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110 Polynomial Approximation of Differential Equations
The proof is completed by approximating f ∈ H k w(I) with a sequence of functions in
C∞ ( Ī) (see section 5.7).
The proof of the above statement was formerly given in maday and quarteroni(1981)
for Legendre and Chebyshev weights. The extension to other ultraspherical weights is
provided in bernardi and maday(1989). In those papers, one also finds estimates of
the same error in other norms. For instance, when −1 < α = β < 1 and k ≥ 1, there
exists a constant C > 0 such that, for any f ∈ H k w(I)
(6.4.10) f − Π 1 w,nf L 2 w (I) ≤ C
k 1
fH k
w n
(I), ∀n > k.
Suitable projections can be also defined in H 1 0,w(I) ⊂ H 1 w(I) (see definition (5.7.6)).
First of all we introduce the space
(6.4.11) P 0 n :=
p ∈ Pn
p(±1) = 0 , n ≥ 2.
Assuming that −1 < α < 1 and −1 < β < 1, we examine two projection operators.
For any n ≥ 2, we consider Π 1 0,w,n : H 1 0,w(I) → P 0 n and ˆ Π 1 0,w,n : H 1 0,w(I) → P 0 n such
that, for f ∈ H 1 0,w(I), we have
(6.4.12)
(6.4.13)
(f − Π
I
1 0,w,nf) ′ φ ′ w dx = 0, ∀φ ∈ P 0 n,
(f −
I
ˆ Π 1 0,w,nf) ′ (φw) ′ dx = 0, ∀φ ∈ P 0 n.
Existence and uniqueness for Π 1 0,w,nf are obtained by recalling definition (5.7.7) and
arguing as in theorem 6.2.1. Furthermore, it is easy to verify
(6.4.14) f − Π 1 0,w,nfH 1
0,w (I) = inf
ψ∈P0 f − ψH1 0,w (I).
n
Error estimates for Π 1 0,w,n can be inferred from estimates of other operators, as illus-
trated by the following proposition.