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66 Polynomial Approximation of Differential Equations
(4.1.2) ( ˜ l (n)
i , ˜ l (n)
j )w,n = δij ˜w (n)
i , 0 ≤ i ≤ n, 0 ≤ j ≤ n.
This is not true anymore when using the inner product given by (2.1.8). In fact, using
(3.8.15), one obtains
(4.1.3)
1
−1
˜ l (n)
i ˜ l (n)
j wdx = δij ˜w (n)
i − un 2 w,n − un 2 w
un 4 w,n
where un = P (α,β)
n , α > −1, β > −1.
un(η (n)
i )un(η (n)
j ) ˜w (n)
i ˜w (n)
j ,
0 ≤ i ≤ n, 0 ≤ j ≤ n,
In general, the last term on the right-hand side of (4.1.3) tends to zero (when n diverges)
faster than the first term, so that the polynomials considered are almost orthogonal.
Thus, for a fixed n ≥ 1, two orthogonal bases are available in Pn−1, i.e., {uk}0≤k≤n−1
and {l (n)
j }1≤j≤n. Therefore, we can define a linear transformation Kn : Pn−1 → Pn−1,
mapping the vector at the point values of a polynomial p into the vector of its Fourier
coefficients. This application is called discrete Fourier transform, and is represented by
a n × n matrix. Each Fourier coefficient is explicitly determined recalling (2.3.7) and
(3.4.1):
(4.1.4) ci =
1
ui 2 w
n
j=1
Therefore, we have the matrix relation
p(ξ (n)
j ) ui(ξ (n)
j ) w (n)
j , 0 ≤ i ≤ n − 1.
(4.1.5) Kn := {kij} 0≤i≤n−1 , where kij :=
0≤j≤n−1
(n)
ui(ξ j+1 ) w(n) j+1
ui2 w
Obviously Kn is invertible. We express its inverse using (2.3.1):
(4.1.6) p(ξ (n)
i ) =
and obtain
(4.1.7) K −1
n−1
j=0
cj uj(ξ (n)
i ), 1 ≤ i ≤ n,
n = {uj(ξ (n)
i+1
)} 0≤i≤n−1 .
0≤j≤n−1
.