Untitled - Cdm.unimo.it

8 Polynomial Approximation of Differential Equations
Furthermore, (1.3.9) leads to
(1.4.4) |Pn(x)| ≤ 1 , |x| ≤ 1, n ∈ N.
Combining (1.3.6) and (1.3.9), we can also estimate the derivative according to
(1.4.5) |P ′ n(x)| ≤ 1
2n(n + 1) , |x| ≤ 1 , n ∈ N.
Another useful relation can be easily established by taking x = 0 in (1.4.2), i.e.
⎧
⎨0
if n is odd,
(1.4.6) Pn(0) =
⎩
n! 2−n
n −2
2 ! if n is even.
Therefore, one deduces that lim
n→+ ∞ Pn(0) = 0.
In Figure 1.4.1 the polynomials Pn, 1 ≤ n ≤ 6, are plotted, while Figure 1.4.2
shows the behavior of P11.
In order to give some general ideas of the behavior of Legendre polynomials, we
state two other results (see szegö(1939) and jackson(1930) respectively).
Theorem 1.4.1. - For any n ≥ 5, when x increases from 0 to 1, the successive relative
maximum values of |Pn(x)| also increase.
Theorem 1.4.2. - For any n ∈ N and any x ∈ Ī, one has
(1.4.7) Pn(x) = 1
π
π
0
x + i 1 − x2 n cos θ
Though the right hand side in (1.4.7) contains the imaginary unity (i.e., i = √ −1), the
global integral is real. Taking the absolute value on both sides of (1.4.7), we can prove
the estimate:
dθ.