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Results in Approximation Theory 115
6.6 Estimates for the interpolation operator
The interpolation operators Iw,n and Ĩw,n have been introduced in section 3.3. In
this section, we are concerned with estimating the rate of convergence to zero of the
error between a function and its interpolant. We first study the case of ultraspherical
polynomials (ν := α = β).
Theorem 6.6.1 - Let k ≥ 1 and −1 < ν ≤ 0. Then we can find a constant C > 0
such that, for any f ∈ H k w(I), we have
(6.6.1) f − Iw,nf L 2 w (I) ≤ C
The same relation holds for the error f − Ĩw,nf.
k−1/2 1
(1
2 −ν/2
− x )
n
dkf dxk
Proof - We follow the proof of theorem 6.5.1. One has
(6.6.2) f − Iw,nf L 2 w (I) ≤ f − Ψ∞,n−1(f) L 2 w (I)
+
⎛
⎝
n
(Ψ∞,n−1(f) − Iw,nf)
j=1
2 (ξ (n)
j ) w (n)
j
⎞
⎠
1
2
L 2 w (I)
≤ 2 f −Ψ∞,n−1(f) C 0 (Ī)
Thus, recalling (6.1.7), we can find a constant C ∗ > 0 such that
(6.6.3) f − Iw,nf L 2 w (I) ≤ C ∗
≤ C ∗
= C ∗
k−1 1
n
k−1 1
n
sup
|x1−x2|