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4
TRANSFORMS
The aim of the previous chapters was to show that a polynomial admits a double
representation. One is given by its Fourier coefficients with respect to set of orthogonal
basis functions, the other by the values at the nodes associated with a suitable high
accuracy integration formula. Clearly, it is possible to switch from one representation
to the other. The way to do it is the subject of this chapter.
4.1 Fourier transforms
One of the main results of chapter two was to show that the polynomials introduced in
chapter one were orthogonal in relation to a suitable weight function. In addition, other
polynomials, i.e., the Lagrange polynomials with respect to a certain set of nodes, have
been presented in chapter three. These too are in general orthogonal. Actually, due to
(3.4.1) and theorem 3.4.1, we have for any fixed n ≥ 1,
(4.1.1)
I
l (n)
i l (n)
j
wdx =
n
k=1
l (n)
i (ξ (n)
k
j (ξ (n)
k ) w(n)
k =
) l(n)
⎧
⎨0
if i = j,
⎩
w (n)
i
if i = j.
A similar relation holds when Gauss-Radau nodes are considered. When Gauss-Lobatto
points are used, we loose orthogonality. However, in terms of the discrete inner product
introduced in section 3.8, we still get orthogonality. In fact, it is trivial to show that,
for any n ≥ 1: