Untitled - Cdm.unimo.it

Results in Approximation Theory 111
Theorem 6.4.3 - There exists a constant C > 0 such that, for any f ∈ H 1 0,w(I), we
have
(6.4.15) f − Π 1 0,w,nf H 1 0,w (I) ≤ C f − Π 1 w,nf H 1 w (I), ∀n ≥ 2.
Proof - We define
(6.4.16) ψ(x) := (Π 1 w,nf)(x) + 1
2
(1 + x)(f − Π 1 w,nf)(1)
+ (1 − x)(f − Π 1
w,nf)(−1) , x ∈ I.
Recalling that f(±1) = 0, we have ψ ∈ P 0 n. Hence, substituting in (6.4.14), we get
(6.4.17) f − Π 1 0,w,nf H 1 0,w (I) ≤ f − Π 1 w,nf H 1 0,w (I)
+ 1
2
d
dx
(1 + x)(f − Π 1 w,nf)(1) + (1 − x)(f − Π 1
L
w,nf)(−1)
2
w (I)
≤ f − Π 1 w,nf H 1 w (I) + C ∗ f − Π 1 w,nf C 0 (Ī) ,
where C ∗ > 0 is a constant. Now, recalling (5.7.4), we immediately conclude the proof.
The analysis of the operator ˆ Π 1 0,w,n is more involved and will be considered in
section 9.4. Of course, when w ≡ 1 is the Legendre weight function, the relations
(6.4.12) and (6.4.13) are identical.
6.5 Convergence of the Gaussian formulas
High order integration formulas were introduced in sections 3.4, 3.5 and 3.6. Here we
consider their asymptotic properties for increasing number of nodes. In the case of
Jacobi weights, we begin with the corollary of a convergence result due to Stekloff and
Fejér (see szegö(1939), p.342).