Untitled - Cdm.unimo.it

120 Polynomial Approximation of Differential Equations
polynomial cannot give accurate results (see gottlieb and orszag(1977), p.45). It is
thus necessary to introduce new families of approximating functions. We set v(x) :=
x α e x , α > −1, x ∈ I. For n ∈ N, we define Sn to be the space of the functions (named
Laguerre functions) of the form pe −x , where p ∈ Pn. By virtue of (2.2.6), this is an
orthogonal set of functions where the inner product is weighted by v. We present a
preliminary result.
Lemma 6.7.1 - Let f = ge x then, for any k ∈ N, we have
(6.7.1) x m/2 dm f
dx m ∈ L2 w(I), 0 ≤ m ≤ k ⇐⇒ x k/2 dk g
dx k ∈ L2 v(I).
Proof - This is a direct consequence of the formula
(6.7.2)
=
k
m=0
I
k Γ(α + k + 1)
m Γ(α + m + 1)
k d g
dxk 2
x α+k e x dx
which is proven by induction (see funaro(1991)).
I
m d f
dxm 2 x α+m e −x dx,
The operator Π ∗ v,n : L 2 v(I) → Sn, n ∈ N, is defined as follows:
(6.7.3) Π ∗ v,ng := [Πw,n(ge x )]e −x , ∀g ∈ L 2 v(I).
Combining theorem 6.2.5 and relation (6.7.2), we get the following proposition.
Theorem 6.7.2 - Let k ∈ N. Then there exists a constant C > 0 such that, for any g
satisfying dk g
dx k x k/2 ∈ L 2 v(I), one has
(6.7.4) g − Π ∗ v,ng L 2 v (I) ≤ C
k 1
x
k/2 √n
dkg dxk
L 2 v (I)
, ∀n > k.