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28 Polynomial Approximation of Differential Equations
If p = n k=0 ckuk and q = n k=0 bkuk
proven:
, then the following relations are trivially
n
(2.3.2) p + q = (ck + bk)uk,
(2.3.3) λp =
(2.3.4) pq =
k=0
n
(λck)uk, ∀λ ∈ R,
k=0
n
k=0 j=0
n
(ckbj)ukuj.
By integrating (2.3.4) in I, orthogonality implies
n
(2.3.5) (p,q)w =
Hence
(2.3.6) pw =
m=0
n
m=0
cmbm um 2 w.
c 2 m um 2 w
1
2
.
The quantities um 2 w, m ∈ N, have been computed in theorem 2.2.2 for the different
families of polynomials. Formula (2.3.6) allows us to evaluate the weighted norm of p
when its coefficients are known. In particular, if we take q ≡ uk, 0 ≤ k ≤ n, in (2.3.5)
(then bm = δkm), the following explicit expression for the coefficients is available:
(2.3.7) ck = (p,uk)w
uk 2 w
, p ∈ Pn, 0 ≤ k ≤ n.
Given g : R 2 → R , one can try to characterize the Fourier coefficients of g(x,p(x)),
where p is a polynomial in Pn, in terms of the coefficients of p. This turns out to be useful
for the approximation of nonlinear terms in differential equations. For instance, we can
easily handle the case g(x,p(x)) = xp(x). Namely, from the coefficients of the expression
p = n
k=0 ckuk we want to determine those of the expression xp = n+1
k=0 bkuk. First
of all we deduce from (1.1.2)
(2.3.8) xuk = 1
ρk+1
(uk+1 − σk+1uk − τk+1uk−1), 1 ≤ k ≤ n.