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128 Polynomial Approximation of Differential Equations
As proposed in section 2.3, one can study the relation between the Fourier coeffi-
cients of polynomials such as xp ′ (or more involved expressions) and the coefficients of
p. Many examples of this type, for Legendre and Chebyshev expansions, are discussed
in the appendix of gottlieb and orszag(1977).
For Laguerre polynomials, relations taken from szegö(1939), p.98, lead to
(7.1.11)
Therefore, one obtains
(7.1.12) c (1)
i
d
dx L(α)
n−1
n = −
m=0
= −
n
j=i+1
L (α)
m , n ≥ 1, α > −1.
cj, 0 ≤ i ≤ n − 1, and c (1)
n = 0.
The Hermite case is very easy to analyze. Actually, from (1.7.8), we have
(7.1.13) c (1)
i = 2(i + 1) ci+1, 0 ≤ i ≤ n − 1, and c (1)
n = 0.
An initial analysis suggests that the evaluation in the frequency space of the deriva-
tive of the polynomial p ∈ Pn has a computational cost proportional to n 2 , since it cor-
responds to a matrix-vector multiplication. A deeper study reveals an algorithm with a
cost only proportional to n. In fact, one easily proves the following recursion formula.
First we compute c (1)
n = 0 and
(7.1.14) c (1)
n−1
= (2n + 2ν − 1)(n + ν)
(n + 2ν)
Then, we proceed backwards according to the relations
(7.1.15) c (1)
i
(2i + 2ν + 1)(i + ν + 1)
=
i + 2ν + 1
cn, ν > −1, ν = − 1
2 .
i + ν + 2
(2i + 2ν + 5)(i + 2ν + 2) c(1)
i+2 + ci+1 ,
0 ≤ i ≤ n − 2, ν > −1, ν = − 1
2 .