Untitled - Cdm.unimo.it

152 Polynomial Approximation of Differential Equations
we are concerned with finding n + 1 complex polynomials pn,m, 0 ≤ m ≤ n, of degree
at most n, and n + 1 complex numbers λn,m, 0 ≤ m ≤ n, such that
(8.1.1)
⎧
⎨
⎩
p ′ n,m(η (n)
i ) = λn,m pn,m(η (n)
i ), 1 ≤ i ≤ n,
pn,m(η (n)
0 ) = λn,m pn,m(η (n)
0 ),
0 ≤ m ≤ n.
Each pn,m, 0 ≤ m ≤ n (different from the zero function) is determined up to a constant
factor. Let us suppose that pn,m(η (n)
0 ) = 0, for m = 0. Then, it is easy to realize that
λn,0 = 1. On the contrary, the remaining λn,m, 1 ≤ m ≤ n, are eigenvalues of the
n × n matrix, obtained by assuming σ = 0 in (7.4.1) (in (7.4.3) we have an example
for n = 3). Therefore, except for m = 0, the eigenfunctions satisfy pn,m(η (n)
0 ) = 0.
In figures 8.1.1 to 8.1.4, we give the plot of the eigenvalues λn,m, 1 ≤ m ≤ n, in the
window [−4,12] × [−12,12] of the complex plane. We considered respectively the four
cases: α = β = −.7, −.5, 0, .5 . The integer n varies from 4 to 10. The eigenvalues,
corresponding to the same value of n, have been joined together by segments.
In the first three cases, all the eigenvalues lie in the half plane of the complex numbers
with positive real part. In general, we are induced to conjecture that Reλn,m > 0,
1 ≤ m ≤ n, when α > −1, −1 < β ≤ 0. Unfortunately, we do not have the proof of
this fact. The statement has been proven in solomonoff and turkel(1989), for the
Chebyshev case (α = β = −.5), and could be adapted to other situations. Anyway, the
proof is quite technical and we omit it.
We further characterize the eigenvalues with the following proposition.
Theorem 8.1.1 - For 1 ≤ m ≤ n, λn,m satisfies the relation
(8.1.2) λ n n,m +
n
j d
j=1
dx j ˜ l (n)
0 (−1)
where the Lagrange polynomial ˜ l (n)
0 is defined in (3.2.8).
λ n−j
n,m = 0,