Untitled - Cdm.unimo.it

114 Polynomial Approximation of Differential Equations
Proof - We first recall the equality
(6.5.10) Ĩw,nf = [Ia,n−1(fq −1 )] q + f(−1) ˜ l (n)
0 + f(1) ˜ l (n)
n , n ≥ 1,
where q(x) := (1 − x 2 ), x ∈ I, a = qw, and the Lagrange polynomials are defined
in (3.2.8). To check (6.5.10) it is sufficient to note that the relation holds at the points
η (n)
j , 0 ≤ j ≤ n. Actually, due to (1.3.6), we have P (α+1,β+1)
n−1 (η (n)
j ) = 0, 1 ≤ j ≤ n −1.
Hence, the η (n)
j ’s are the nodes of a Gauss formula where α and β are respectively
(n)
replaced by α + 1 and β + 1. Therefore ( Ĩw,ng)(η j ) = (Ia,n−1g)(η (n)
j ) = g(η (n)
j ),
1 ≤ j ≤ n − 1, for any function g. The verification of (6.5.10) at the points x = ±1 is
trivial. Multiplying both sides of (6.5.10) by w and integrating in I, one gets
(6.5.11) lim
n→+∞
[Ia,n−1(fq
I
−1 )] qw dx =
(6.5.12) lim
n→+∞
I
˜(n) l j w dx = lim
n→+∞ ˜w(n) j
I
fq −1 a dx =
I
fw dx,
= 0, j = 0 or j = n.
The first limit is a consequence of theorem 6.5.1, and (6.5.12) is obtained by studying
the behavior for n → +∞ of the weights in (3.5.2).
A statement like that of theorem 6.5.4 also exists for Gauss-Lobatto and Gauss-
Radau formulas. We mention davis and rabinowitz(1984) for the treatment of the
Legendre case and ghizzetti and ossicini(1970) for a more general discussion. Esti-
mates in the Legendre case are also analyzed in ditzian and totik(1987), p.87.
In the case of Laguerre Gauss-Radau formulas (see (3.6.1)), we obtain a proposition like
theorem 6.5.2. This time the proof is based on the following relation:
(6.5.13) Ĩw,nf = [Ia,n−1(fq −1 )] q + f(0) ˜l (n)
0 , n ≥ 1,
where q(x) = x, x ∈]0,+∞[, a = qw, and ˜ l (n)
0 is given in (3.2.11).