Untitled - Cdm.unimo.it

Numerical Integration 61
(3.9.5) p∞ ≤ γ |||p|||w,n, ∀p ∈ Pn−1.
This time, a statement like that of theorem 3.8.2 cannot be proven. In fact, γ ≡ γ(n)
actually grows with n. Following erdös(1961), it is possible to find a constant δ > 0
such that
(3.9.6) γ(n) > 2
π
log(n) − δ.
However, using Chebyshev nodes, we get the relation (see natanson(1965))
(3.9.7) γ(n) < 2
π
log(n) + 1.
Further inequalities of this type will be covered in section 6.3.
Within the context of the never ending properties of Chebyshev polynomials, we
state a result which generalizes theorem 2.5.1. The proof is in duffin and schaef-
fer(1941).
Theorem 3.9.1 - Let w(x) = 1/ √ 1 − x 2 , then for any n ≥ 1, we have
(3.9.8) p ′ ∞ ≤ n 2 |||p||| ∗ w,n, ∀p ∈ Pn.
Another theorem, closely related to theorem 2.5.2, is stated in rivlin(1974), p.104.
Theorem 3.9.2 - Let w(x) = 1/ √ 1 − x 2 , then for any n ≥ 1, we have
(3.9.9) |p ′ (ξ (n)
j )| ≤
n
1 − [ξ (n)
j ] 2
|||p||| ∗ w,n, 1 ≤ j ≤ n, ∀p ∈ Pn.