Untitled - Cdm.unimo.it

108 Polynomial Approximation of Differential Equations
Theorem 6.4.1 - For a given f ∈ L 2 w(I), let ψ ∈ Pn such that
(6.4.1) (f − ψ,φ) L 2 w (I) = 0, ∀φ ∈ Pn.
Then we have ψ = Πw,nf.
Proof - We can write ψ = n
k=0 dkuk for some coefficients dk, 0 ≤ k ≤ n. For any
0 ≤ j ≤ n, choosing φ = uj as test function, (6.4.1) yields
(6.4.2) (f,uj) L 2 w (I) =
n
k=0
dk (uk,uj) L2 w (I) = dj uj 2 L2 w (I), 0 ≤ j ≤ n.
Thus, by (6.2.2), the dk’s are the first n+1 Fourier coefficients of f. Hence ψ = Πw,nf.
This result is the starting point for the definition of new projection operators. For
any m ∈ N and n ≥ m, we introduce the operator Π m w,n : H m w (I) → Pn, such that
(6.4.3) (f − Π m w,nf,φ) H m w (I) = 0, ∀φ ∈ Pn.
The inner product in H m w (I) has been defined in (5.6.2) and (5.7.2). Of course we
have Π 0 w,n = Πw,n. One easily verifies that the polynomial Π m w,nf ∈ Pn is uniquely
determined and represents the orthogonal projection of f in the H m w (I) norm. One
checks that this projection satisfies a minimization problem like (6.2.4), namely
(6.4.4) f − Π m w,nfH m
w (I) = inf f − ψHm w (I).
ψ∈Pn
Moreover, we have for n ≥ m
(6.4.5) Π m w,nf H m w (I) ≤ f H m w (I), ∀f ∈ H m w (I).
As before, the expression in (6.4.4) tends to zero when n → +∞, with a rate depending
on the degree of smoothness of f. For simplicity, we restrict the analysis to the case
m = 1 and we just consider the Jacobi weight w(x) = (1−x) α (1+x) β , x ∈ I =]−1,1[.