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42 Polynomial Approximation of Differential Equations
3.2 Lagrange polynomials
As remarked in section 2.3, the set {uk}0≤k≤n is a basis in the space Pn of polynomials
of degree at most n. When n + 1 distinct points are given in
Ī, another basis in Pn
is generated in a natural way. This is the basis of Lagrange polynomials with respect
to the prescribed points. An element of the basis attains the value 1 at a certain point
and vanishes in the remaining n points.
Let us analyze first the Jacobi case. We examine two interesting examples. On
one hand we have the set of the Lagrange polynomials in Pn−1 relative to the n points
ξ (n)
k
(α,β)
, 1 ≤ k ≤ n, i.e., the zeroes of P n . On the other we have the set of Lagrange
polynomials in Pn relative to the n + 1 points η (n)
k , 0 ≤ k ≤ n, i.e., the zeroes of
d (α,β)
dxP n plus the two points −1 and 1.
In the first case, the elements of the basis are denoted by l (n)
j , 1 ≤ j ≤ n. These
polynomials in Pn−1 are uniquely defined by the conditions
(3.2.1) l (n)
j (ξ (n)
i ) =
1 if i = j
, 1 ≤ j ≤ n.
0 if i = j
They actually form a basis because any polynomial p ∈ Pn−1 can be written as follows:
(3.2.2) p =
n
j=1
p(ξ (n)
j ) l (n)
j .
Therefore, p is a linear combination of the Lagrange polynomials. Such a combination
is uniquely determined by the coefficients p(ξ (n)
j ), 1 ≤ j ≤ n.
The following expression is easily proven:
(3.2.3) l (n)
j (x) =
n
k=1
k=j
x − ξ (n)
k
ξ (n)
j
− ξ(n)
k
, 1 ≤ j ≤ n, x ∈ [−1,1].
For future applications, it is more convenient to consider the alternate expression
⎧
⎪⎨
un(x)
if x = ξ (n)
j ,
(3.2.4) l (n)
j (x) =
⎪⎩
u ′ n(ξ (n)
j ) (x − ξ (n)
j )
1 if x = ξ (n)
j ,