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56 Polynomial Approximation of Differential Equations
3.8 Discrete norms
Two polynomials p,q ∈ Pn−1 are uniquely characterized by the values they take at the
Gauss nodes ξ (n)
j , 1 ≤ j ≤ n. Therefore, since pq is a polynomial of degree at most
2n − 2, the inner product
I
pqw dx can be determined with the help of (3.4.1). The
situation is different when the polynomials p,q ∈ Pn are distinguished by their values at
the points η (n)
j , 0 ≤ j ≤ n. This time pq ∈ P2n and the Gauss-Lobatto formula (3.5.1)
becomes invalid. This justifies the definition of an inner product (·, ·)w,n : Pn×Pn → R,
and a norm · w,n : Pn → R + , given by
(3.8.1) (p,q)w,n :=
(3.8.2) pw,n :=
n
j=0
⎛
⎝
p(η (n)
j ) q(η (n)
j ) ˜w (n)
j , ∀p,q ∈ Pn,
n
j=0
p 2 (η (n)
j ) ˜w (n)
j
⎞
⎠
1
2
, ∀p ∈ Pn.
These expressions are called discrete inner product and discrete norm respectively.
We first compute the discrete norm of un. For the sake of simplicity we restrict ourselves
to the ultraspherical case.
Theorem 3.8.1 - For any n ≥ 1, we have
(3.8.3) un 2 w,n = 2 2α+1 Γ 2 (n + α + 1)
n n! Γ(n + 2α + 1) ,
where un = P (α,α)
n , α > −1.
Proof - The key point is the relation
(3.8.4) u ′ n−1(η (n)
j ) = −
n(n + 2α)
n + α
un(η (n)
j ), 1 ≤ j ≤ n − 1.
To obtain (3.8.4), we refer to szegö(1939), p.71, formula (4.5.7). We substitute n − 1
for n in the second part of that formula, and then we eliminate un−1 with the help of