Untitled - Cdm.unimo.it

106 Polynomial Approximation of Differential Equations
(6.3.6) p ′ L 2 w (I) ≤ Cn 2 p L 2 w (I), ∀p ∈ Pn.
Proof - We set p = n
k=0 ckuk, where {uk}k∈N is the sequence of Jacobi polynomials
relative to the weight function w. From relations (2.2.15) and (2.2.8), we get
(6.3.7) p ′ 2 L2 a (I) =
=
n
k=1
c 2 kλk uk 2 L2 w (I) ≤ λn
n
k=1
n
k=0
c 2 k u ′ k 2 L 2 a (I)
c 2 k uk 2 L 2 w (I) ≤ C2 n 2 p 2 L 2 w (I),
where we observed that the eigenvalue λn behaves like n 2 . To obtain (6.3.6), we first
apply lemma 6.3.1 to the polynomial p ′ and then use (6.3.5).
As a byproduct of the previous theorem, it is not difficult to get, for any n ≥ 1
(6.3.8) p H m w (I) ≤ Cn 2(m−k) p H k w (I), ∀p ∈ Pn, ∀m,k ∈ N, m ≥ k.
When w is the Legendre weight function, relation (6.3.8) is the Schmidt inequality (see
bellman(1944)). Using different arguments, (6.3.8) was established in canuto and
quarteroni (1982a) for Chebyshev weights.
We study now the Laguerre case (a(x) = xw(x) = x α+1 e −x , α > −1, x ∈ I =]0,+∞[).
Lemma 6.3.3 - We can find two constants C1,C2 > 0 such that, for any n ≥ 1
(6.3.9)
C1
√
√ pL2 a (I) ≤ pL2 w (I) ≤ C2 n pL2 a (I), ∀p ∈ Pn.
n
Proof - As done in the proof of lemma 6.3.1, let ξ (m)
j , 1 ≤ j ≤ m, be the zeroes of
L (α)
m , m = n + 1. Then one has the inequality C ∗ 1 1
n
≤ ξ(m)
j
≤ C∗ 2n, 1 ≤ j ≤ m (see
section 3.1), where C ∗ 1,C ∗ 2 > 0 do not depend on n. Now we can easily deduce (6.3.9).