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Results in Approximation Theory 107
Using the same proof of theorem 6.3.2, we also deduce the the following proposition.
Theorem 6.3.4 - We can find a constant C > 0 such that, for any n ≥ 1
(6.3.10) p ′ L 2 a (I) ≤ C √ n p L 2 w (I), ∀p ∈ Pn,
(6.3.11) p ′ L 2 w (I) ≤ Cn p L 2 w (I), ∀p ∈ Pn.
Finally, we are left with the Hermite case (i.e., a(x) = w(x) = e−x2, x ∈ R). Since
we have the identity a ≡ w, lemma 6.3.1 is not useful. We also have the counterpart
of theorem 6.3.2, which states:
Theorem 6.3.5 - We can find a constant C > 0 such that, for any n ≥ 1
(6.3.12) p ′ L 2 w (R) ≤ C √ n p L 2 w (R), ∀p ∈ Pn.
Proof - We argue exactly as in (6.3.7).
The problem of giving a bound to the norm in C0 ( Ī), by means of the norm in
L 2 w(I), has arisen in section 3.9, where other inverse inequalities are presented.
6.4 Other projection operators
We focus our attention on theorem 6.2.2. As we can see from the next proposition, the
expression (6.2.5) can actually be used to define the polynomial Πw,nf.