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122 Polynomial Approximation of Differential Equations
Taking into account lemma 6.7.4, theorems 6.2.6 and 6.3.5 are respectively generalized
to the case of Hermite functions as follows.
Theorem 6.7.5 - Let k ∈ N, then there exists a constant C > 0 such that, for any
g ∈ H k v (R), one has
(6.7.9) g − Π ∗ v,ng L 2 v (R) ≤ C
k 1
√n
g H k v (R), ∀n > k.
Theorem 6.7.6 - We can find a constant C > 0 such that, for any n ≥ 1
(6.7.10) q ′ L 2 w (R) ≤ C √ n q L 2 w (R), ∀q ∈ Sn.
We finally define I ∗ v,n : C 0 (R) → Sn−1, n ≥ 1, such that
(6.7.11) I ∗ v,n := [Iw,n(gv)]w, ∀g ∈ C 0 (R).
A theoretical convergence analysis for the operator I ∗ v,n for n → +∞ is considered in
funaro and kavian(1988). Here we state the main result (we recall that the Sobolev
spaces with real exponent have been introduced in section 5.6).
Theorem 6.7.7 - Let ǫ > 0 and s ≥ 1 + ǫ. Then there exists a constant C > 0 such
that, for any g ∈ H s v(R), we have
(6.7.12) g − I ∗ v,ng L 2 v (R) ≤ C
s−1−ǫ 1
√n gHs v (R), ∀n ≥ 1.
Other results, concerning the rate of convergence of the so called normalized Hermite
functions, are provided in boyd(1984).