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Results in Approximation Theory 113
The rate of convergence in (6.5.1) is measured by the degree of smoothness of the
function f. If f fulfills certain regularity requirements, we can analytically express the
error between the integral and the quadrature. The proof of the following theorem does
not present difficulties and is given for instance in todd(1963).
Theorem 6.5.4 - Let n ≥ 1 and f ∈ C2n ( Ī). Let us define
(6.5.3)
En(f) := fw dx −
n
I
j=1
f(ξ (n)
j ) w (n)
j .
Then we can find ξ ∈ I such that
2
(6.5.4) (Jacobi) En(f) =
2n+α+β+1 n! Γ(n + α + 1) Γ(n + β + 1)
(2n)! Γ(2n + α + β + 1)
× Γ(n + α + β + 1)
Γ(2n + α + β + 2)
(6.5.5) (Legendre) En(f) =
(6.5.6) (Chebyshev) En(f) =
(6.5.7) (Laguerre) En(f) =
d2nf (ξ), α > −1, β > −1,
dx2n 2 2n+1 [n!] 4
(2n + 1) [(2n)!] 3
π
2 2n−1 (2n)!
n! Γ(n + α + 1)
(2n)!
(6.5.8) (Hermite) En(f) = n! √ π
2 n (2n)!
d2nf (ξ),
dx2n d2nf (ξ),
dx2n d2nf (ξ), α > −1,
dx2n d2nf (ξ).
dx2n Concerning Gauss-Lobatto formulas, we can recover a convergence result for f ∈
C0 ( Ī), I =] − 1,1[, with the same proof given for theorem 6.5.1. A more general
theorem, based on weaker assumptions on f, also holds in the Jacobi case:
Theorem 6.5.5 - Let f : Ī → R, such that fw is Riemann integrable. Then we have
(6.5.9)
n
lim f(η
n→+∞
(n)
j ) ˜w (n)
j
= lim
n→+∞
Ĩw,nf w dx = fw dx.
j=0
I
I