Untitled - Cdm.unimo.it

44 Polynomial Approximation of Differential Equations
In analogy with (3.2.4), we prove the following proposition.
Theorem 3.2.1 - For any n ≥ 1, we have
(3.2.8) ˜ l (n)
j (x) =
with x = η (n)
⎧
⎪⎨
⎪⎩
(−1) n (n − 1)! Γ(β + 2)
(n + α + β + 1) Γ(n + β + 1) (x − 1)u′ n(x) if j = 0,
(x 2 − 1)u ′ n(x)
n(n + α + β + 1) un(η (n)
j ) (x − η (n)
j )
if 1 ≤ j ≤ n − 1,
(n − 1)! Γ(α + 2)
(n + α + β + 1) Γ(n + α + 1) (x + 1)u′ n(x) if j = n,
j and un = P (α,β)
n . Besides, we have ˜l (n)
j (η (n)
j ) = 1, 0 ≤ j ≤ n.
Proof - If j = 0, then γ(x − 1)u ′ n, γ ∈ R, is a polynomial in Pn vanishing at
η (n)
k , 1 ≤ k ≤ n. After evaluating in x = −1, the constant γ has to be determined
so that −2γu ′ n(−1) = 1. Finally, the value u ′ n(−1) is found from (3.1.18). A similar
procedure applies when j = n (this time recalling (3.1.17)). If 1 ≤ j ≤ n − 1, for some
γ ∈ R we have
(3.2.9) lim
x→η (n)
j
γ (x2 − 1)u ′ n(x)
x − η (n)
j
= lim
x→η (n)
j
γ[(x 2 − 1)u ′′ n(x) + 2xu ′ n(x)]
= γ([η (n)
j ] 2 − 1)u ′′ n(η (n)
j ) = γn(n + α + β + 1)un(η (n)
j ).
The last equality in (3.2.9) is a consequence of (1.3.1). The required γ is then obtained
by equating the last expression in (3.2.9) to 1.
Recalling (1.5.1), (1.5.4) and (3.1.15), relation (3.2.8) becomes in the Chebyshev
case (see also gottlieb, hussaini and orszag(1984)):