Untitled - Cdm.unimo.it

Orthogonality 27
Let us now define
(2.2.17) ũn := γ −1
n un, where γn := lim
x→+∞
un(x)
.
xn Of course, we get ũn(x) = x n +{lower degree terms}. The values of γn can be deduced in
the different cases from (1.3.5), (1.5.5), (1.6.4) and (1.7.5). Then, we have the following
minimizing property.
Theorem 2.2.3 - For any n ∈ N and for any polynomial p of degree n such that
p(x) = x n + {lower degree terms}, we have
(2.2.18) ũnw ≤ pw.
Proof - Let us note that ũn − p has degree at most n − 1. Consequently, ũn − p is
orthogonal to ũn. Hence
(2.2.19) ũn 2 w = (ũn,p)w.
Applying the Schwarz inequality (2.1.7) to the right-hand side of (2.2.19) we get (2.2.18).
2.3 Fourier coefficients
Hereafter, Pn denotes the linear space of polynomials whose degree is at most n. The
dimension of Pn is clearly n + 1. The fact that the polynomials uk, 0 ≤ k ≤ n,
are orthogonal with respect to some inner product implies that they form a basis for
Pn. Therefore, for any p ∈ Pn, one can determine in a unique way n + 1 coefficients
ck, 0 ≤ k ≤ n, such that
(2.3.1) p =
n
k=0
ckuk.
The ck’s are called the Fourier coefficients of p with respect to the designated basis.
By knowing the coefficients of p, the value of p(x) at a given x can be efficiently evaluated
by (1.1.2) and (2.3.1) with a computational cost proportional to n.