Untitled - Cdm.unimo.it

164 Polynomial Approximation of Differential Equations
Similar arguments hold for the case of unbounded domains. Let us see some exam-
ple. Here, we denote by λn,m and pn,m, 0 ≤ m ≤ n − 1, respectively the eigenvalues
and the eigenfunctions related to problem (7.4.26). In particular, the pn,m’s are com-
plex polynomials of degree at most n − 1. Again, we have λn,0 = 1 and pn,m(0) = 0,
for 1 ≤ m ≤ n − 1.
We recall that the space of Laguerre functions Sn has been introduced in section 6.7.
As in the previous cases, we need a preparatory result.
Lemma 8.2.7 - Let −1 < α < 1 and v(x) := x α e x , x ∈]0,+∞[. Then, we can find
a constant C > 0 such that, for any n ≥ 1 and P ∈ Sn−1, satisfying P(0) = 0, one
has
(8.2.22)
+∞
where µ > 1
2 max{1, 1
1−α }.
0
P ′ (Pv) ′ +∞
dx + µ P
0
2 v dx ≥ C
+∞
[(P
0
′ ) 2 + P 2 ] v dx,
For the proof we refer to kavian(1990). Combining this lemma with (7.2.25) and the
experience we have gained in the previous examples, we obtain the following result.
Theorem 8.2.8 - Let −1 < α < 1 and µ > 1
2 max{1, 1
1−α }. Then, for any n ≥ 1,
the eigenvalues relative to the system (7.4.26) satisfy Reλn,m > 0, 0 ≤ m ≤ n − 1.
Actually, in the Laguerre case, the eigenvalues of the operator considered are real.
Nevertheless, there is no proof of this.
Concerning problem (7.5.4), where the unknowns are the point values of a scaled
function, the eigenvalues are identical to those of problem (7.4.26). Actually, definition
(7.5.1) corresponds to a linear change of variables which does not alter the spectrum of
the derivative matrix.
For the matrix in (7.2.15), with the vanishing boundary condition at the point
x = 0, the eigenvalues are complex, but their real part is positive provided that α ≤ 0.