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90 Polynomial Approximation of Differential Equations
5.7 Sobolev spaces in intervals
For any integer k ≥ 1, the weighted Sobolev space of order k in I ⊂ R is defined by
(5.7.1) H k
w(I) :=
u differentiable k times and
Cu
dmu dxm ∈ L2
w(I), for 0 ≤ m ≤ k .
Here, w : I → R is a weight function. Moreover H 0 w(I) := L 2 w(I). For every
k ∈ N, H k w(I) is a Hilbert space with the inner product and the norm
(5.7.2) (u,v) H k w (I) :=
(5.7.3) u H k w (I) :=
k
m=0
m d u
dxm , dmv dxm
L2 w (I)
, ∀u,v ∈ H k w(I),
k
dmu dxm
m=0
2
L2 w (I)
1
2
, ∀u ∈ H k w(I).
Many fundamental properties about these spaces are given in kufner(1980). We recall
that, for a function u ∈ H k w(I), there exists a sequence of regular functions {vn}n∈N
(we can assume for example vn ∈ C∞ ( Ī), ∀n ∈ N), converging to u in the norm of
H k w(I), i.e., limn→+∞ vn − u H k w (I) = 0. In the following, using a standard functional
analysis technique, many theorems will be proven with the help of this property. The
statement to be proven is checked for regular functions and successively extended to
Sobolev spaces with a limit procedure. For simplicity, we shall often omit the details
when applying this kind of technique.
Let us examine the Jacobi weight function w(x) = (1−x) α (1+x) β , x ∈ I =]−1,1[,
where the parameters satisfy the conditions −1 < α < 1 and −1 < β < 1. In this case,
one verifies that H 1 w(I) ⊂ C 0 ( Ī). Moreover, we can find two constants K1, K2 > 0,
such that
(5.7.4) u L 2 w (I) ≤ K1 u C 0 (Ī) ≤ K2 u H 1 w (I), ∀u ∈ H 1 w(I).
As a byproduct of Hardy’s inequality (see for instance lions and magenes(1972), p.49),
one obtains the relation