Untitled - Cdm.unimo.it

Results in Approximation Theory 109
Theorem 6.4.2 - Let −1 < α < 1, −1 < β < 1 and k ≥ 1, then we can find a
constant C > 0 such that, for any f ∈ H k w(I), one has
(6.4.6) f − Π 1 w,nf H 1 w (I) ≤ C
k−1
1
(1 − x
n
2 ) (k−1)/2 dkf dxk
Proof - We first consider f ∈ C ∞ ( Ī). Let x0 ∈ I and set
(6.4.7) qn(x) :=
From (6.4.4) we can write
f(x0) +
x
x0
(Πw,n−1f ′
)(t) dt
(6.4.8) f − Π 1 w,nf H 1 w (I) ≤ f − qn H 1 w (I)
=
L 2 w (I)
f − qn 2 L2 w (I) + f ′ − Πw,n−1f ′ 2 L2 w (I)
1
2
.
On the other hand, we have by the Schwarz inequality
(6.4.9) f − qn 2 L2 w (I) =
≤
I
=
I
x
x0
I
f(x) − f(x0) −
x
(f ′ − Πw,n−1f ′ )(t) w(t)
x
(f ′ − Πw,n−1f ′ ) 2 (t) w(t) dt
x0
≤ f ′ − Πw,n−1f ′ 2 L 2 w (I)
I
·
x
x0
x0
x
x0
∈ Pn, n ≥ 1.
, ∀n > k.
(Πw,n−1f ′ 2 )(t)dt w(x)dx
2 dt
w(x) dx
w(t)
w −1
(t) dt
w(x) dx
w −1
(t)dt
w(x) dx.
We note that the last integral in (6.4.9) is finite when −1 < α < 1 and −1 < β < 1.
At this point we can conclude after substituting (6.4.9) in (6.4.8) and recalling theorem
6.2.4.