Untitled - Cdm.unimo.it

112 Polynomial Approximation of Differential Equations
Theorem 6.5.1 - Let f be such that fw is Riemann integrable in I =] − 1,1[. Then
we have
(6.5.1) lim
n→+∞
n
j=1
f(ξ (n)
j ) w (n)
j
= lim Iw,nf w dx =
n→+∞
I
I
fw dx.
Proof - We show a simplified proof for f ∈ C0 ( Ī). The general case is studied in
stekloff(1916). We use the polynomial of best uniform approximation of f (see section
6.1). For any n ≥ 1, taking into account that [Ψ∞,n−1(f) − Iw,nf] ∈ Pn−1, we have
(6.5.2)
(f −Iw,nf)wdx
≤
(f −Ψ∞,n−1(f))wdx
+
I
≤ f −Ψ∞,n−1(f) C 0 (Ī)
I
⎛
⎝
I
w dx +
n
j=1
w (n)
j
⎞
n
(Ψ∞,n−1(f) −Iw,nf)(ξ
j=1
(n)
j ) w (n)
j
⎠ = 2 f −Ψ∞,n−1(f) C 0 (Ī)
I
w dx,
where we made use of Iw,nf(ξ (n)
j ) = f(ξ (n)
j ), 1 ≤ j ≤ n. Now, because of (6.1.5), the
last term in (6.5.2) tends to zero. Moreover, when f is smooth, we can establish the
rate of convergence using (6.1.7).
For Laguerre and Hermite quadrature formulas we can state similar propositions (see
uspensky(1928) and davis and rabinowitz(1984)).
Theorem 6.5.2 - Let f be such that fw is Riemann integrable in I =]0,+∞[, where
w is the Laguerre weight function. Assume the existence of x0 ∈ I and ǫ > 0 such that
|f(x)w(x)| ≤ x −1−ǫ , ∀x > x0. Under these assumptions (6.5.1) holds.
Theorem 6.5.3 - Let f be such that fw is Riemann integrable in I = R, where w
is the Hermite weight function. Assume the existence of x0 ∈ I and ǫ > 0 such that
|f(x)w(x)| ≤ x −1−ǫ , ∀|x| > x0. Under these assumptions (6.5.1) holds.