!
0. Just how large the
region of analyticity needs to be? it depends on the asymptotic distribution
of the nodes as
n
! 1
.
In the limit as
n
! 1
, we can think of the nodes as a continuum with a
density
⇢
so that
(
n
+ 1)
Z
x
a
⇢
(
t
)
dt
(3.51)
is the total number of nodes in [
a, x
].
Take for example, [

1
,
1].
For
equidistributed nodes
⇢
(
x
) = 1
/
2 and for the Chebyshev nodes
⇢
(
x
) =
1
/
(
⇡
p
1

x
2
).
It turns out that the relevant domain of analyticity is given in terms of
the function
φ
(
z
) =

Z
b
a
⇢
(
t
) ln

z

t

dt.
(3.52)
3.8.
PIECEWISE LINEAR INTERPOLATION
45
Let
Γ
c
be the level curve consisting of all the points
z
2
C
such that
φ
(
z
) =
c
for
c
constant. For very large and negative
c
,
Γ
c
approximates a large circle.
As
c
is increased,
Γ
c
shrinks. We take the “smallest” level curve,
Γ
c
0
, which
contains [
a, b
]. The relevant domain of analyticity is
R
c
0
=
{
z
2
C
:
φ
(
z
)
≥
c
0
}
.
(3.53)
Then, if
f
is analytic in
R
c
0
,
k
f

p
n
k
1
!
0, not only in [
a, b
] but for at
point in
R
c
0
. Moreover,

f
(
x
)

p
n
(
x
)

Ce

N
(
φ
(
x
)

c
0
)
,
(3.54)
for some constant
C
. That is
p
n
converges exponentially fast to
f
. For
the Chebyshev nodes
R
c
0
approximates [
a, b
], so if
f
is analytic in any region
containing [
a, b
], however thin this region might be,
p
n
will converge uniformly
to
f
. For equidistributed nodes,
R
c
0
looks like a football, with [
a, b
] as its
longest axis.
In the Runge example, the function is singular at
z
=
±
i/
5
which happens to be inside this footballlike domain and this explain the
observed lack of convergence for this particular function.