Untitled - Cdm.unimo.it

Numerical Integration 43
where 1 ≤ j ≤ n and un = P (α,β)
n . Of course, we have
(3.2.5) lim
x→ξ (n)
j
l (n)
j (x) = lim
x→ξ (n)
j
u ′ n(x)
u ′ n(ξ (n)
j )
= 1, 1 ≤ j ≤ n.
Plots of the Lagrange basis with respect to the Legendre and Chebyshev zeroes are
given in figures 3.2.1 and 3.2.2 for n = 7.
Figure 3.2.1 - Legendre Lagrange basis Figure 3.2.2 - Chebyshev Lagrange basis
with respect to ξ (7)
i , 1 ≤ i ≤ 7. with respect to ξ (7)
i , 1 ≤ i ≤ 7.
The other basis we mentioned above is denoted by { ˜ l (n)
j }0≤j≤n ⊂ Pn, and it is
defined by
(3.2.6)
˜ l (n)
j (η (n)
j ) = δij, 0 ≤ i ≤ n, 0 ≤ j ≤ n.
Indeed, for any polynomial p ∈ Pn, one has
(3.2.7) p =
n
j=0
p(η (n)
j ) ˜ l (n)
j .