Untitled - Cdm.unimo.it

Numerical Integration 45
(3.2.10)
˜ l (n)
j (x) =
⎧
(−1)
⎪⎨
n
2n2 (x − 1)T ′ n(x) if j = 0,
(−1) j+n
n2 (x2 − 1)T ′ n(x)
if 1 ≤ j ≤ n − 1,
⎪⎩
x − η (n)
j
1
2n 2 (x + 1)T ′ n(x) if j = n.
The values η (n)
j , 0 ≤ j ≤ n, are given in (3.1.11).
Lagrange bases with respect to zeroes of derivatives of Legendre and Chebyshev poly-
nomials are shown for n = 7, respectively in figures 3.2.3 and 3.2.4.
Figure 3.2.3 - Legendre Lagrange basis Figure 3.2.4 - Chebyshev Lagrange basis
with respect to η (7)
i , 0 ≤ i ≤ 7. with respect to η (7)
i , 0 ≤ i ≤ 7.
Application of a similar procedure to the case of Laguerre (or Hermite) polynomials
is straightforward. Again, we denote by {l (n)
j }1≤j≤n the Lagrange basis with respect to
the Laguerre (respectively Hermite) zeroes. Relations (3.2.2) and (3.2.3) are still valid.
Formula (3.2.4) also holds provided un = L (α)
n (respectively un = Hn).