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Results in Approximation Theory 97
Proof - A polynomial ψ ∈ Pn can be written in the form ψ = n
k=0 dkuk, for some
real coefficients dk, 0 ≤ k ≤ n. Minimizing f −ψ L 2 w (I), or equivalently f −ψ 2 L 2 w (I),
requires the derivatives
(6.2.3)
+
n
k=0
∂
∂dj
f − ψ 2 L2
∂
w (I) = f
∂dj
2 L2 w
d 2 kuk 2 L 2 w (I)
(I) − 2
n
k=0
dk(f,uk) L 2 w (I)
= −2 (f,uj) L 2 w (I) + 2dj uj 2 L 2 w (I),
0 ≤ j ≤ n.
We deduce that the unique minimum is attained when dj = cj, 0 ≤ j ≤ n, where the
cj’s are the Fourier coefficients of f (see (6.2.2)). This ends the proof.
In short, we can write
(6.2.4) f − Πw,nf L 2 w (I) = inf
ψ∈Pn
f − ψ L 2 w (I).
Another interesting characterization is given in the following theorem.
Theorem 6.2.2 - For any f ∈ L 2 w(I) and n ∈ N, we have
(6.2.5)
I
(f − Πw,nf)φw dx = 0, ∀φ ∈ Pn.
Proof - We fix φ ∈ Pn and define G : R → R by
G(ν) := f − Πw,nf + νφ 2 L 2 w (I).
We know from theorem 6.2.1 that ν = 0 is a minimum for G. Therefore
(6.2.6) G ′ (ν) = 2
(f − Πw,nf)φw dx + 2νφ
I
2 L2 w (I), ∀ν ∈ R.
Imposing that G ′ (0) = 0, we get (6.2.5).