Untitled - Cdm.unimo.it

58 Polynomial Approximation of Differential Equations
(3.8.9) p 2 w,n ≤ un 2 w,n
un 2 w
p 2 w, ∀p ∈ Pn.
Again, from a direct computation, we have that the ratio on the right-hand side of
(3.8.9) is bounded by max {2,3 + 2α} = γ −2
1 .
It is possible to compute the norm pw of a polynomial p ∈ Pn as a function of
the values it attains at the points η (n)
j , 0 ≤ j ≤ n. This is shown by the following
proposition.
Theorem 3.8.3 - For any n ≥ 1, we have
(3.8.10) p 2 w = p 2 w,n − un 2 w,n − un 2 w
un 4 w,n
where un = P (α,β)
n , α > −1, β > −1.
Proof - By (2.3.7) and the exactness of (3.5.1) in P2n−1, one has
[(p,un)w,n] 2 , ∀p ∈ Pn,
(3.8.11) cn un 2 w = (p,un)w = (p,un)w,n + cn(un 2 w − un 2 w,n), ∀p ∈ Pn.
Recovering cn, one gets the interesting property
(3.8.12) cn = (p,un)w,n
un2 , ∀p ∈ Pn.
w,n
The proof is concluded after substitution of cn in (3.8.7).
The right-hand side of (3.8.10) actually depends only on the values of p at the
nodes. Using (2.2.10) and (3.8.3), equation (3.8.10) becomes
(3.8.13) pw =
p 2 w,n −
n(n + 1)
2(2n + 1)
[(p,Pn)w,n] 2
1
2
, ∀p ∈ Pn,