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Results in Approximation Theory 121
Furthermore, we have the following inverse inequality.
Theorem 6.7.3 - We can find a constant C > 0 such that, for any n ≥ 1
(6.7.5) q ′ L 2 v (I) ≤ Cn q L 2 v (I), ∀q ∈ Sn.
Proof - Set q = pe −x , p ∈ Pn, and recall (6.3.11).
Next, we define the operator I ∗ v,n : C 0 ( Ī) → Sn−1, n ≥ 1, such that
(6.7.6) I ∗ v,ng := [Iw,n(ge x )]e −x , ∀g ∈ C 0 ( Ī).
In a similar way, one defines the interpolation operator Ĩ∗ v,n, based on the Gauss-Radau
nodes. On the basis of theorem 6.6.4, one recovers estimates for the error g − I ∗ v,ng .
Now let us set v(x) := w−1 (x) = ex2, x ∈ R, and define for any n ∈ N, the space Sn of
the so called Hermite functions. Any element of Sn is of the form pw, where p ∈ Pn.
Furthermore, we introduce the operator Π ∗ v,n : L 2 v(R) → Sn, n ∈ N, such that
(6.7.7) Π ∗ v,ng := [Πw,n(gv)]w, ∀g ∈ L 2 v(R).
As in the previous case, we begin with a basic result.
Lemma 6.7.4 - The following implication: f ∈ H k w(R) ⇔ g := fw ∈ H k v (R), is
satisfied for any k ∈ N. Moreover, the corresponding norms are equivalent.
Proof - It is possible to find real coefficients γ (k)
m , 0 ≤ k ≤ m (see funaro(1991)),
such that
(6.7.8)
R
k d g
dxk 2
v dx =
k
m=0
γ (k)
m
This completes the proof by virtue of (5.5.10) and (5.6.4).
R
m d f
dxm 2 w dx.