Untitled - Cdm.unimo.it

Numerical Integration 57
the first part. Substituting x = η (n)
j , 1 ≤ j ≤ n − 1, the terms containing u ′ n vanish.
When (3.8.4) is achieved, the proof is straightforward with the help of (3.5.2), (3.1.17)
and (3.1.18).
In particular, when α = −1 2 , (3.8.4) yields
(3.8.5) Tn 2 w,n = π, ∀n ≥ 1.
For any fixed n ≥ 1, the two norms · w and · w,n defined in Pn are equivalent.
This means that it is possible to find two constants γ1 > 0,γ2 > 0, such that
(3.8.6) γ1 pw,n ≤ pw ≤ γ2 pw,n, ∀p ∈ Pn.
This is not surprising, since in a finite dimensional space (such as Pn) any two norms
are always equivalent. More important is the following statement, which we prove in
the ultraspherical case.
Theorem 3.8.2 - The constants γ1 and γ2 in (3.8.6) do not depend on n.
Proof - Expanding p with respect to the orthogonal basis {uk}0≤k≤n, we can write
p = cnun + r, where r ∈ Pn−1. Using that (3.5.1) is true in P2n−1, one obtains
(3.8.7) p 2 w,n = c 2 n un 2 w,n + 2cn (un,r)w,n + r 2 w,n
= c 2 n un 2 w,n + 2cn(un,r)w + r 2 w = c 2 n
un 2 w,n − un 2 2
w + pw, ∀p ∈ Pn.
By direct comparison of (2.2.10) and (3.8.3) we get un 2 w,n ≥ un 2 w, ∀n ≥ 1. This
implies γ2 = 1. On the other hand by (2.3.7) and by the Schwarz inequality (2.1.7), we
get
(3.8.8) c 2 n =
Thus, substituting in (3.8.7),
(p,un)w
un 2 w
2
≤ p2 w
un2 , ∀p ∈ Pn.
w