18.1 Basic Properties and Formulae
217
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Cebyshev Polynomial of 1st kind
T0
T1
T2
T3
-1
-0.5
0
0.5
1
-4
-2
0
2
4
Cebyshev Polynomial of 2nd kind
U0
U1
U2
U3
-1
-0.5
0
0.5
1
-10
-5
0
5
Cebyshev Polynomial of 3rd kind
V0
V1
V2
V3
-1
-0.5
0
0.5
1
-2
0
2
4
6
8
Cebyshev Polynomial of 4th kind
W0
W1
W2
W3
Fig. 18.1 Visualization of the Chebyshev polynomials of degree 0 · · · 3. The plots were created
via chebyx(n,x,’T’).plot, grid on, shg with n = 3 and −1 ≤ x ≤ 1
The Chebyshev polynomials are orthogonal
+1
−1
C n (x)C m (c)w(x) = δ n,m n
2
(18.3a)
with respect to the following weight functions w(x) and norms n 2 :
for T n (x) : w(x) =
1
√
1 − x 2
and n
2
=
π : n = 0
π/2 : n > 0
(18.3b)
for U n (x) : w(x) =
1 − x 2 and n
2
=
1
2
π
(18.3c)
for V n (x) : w(x) =
√
1 + x
√
1 − x
and n
2
= π
(18.3d)
for W n (x) : w(x) =
√
1 − x
√
1 + x
and n
2
= π.
(18.3e)