Untitled - Cdm.unimo.it

102 Polynomial Approximation of Differential Equations
(6.2.19) f − Πw,nf 2 L2 w (I) =
≤
=
max
j≥n+1
≤
∞
j=n+1
c 2 j
k−1
k−1
i=0
∞
j=n+1
c 2 j uj 2 L 2 w (I)
[(j − i)(j + α + β + 1 + i)] −1
dkuj dxk
[(j − i)(j + α + β + 1 + i)] −1
∞
i=0
k−1
j=k
[(n + 1 − i)(n + α + β + 2 + i)] −1
dkf dxk
i=0
≤ C 2 n −2k
(1 − x 2 ) k/2 dk f
c 2 j
dxk
where C depends on α, β and k. Thus, we obtained (6.2.15).
dk uj
dx k
2
L 2 v (I)
2
L 2 w (I),
2
L2 v (I)
2
L2 v (I)
The hypotheses of the previous theorem can be weakened by requiring only that f
satisfies dm f
dx m (1 − x 2 ) m/2 ∈ L 2 w(I), 0 ≤ m ≤ k.
A proof based on similar arguments has been provided in bernardi and maday(1989)
for ultraspherical polynomials. As pointed out in gottlieb and orszag(1977), p.33, a
strict requirement to obtain estimates like (6.2.15), is that the polynomial basis consid-
ered derives from a Sturm-Liouville problem, which is singular at the boundary points
of the domain (see section 1.1).
Equations (6.2.16) and (6.2.17) lead to the following remarkable relation, ∀n ≥ 1:
(6.2.20) (Πw,nf) ′ = Πa,n−1f ′ , ∀f ∈ L 2 w(I) ∩ H 1 a(I).
Inequality (6.2.15) is a typical example of a spectral error estimate. The rate of
convergence depends only on the smoothness of the function f. In contrast to other
approximation techniques, for instance those based on splines where the rate of con-
vergence is controlled by the regularity of the approximating functions in the domain,