Untitled - Cdm.unimo.it

176 Polynomial Approximation of Differential Equations
In figure 8.6.1, for the Legendre case, in the interval [0,600], we show the exact eigen-
values λm, 1 ≤ m ≤ 15 (first line) and the computed eigenvalues λn,m, 1 ≤ m ≤ n−1
(next lines), when n varies from 6 to 11.
Figure 8.6.1 - Behavior of the eigenvalues of problem
(8.2.1) in the Legendre case when 6 ≤ n ≤ 11.
In order to check (8.3.6), we first need the following result. We recall that the
polynomial space P 0 n, n ≥ 2, is defined in (6.4.1).
Lemma 8.6.1 - Let −1 < α < 1 and −1 < β < 1. Then we can find two constants
C1 > 0, C2 > 0, such that, for any n ≥ 2 and p ∈ P 0 n, one has
(8.6.3)
p
1 − x2
w
≤ C1 p ′ w,