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116 Polynomial Approximation of Differential Equations
where we used the Schwarz inequality. This proves the statement. Concerning the error
f − Ĩw,nf, we apply the same arguments. This time, the quadrature formula is not
exact for polynomials in P2n. However, we obtain an inequality like (6.6.2), thanks to
theorem 3.8.2.
For the Chebyshev case (ν = −1/2), a sharper estimate of the error is readily available
(6.6.4) f − Iw,nf L 2 w (I) ≤ C
k 1
n
f H k w (I),
where k ≥ 1 and w(x) = 1/ √ 1 − x 2 , x ∈ I. The proof of this inequality relies on
the possibility of using results in approximation theory for trigonometric polynomials
(see jackson(1930) or zygmund(1988)), related to Chebyshev polynomials via (1.5.6).
The same procedure applies for the interpolation operator corresponding to Chebyshev
Gauss-Lobatto nodes.
To obtain convergence results for Iw,n when ν > 0, we recall the expression
(6.6.5) Iw,nf = [ Ĩv,n+1(fq)] q −1 , ∀n ∈ N,
where q(x) := (1 − x 2 ), x ∈ I, and v := q −1 w. Actually, both the terms in (6.6.5)
are polynomials in Pn−1, coinciding at the nodes ξ (n)
j , 1 ≤ j ≤ n, as a consequence of
relation (1.3.6). Therefore, one gets
(6.6.6) f − Iw,nf L 2 w (I) = (fq) − Ĩv,n+1(fq) L 2 v (I).
Thus, an estimate for 0 < ν ≤ 1, can be derived from theorem 6.6.1, by noting that v
is an ultraspherical weight function, whose exponent is negative.
Convergence results for Ĩw,n in the case −1 < ν < 1 are presented in bernardi and
maday(1989). Proofs are based on techniques borrowed from the theory of interpolation
of Sobolev spaces (see section 5.7).