Untitled - Cdm.unimo.it

Results in Approximation Theory 105
Lemma 6.3.1 - We can find a constant C > 0 such that, for any n ≥ 1
(6.3.1) p L 2 w (I) ≤ Cn p L 2 a (I), ∀p ∈ Pn.
Proof - Let m = n + 2. Since (1 − x 2 )p 2 ∈ P2m−2, formula (3.4.1) and theorem 3.4.1
lead to
(6.3.2)
(6.3.3)
I
I
p 2 a dx =
p 2 w dx =
m
j=1
m
j=1
p 2 (ξ (m)
j ) w (m)
j , ∀p ∈ Pn,
p 2 (ξ (m)
j )(1 − [ξ (m)
j ] 2 ) w (m)
j , ∀p ∈ Pn,
where ξ (m)
j , 1 ≤ j ≤ m, are the zeroes of P (α,β)
m .
Now, we can easily conclude the proof by noting that there exists a constant C ∗ > 0,
depending only on α and β, such that, for any n ≥ 1
(6.3.4) 1 − [ξ (m)
j ] 2 ≥ C∗ C∗
≥ 2
m
, 1 ≤ j ≤ m.
9n2 In fact, from (1.3.2) and (3.1.17), the straight-line tangent to P (α,β)
m
at the point
2(α+1)
(α,β)
x = 1 vanishes at ˆx := 1 − m(m+α+β+1) . It is easy to see that P m is convex in
the interval [ξ (m)
m ,1], hence ˆx is an upper bound for the zeroes of P (α,β)
m . A similar
argument is valid near the point x = −1 which proves (6.3.4).
Relation (6.3.1) is an example of inverse inequality. Actually, the norm in L 2 w(I)
is in general bigger than the norm in L 2 a(I), because w ≥ a, but the inequality in the
opposite direction holds in the finite dimensional space Pn, for any fixed n ∈ N.
More interesting cases are obtained for inverse inequalities that involve derivatives
of polynomials. Examples have been given in section 2.5. Using Sobolev norms, we can
establish similar results.
Theorem 6.3.2 - We can find a constant C > 0 such that, for any n ≥ 1
(6.3.5) p ′ L 2 a (I) ≤ Cn p L 2 w (I), ∀p ∈ Pn,