Untitled - Cdm.unimo.it

118 Polynomial Approximation of Differential Equations
Proof - Expand all the squared norms in (6.6.10) (or (6.6.11)) in terms of the inner
product in L 2 w(I) and recall (6.2.5).
Another characterization of the aliasing error is given by (see (3.3.1) and (6.2.11))
∞
∞
=
(6.6.12) Aw,nf = Iw,n(f − Πw,n−1f) = Iw,n
k=n
ckuk
k=n+1
ck(Iw,nuk),
where ck, k ∈ N, are the Fourier coefficients of f and uk = P (α,β)
k , k ∈ N. The
last equality in (6.6.12) follows by observing that, for any n ≥ 1, Iw,n is a continuous
operator in C 0 ( Ī) (see section 6.8). A similar relation holds for Ãw,nf. In the Chebyshev
case, the formula (6.6.12) takes a simplified form, thanks to the periodicity of the cosine
function, i.e.
(6.6.13) Aw,nf =
n−1
∞
m=1 j=1
(c2jn+m − c2jn−m)Tm.
Moreover, we note that Aw,nTjn ≡ 0, ∀j ≥ 1. A discussion of the effects of the aliasing
error in the computation of solutions of boundary value problems are given in canuto,
hussaini, quarteroni and zang(1988).
We analyse now the case when I is not bounded. Let us start with the Laguerre
approximations (w(x) = x α e −x , α > −1, x ∈ I =]0,+∞[).
Theorem 6.6.3 - Let δ ∈]0,1[ and let f ∈ C 0 ( Ī) satisfy limx→+∞ f(x)e −δx = 0.
Then we have
(6.6.14) lim
n→+∞ f − Iw,nf L 2 w (I) = 0.
The same relation holds for the error f − Ĩw,nf, relative to the Gauss-Radau interpo-
lation operator.
Proof - For any pn ∈ Pn−1, we have (compare with (6.6.2))
(6.6.15) f − Iw,nf L 2 w (I) ≤ f − pn L 2 w (I) +
⎛
⎝
n
(pn − f)
j=1
2 (ξ (n)
j ) w (n)
j
⎞
⎠
1
2
≤