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Results in Approximation Theory 117
We observe that the inequality (6.6.2) is satisfied for all the Jacobi weights. There-
fore, by (6.1.7), the rate of convergence can be estimated in the general case. For
instance, for any f ∈ Ck ( Ī), k ≥ 1, the following inequality holds:
(6.6.7) f − Iw,nf L 2 w (I) ≤ C
k 1
d
n
kf dxk
C 0 (Ī)
, ∀n ≥ k,
where w(x) = (1 − x) α (1 + x) β , α > −1, β > −1, x ∈ I. The same statement applies
for the error f − Ĩw,nf. We expect that (6.6.7) can be further refined by weakening the
assumptions on f and, with the same rate of convergence, replacing the norm on the
right-hand side of (6.6.7) with a weaker one (see also section 6.8).
By means of inverse inequalities, we also give a bound to the derivative of the error
as follows:
(6.6.8) (f − Iw,nf) ′ L 2 w (I) ≤ (f − Π 1 w,nf) ′ L 2 w (I) + (Π 1 w,nf − Iw,nf) ′ L 2 w (I)
≤ f − Π 1 w,nf H 1 w (I) + Cn 2 Π 1 w,nf − Iw,nf L 2 w (I),
where Π 1 w,n is defined in (6.4.3) and where we used the inequality (6.3.6) for the
polynomial p := Π 1 w,nf − Iw,nf ∈ Pn. Assuming a sufficient regularity for f, the last
terms in (6.6.8) tend to zero for n → +∞.
It is clear now that the aliasing errors Aw,nf and Ãw,nf, introduced in section
4.2, decay to zero for n → +∞, with a rate depending on the smoothness of f. From
(6.2.4) we know that
(6.6.9) f − Πw,n−1fL2 w (I) ≤ f − Iw,nfL2 w (I), ∀f ∈ C 0 ( Ī), ∀n ≥ 1.
We can be more precise about this estimate.
Theorem 6.6.2 - For any n ≥ 1 and any f ∈ C0 ( Ī), we have
(6.6.10) f − Iw,nf 2 L 2 w (I) = f − Πw,n−1f 2 L 2 w (I) + Aw,nf 2 L 2 w (I),
(6.6.11) f − Ĩw,nf 2 L 2 w (I) = f − Πw,nf 2 L 2 w (I) + Ãw,nf 2 L 2 w (I).