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50 Polynomial Approximation of Differential Equations
The set of the Chebyshev zeroes is the only distribution of Jacobi nodes for which
the corresponding Gaussian weights are all the same. This further sets Chebyshev
polynomials apart from the other families of orthogonal polynomials.
Laguerre case - For α > −1 and n ≥ 1, we have
(3.4.7) w (n)
j
Γ(n + α)
= −
n!
This time the counterpart of (3.4.3) is
(3.4.8)
un(x)
x − ξ (n)
j
L (α)
n−1 (ξ(n) j ) d
dx L(α) n (ξ (n)
−1 j )
= − 1
n un−1(x) + qj(x), 1 ≤ j ≤ n,
, 1 ≤ j ≤ n.
where qj ∈ Pn−2, 1 ≤ j ≤ n, and un = L (α)
n . The proof proceeds like that of the Jacobi
case after recalling (2.2.13).
Hermite case - We have for n ≥ 1
(3.4.9) w (n)
j
= √ π 2 n+1 n!
H ′ n(ξ (n)
−2 j ) , 1 ≤ j ≤ n.
For the proof we can argue as in the previous cases. We only note that, by (1.7.8) one
has H ′ n(ξ (n)
j ) = 2nHn−1(ξ (n)
j ), 1 ≤ j ≤ n.
We observe that for all the cases the weights are strictly positive. This is clear if
we consider that
w (n)
j
=
n
i=1
[l (n)
j ] 2 (ξ (n)
i ) w (n)
i
=
I
[l (n)
j ] 2 w dx > 0, 1 ≤ j ≤ n.
Once the zeroes have been computed, the weights can be evaluated using the recurrence
formula (1.1.2) to recover the values of un−1 and u ′ n.
Due to their high accuracy, Gauss formulas play a fundamental role in the theoret-
ical analysis of spectral methods. The possibility of integrating polynomials of degree
2n − 1 just by knowing their values at n points will be widely used. The reader inter-
ested in collecting more results may consult szegö(1939), stroud and secrest(1966),
ghizzetti and ossicini(1970), davis and rabinowitz(1984), where other explicit ex-
pressions of the weights are also given.