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136 Polynomial Approximation of Differential Equations
determined according to the following relation:
(7.3.1) c0 = σ −
n
k=1
ck uk(ξ).
This comes from evaluating (2.3.1) at the point x = ξ. Many other conditions are
possible. For instance, one can impose
pdx = σ, σ ∈ R. In the Chebyshev case, by
(2.6.4), c0 is then obtained from
(7.3.2) c0 = σ
2 −
I
n
k=2
k even
ck
.
1 − k2 Let us analyze a more general situation. We just consider the ultraspherical case.
Let A and σ be real constants and ξ ∈ Ī. Given the polynomial q ∈ Pn−1, we are
concerned with finding p ∈ Pn, such that
(7.3.3)
⎧
⎨
p ′ + Πw,n−1(Ap) = q,
⎩
p(ξ) = σ.
The projection operator Πw,n, n ∈ N, has been introduced in section 2.4.
For example, when n = 4, problem (7.3.3) is equivalent to the following system:
(7.3.4)
⎡
A ∗ 0 ∗
⎤⎡
0
⎢ 0
⎢ 0
⎣
0
A
0
0
∗
A
0
0
∗
A
∗⎥
⎢
⎥⎢
0⎥
⎢
⎦⎣
∗
∗ ∗ ∗ ∗ ∗
c0
c1
c2
c3
c4
⎤
⎥
⎦ =
where the dk’s are the Fourier coefficients of q, and the symbol ∗ denotes the non zero
entries. Due to the special structure of the matrix in (7.3.4), the polynomial p in (7.3.3)
can be computed by Gauss elimination with a cost proportional to n 2 . However, the
situation gets far more complicated when A : I → R is a non constant function. In
this case the matrix associated to the system is full (see section 2.4).
⎡
⎢
⎣
d0
d1
d2
d3
σ
⎤
⎥
⎦ ,