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88 Polynomial Approximation of Differential Equations
From now on, let us assume that w(x) = ex2, x ∈ R, i.e. the inverse of the Hermite
weight function. A first characterization is immediately obtained by virtue of relation
(5.5.10)
(5.6.4) H k w(R) =
Cu
ρ k F(u √ w) ∈ L 2
(R;C) ,
where ρ(t) = (1 + t 2 ) 1/2 , t ∈ R. This new point of view suggests the introduction of
other Sobolev spaces. Namely, let s ≥ 0 be a real number, we define
(5.6.5) H s w(R) :=
Cu
ρ s F(u √ w) ∈ L 2
(R;C) .
We note that ρ s , s ≥ 0 is a smooth function in R. This justifies the use of ρ(t) in place
of t.
Of course, when s is integer, the two definitions (5.6.1) and (5.6.5) coincide. It is clear
that we have the inclusion H s1
w (R) ⊂ H s2
w (R), if s1 > s2 ≥ 0. In this way, although we
cannot speak of derivatives of order s when s is not an integer, we can instead decide
when a function u belongs to H s w(R). In this case, u will be more regular than a function
in H [s]
w (R) and less regular than a function in H [s]+1
w (R), where [s] denotes the integer
part of s. For instance, let u(x) = |x|e−x2, x ∈ R. Then, one can easily check that
u ∈ H 1 w(R). On the other hand, for any s such that 0 ≤ s < 1, we have u ∈ H s w(R).
By (5.5.2), a norm in H s w(R), s ≥ 0, is defined by
(5.6.6) |||u||| H s w (R) := ρ s F(u √ w) L 2 (R;C), ∀u ∈ H s w(R).
One can prove that, when s is an integer, the norms (5.6.3) and (5.6.6) are equivalent (see
triebel(1978), p.177). Actually, for any k ∈ N, we can find two constants C1,C2 > 0,
such that
(5.6.7) C1 u H k w (R) ≤ |||u||| H k w (R) ≤ C2u H k w (R), ∀u ∈ H k w(R).
When s is not integer, it is possible to show (see triebel(1978), p.189) that the norm
||| · ||| H s w (R) is equivalent to: