Untitled - Cdm.unimo.it

162 Polynomial Approximation of Differential Equations
Let us investigate the case of fourth-order operators. Here, λn,m, 1 ≤ m ≤ n − 1,
and rn,m, 1 ≤ m ≤ n − 1, are respectively the eigenvalues and the eigenfunctions
associated to problem (7.4.22). Therefore, we can write r IV
n,m(η (n)
i ) = λn,mrn,m(η (n)
i ),
1 ≤ i ≤ n − 1, and rn,m(±1) = r ′ n,m(±1) = 0.
Before proceeding with our analysis, we state the following proposition.
Lemma 8.2.5 - Let ν := α = β with −1 < ν ≤ 1. Then, we can find a constant
C > 0, such that, for any n ≥ 2 and p ∈ Pn+2 satisfying p(±1) = p ′ (±1) = 0, one
has
(8.2.15)
1
p
−1
′′ (pw) ′′ 1
dx ≥ C [p
−1
′′ ] 2 w dx.
The proof of the above result can be found in bernardi and maday(1990). It is quite
technical, but very similar to that of theorem 8.2.1. The same proof, adapted to the
case of complex polynomials, yields the inequality
1
(8.2.16) Re p ′′ (¯pw) ′′
dx ≥ C
We are now ready to prove the following proposition.
−1
1
|p
−1
′′ | 2 w dx.
Theorem 8.2.6 - Let ν := α = β with −1 < ν ≤ 1. Then, for any n ≥ 2,
the eigenvalues relative to the system (7.4.22) satisfy Reλn,m > 0, 1 ≤ m ≤ n − 1.
Moreover, when ν = 0, we have λn,m ∈ R, 1 ≤ m ≤ n − 1.
Proof - One has
n
(8.2.17)
i=0
r IV
n,m(η (n)
i )¯rn,m(η (n)
i ) ˜w (n)
i
= λn,m
n
i=0
[|rn,m|(η (n)
i )] 2 ˜w (n)
i .
We must show that the left-hand side of (8.2.17) has positive real part. Now, for
1 ≤ m ≤ n − 1, we can find a complex number ζn,m ∈ C such that
(8.2.18)
1
1 − x 2 rn,m(x) = ζn,mun(x) + {lower degree terms},