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46 Polynomial Approximation of Differential Equations
In particular, for the Laguerre case, another set of Lagrange polynomials in Pn−1 will
be useful in what follows. We first define η (n)
0 := 0, and then we construct the Lagrange
basis { ˜ l (n)
j }0≤j≤n−1 with respect to η (n)
i , 0 ≤ i ≤ n − 1. With a proof similar to that
of theorem 3.2.1 (this time using (1.6.1) and (3.1.19)), one gets, for n ≥ 1,
(3.2.11)
˜ l (n)
j (x) =
where un = L (α)
n and x ∈ [0,+∞[.
3.3 The interpolation operators
⎧
(n − 1)! Γ(α + 2)
⎪⎨
− u
Γ(n + α + 1)
⎪⎩
′ n(x) if j = 0,
−
x u ′ n(x)
if 1 ≤ j ≤ n − 1,
n un(η (n)
j ) (x − η (n)
j )
Following the guideline of section 2.4, we introduce some new operators defined in the
space of continuous functions. They are named interpolation operators. We first begin
by considering the set of polynomials {uk}k∈N , orthogonal with respect to the weight
w. Next, we choose n ≥ 1 and evaluate the zeroes ξ (n)
k , 1 ≤ k ≤ n, of un. At this point,
we define Iw,n : C 0 (I) → Pn−1 to be the operator mapping a continuous function f
to the unique polynomial pn ∈ Pn−1 satisfying pn(ξ (n)
j ) = f(ξ (n)
j ), 1 ≤ j ≤ n. This
polynomial is the interpolant of f with respect to the zeroes of un. The operator Iw,n
is linear. Moreover, one can check that
(3.3.1) Iw,np = p, ∀p ∈ Pn−1.
A question arose in sections 2.3 and 2.4 on how to characterize a certain expression
g(x,p(x)) in terms of a given polynomial p. A very simple answer is obtained when p
is determined by the values attained in a set of given points. For example, we have
(3.3.2) [Iw,ng(x,p(x))] x=ξ (n)
j
for any polynomial p ∈ Pn−1.
= g(ξ (n)
j ,p(ξ (n)
j )), 1 ≤ j ≤ n,