Graphics Reference
In-Depth Information
Definition.
The orthogonal polynomials T
i
(x) defined by formulas (F.6) are called
the
Chebyshev polynomials
.
The first few Chebyshev polynomials are
2
3
4
2
121438 81
,
xx
,
-
,
x
-
xx
,
-
x
+
, ....
One can show that these polynomials give especially good approximations. Particu-
larly relevant for minimizing the error function E(x) in (F.3) is the following fact:
F.2.4. Theorem.
Of all nth degree polynomials with leading coefficient 1, the poly-
nomials 2
1-n
T
n
(x) have the smallest absolute value 2
1-n
on [-1,1].
Proof.
See [ConD72]. Note that by formula (F.6a) the polynomials T
i
(x) are bounded
by 1 on [-1,1] since the cosine function is.
Here is how one would use Chebyshev polynomials to get an efficient polynomial
approximation to a function f(x) on [-1,1] that is within some e of the function.
Step 1:
Begin with the straightforward approach which is to use the Taylor
polynomial
()
=
2
n
px
a
+
a x
+
a x
+ ◊◊◊+
a x
n
0
1
2
for f(x) of smallest degree n so that the remainder is less than e
n
for an e
n
<e. See
Theorem D.2.4.
Step 2:
Since the monomials x
i
are linear combinations of the Chebyshev
polynomials T
i
(x), replace them by the T
i
(x) and express p(x) in the form
()
=+
()
+
()
+ ◊◊◊+
()
px
b
b T x
b T x
b T x
.
0
1
1
2
2
nn
Step 3:
Choose k to be the smallest integer so that
e
+
b
+ ◊◊◊+
b
£
e
.
n
k
+1
n
Then
()
=+
()
+
()
+ ◊◊◊+
()
qx
b
bT x
bT x
b T x
0
1
1
2
2
kk
will approximate to within e.
It turns out that this use of Chebyshev polynomials will often produce approxima-
tions of degree k of much smaller degree than the degree n of the Taylor polynomial.
We return to the problem of this section. Using the trapezoidal and Simpson's rule
as an approximation to the integral I in (F.1) we will get the correct answer if f(x) is
a linear or quadratic polynomial respectively. This fact is easy to establish by looking
