Untitled - Cdm.unimo.it

52 Polynomial Approximation of Differential Equations
Theorem 3.5.2 - For any n ≥ 2, we have
(3.5.2)
˜w (n)
j =
⎧
⎪⎨
⎪⎩
2 α+β (β+1) Γ 2 (β+1)(2n+α+β)(n−2)! Γ(n+α)
Γ(n+α+β+2) Γ(n+β+1)
−2α+β
(2n+α+β)Γ(n+α)Γ(n+β)
(n+α+β+1) n! Γ(n+α+β)
P (α,β)
n
2 α+β (α+1) Γ 2 (α+1)(2n+α+β)(n−2)! Γ(n+β)
Γ(n+α+β+2) Γ(n+α+1)
n−1
(1 − η (n)
m ) if j = 0,
m=1
d (α,β)
dxP n−1
(η (n)
−1 j )
if 1 ≤ j ≤ n − 1,
n−1
(1 + η (n)
m ) if j = n.
m=1
Proof - As in section 3.4 for the Gauss weights, we start with the relation
(3.5.3)
u ′ n(x)
x − η (n)
j
= (2n + α + β)(2n + α + β − 1)
2(n − 1)(n + α + β)
u ′ n−1(x) + q ′ j(x), 1 ≤ j ≤ n − 1,
where qj ∈ Pn−2 and un = P (α,β)
n . To check (3.5.3), it is sufficient to compare the highest
degree terms and recall (1.3.5). Thus, if 1 ≤ j ≤ n − 1, by (3.2.8) and (2.2.15) we get
(3.5.4)
˜w (n)
j
=
= −
1
u ′ n−1 (η(n)
j )
=
n
i=0
( ˜l (n)
j u′ n−1)(η (n)
i ) ˜w (n)
i
−1
n(n + α + β + 1) (u ′
(n)
n−1un)(η j )
=
1
−1
1
u ′ n−1 (η(n)
j )
a
u ′ n
x − η (n)
j
1
−1
˜ l (n)
j u′ n−1 wdx
u ′ n−1 dx
(2n + α + β)(2n + α + β − 1)
2n(n − 1)(n + α + β)(n + α + β + 1) (u ′ u
(n)
n−1un)(η j )
′ n−1 2 a
= −
(2n + α + β)(2n + α + β − 1)
2n(n + α + β + 1) (u ′ (n)
n−1un)(η j )
un−1 2 w.