n
π
, k
= 1
,
2
, . . . , n
that are located in (

1
,
1);
(2)
at
z
i
= cos
iπ
n
, i
= 0
,
1
,
2
, . . . , n
the normalized Chebyshev polynomial
˜
T
n
(
x
) takes
its extremal values
±
1
2
n

1
alternatingly (
n
+ 1) times, i.e.,
˜
T
n
(
z
i
) =
(

1)
i
2
n

1
, i
= 0
,
1
, . . . , n.
(3)
The maximum value of
˜
T
n
(
x
) in [

1
,
1] is
1
2
n

1
, i.e.,
max
x
∈
[

1
,
1]

˜
T
n
(
x
)

=
1
2
n

1
.
27
Minimization of the approximation error of Lagrange polynomial interplant with
respect to the
max
distance in the interval
[

1
,
1]
.
If we use Lagrange interpolating
polynomials as approximants, then a natural question is to try to choose the interpolat
ing nodes such that the approximation error is minimal.
Suppose that we interpolate a
function
f
(
x
) by its
n
th Lagrange interpolating polynomial
P
n
(
x
) based on
n
+ 1 distinct
interpolating nodes
x
0
, x
1
, . . . , x
n
in [1,1]. Then the approximation (interpolating) error is
max
x
∈
[

1
,
1]

f
(
x
)

P
n
(
x
)
 ≤
M
n
+1
(
n
+ 1)!
max
x
∈
[

1
,
1]

(
x

x
0
)(
x

x
1
)
· · ·
(
x

x
n
)

,
where

f
(
n
+1
(
x
)
 ≤
M
n
+1
,
x
∈
[

1
,
1].
Now, our goal is to choose the interpolating nodes
x
*
0
, x
*
1
, . . . , x
*
n
such that
min
x
0
,x
1
,...,x
n
max
x
∈
[

1
,
1]

(
x

x
0
)(
x

x
1
)
· · ·
(
x

x
n
)

=
max
x
∈
[

1
,
1]

(
x

x
*
0
)(
x

x
*
1
)
· · ·
(
x

x
*
n
)

.
In other words, we wish to find (to determine) interpolating nodes that minimize
the approximation error and to use them in interpolating processes in order to
obtain Lagrangian interpolating approximants having a minimal approximation
error.
Theorem 5.1.
Given a function
f
(
x
)
defined on the interval
[

1
,
1]
. Consider
˜
T
n
+1
(
x
)
and
denote its
n
+ 1
zeros by
x
*
0
, x
*
1
, . . . , x
*
n
. Denote by
P
*
n
(
x
)
the
n
th Lagrange interpolating
polynomial to
f
(
x
)
by using as interpolating nodes the zeros of
˜
T
n
+1
(
x
)
:
x
*
0
, x
*
1
, . . . , x
*
n
.
Then,
P
*
n
(
x
)
minimizes the approximation error in the interval
[

1
,
1]
and
max
x
∈
[

1
,
1]

f
(
x
)

P
*
n
(
x
)
 ≤
M
n
+1
(
n
+ 1)!
max
x
∈
[

1
,
1]

(
x

x
*
0
)(
x

x
*
1
)
· · ·
(
x

x
*
n
)

=
M
n
+1
(
n
+ 1)!
max
x
∈
[

1
,
1]

˜
T
n
+1
(
x
)

=
M
n
+1
(
n
+ 1)!
1
2
n
or in a concise form
max
x
∈
[

1
,
1
]

f
(
x
)

P
*
n
(
x
)
 ≤
M
n
+
1
(
n
+
1
)!
1
2
n
,
where

f
(
n
+
1
)
(
x
)
 ≤
M
n
+
1
for all
x
∈
[

1
,
1
]
.
Remark. The zeros
x
*
0
, x
*
1
, . . . , x
*
n
of
T
n
+1
(
x
)
are called Chebyshev interpolating
nodes in the interval
[

1
,
1]
and the interpolation by using the Chebyshev in
terpolating nodes is called Chebyshev interpolation.
28
Proof of Theorem 5.1.
Suppose to the contrary, that for some other
n
+ 1 interpolating
nodes
x
0
, x
1
, . . . , x
n
the interpolating error is smaller, i.e.,
max
x
∈
[

1
,
1]

(
x

x
0
)(
x

x
1
)
· · ·
(
x

x
n
)

<
max
x
∈
[

1
,
1]

(
x

x
*
0
)(
x

x
*
1
)
· · ·
(
x

x
*
n
)

=
1
2
n
,
(1)
where
(
x

x
*
0
)(
x

x
*
1
)
· · ·
(
x

x
*
n
) =
˜
T
n
+1
(
x
) =
x
n
+
q
n
(
x
)
.
Obviously
Q
n
+1
(
x
) = (
x

x
0
)(
x

x
1
)
· · ·
(
x

x
n
) =
x
n
+1
+
r
n
(
x
)
˜
T
n
+1
(
x
) = (
x

x
*
0
)(
x

x
*
1
)
· · ·
(
x

x
*
n
) =
˜
T
n
+1
(
x
) =
x
n
+1
+
q
n
(
x
)
.
Then
(
x

x
*
0
)(
x

x
*
1
)
· · ·
(
x

x
*
n
)

(
x

x
0
)(
x

x
1
)
· · ·
(
x

x
n
)
=
˜
T
n
+1
(
x
)

Q
n
+1
(
x
) =
q
n
(
x
)

r
n
(
x
)
is a polynomial of degree
≤
n
and at the
n
+ 2 extremal points of
˜
T
n
+1
(
x
) it takes alternat
ingly positive and negative value. By the IVT
˜
T
n
+1
(
x
)

Q
n
+1
(
x
) has at least
n
+ 1 zeros
and being a polynomial of degree
≤
n
from here
˜
T
n
+1
(
x
)

Q
n
+1
(
x
)
≡
0 or equivalently
˜
T
n
+1
(
x
)
≡
Q
n
+1
(
x
) that is not possible because of (1).