Untitled - Cdm.unimo.it

60 Polynomial Approximation of Differential Equations
A discrete inner product with its relative norm, can also be conveniently defined
for polynomials in Pn−1 distinguished by their values at the n nodes including the n−1
d zeroes of dxL(α) n
and the point x = 0. However, this definition would be redundant.
Indeed, when r,s ∈ Pn−1 we have p := rs ∈ P2n−2; therefore, by virtue of (3.6.1),
the new inner product coincides with the usual one, i.e. · w, where w is the Laguerre
weight function.
3.9 Discrete maximum norms
Discrete equivalents of the maximum norm, introduced in section 2.5, can be defined.
We study two cases. For any n ≥ 1, we first consider ||| · |||w,n : Pn−1 → R + given by
(3.9.1) |||p|||w,n := max
1≤j≤n |p(ξ(n)
j )|, p ∈ Pn−1,
where the ξ (n)
j ’s are the zeroes of P (α,β)
n .
Similarly, for any n ≥ 1, we define ||| · ||| ∗ w,n : Pn → R + as
(3.9.2) |||p||| ∗ w,n := max
0≤j≤n |p(η(n)
j )|, p ∈ Pn,
where the η (n)
j are the zeroes of d (α,β)
dxP n plus the endpoints of the interval
The following two results are easy to derive:
(3.9.3) |||p|||w,n ≤ p∞, ∀p ∈ Pn−1,
(3.9.4) |||p||| ∗ w,n ≤ p∞, ∀p ∈ Pn.
Ī = [−1,1].
Of course, working within finite dimensional spaces, the inequalities (3.9.3) and (3.9.4)
can be reversed. For instance, ∀n ≥ 1, we can find γ such that