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7
DERIVATIVE MATRICES
The derivative operator is a linear transformation from the space of polynomials into
itself. In addition, the exact derivative of a given polynomial can be determined in
a finite number of operations. Depending on the basis in which the polynomial is
expressed, we will construct the appropriate matrices that allow the computation of its
derivative.
7.1 Derivative matrices in the frequency space
Let p be a polynomial in Pn, where n ≥ 1 is given. From (2.3.1), and u ′ 0 ≡ 0, we can
write
(7.1.1) p ′ =
n
k=1
cku ′ k =
n
k=0
c (1)
k uk.
In (7.1.1), the constants c (1)
k , 0 ≤ k ≤ n, are the Fourier coefficients of p′ . We would like
to establish a relation between these new coefficients and the ck’s. Let us begin with the
Jacobi case (uk = P (α,β)
k , k ∈ N). For simplicity, we just discuss the ultraspherical case
(ν := α = β). Thus, we start by considering the following relation (see szegö(1939),
p.84):
(7.1.2) un = d
dx
(n + 2ν + 1) un+1
(n + ν + 1)(2n + 2ν + 1) −
(n + ν) un−1
, n ≥ 1,
(n + 2ν)(2n + 2ν + 1)