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10 Polynomial Approximation of Differential Equations
In general, we obtain Tn(±1) = (±1) n , n ∈ N. Evaluating at the points x = ±1,
(1.5.2) yields
(1.5.4) T ′ n(±1) = ±(±1) n n 2 , n ∈ N.
Moreover, Tn is even (odd) if and only if n is even (odd). An explicit expression for the
Chebyshev polynomial of degree n is
(1.5.5) Tn(x) =
[n/2]
k=0
⎛
⎝(−1) k
[n/2]
m=k
n m
2m k
⎞
⎠ x n−2k
= 2 n−1 x n − n2 n−3 x n−2 + 1
2 n(n − 3) 2n−5 x n−4 + · · · , x ∈
In (1.5.5), [•] denotes the integer part of •.
The most remarkable characterization is given by the simple relation
(1.5.6) Tn (cos θ) = cos nθ, θ ∈ [0, π], n ∈ N.
Ī, n ∈ N.
The above expression relates algebraic and trigonometric polynomials. Such an impor-
tant peculiarity can be proven by noting that
(1.5.7)
(1.5.8)
dTn
dx
(cos θ) = n sinnθ
sin θ ,
d2 Tn n sinnθ cos θ
(cos θ) =
dx2 sin 3 θ
− n2 cos nθ
sin 3 ,
θ
where x = cos θ , θ ∈ [0,π], and n ∈ N. Thus, (1.5.2) is satisfied by replacing x by
cos θ.
A lot of properties are direct consequence of (1.5.6). In particular we have |Tn(x)| ≤ 1,
∀n ∈ N, ∀x ∈ Ī. Moreover, Tn vanishes n times in Ī, therefore the derivative T ′ n
vanishes n − 1 times in Ī. The function |Tn| attains the maximum value of 1, n + 1
times in Ī. We give the plot in Figure 1.5.1 of the Tn’s for 1 ≤ n ≤ 6. In addition, T11
is plotted in Figure 1.5.2.