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15.2 Legendre Polynomials 525

15.2.3 Fourier-Legendre Expansions
Because the Legendre polynomials are eigenfunctions of a Sturm-Liouville problem, we can
∞
write an eigenfunction expansion c n P n (x) of a function f that is piecewise smooth on
n=0
[−1,1].The c n ’s are given by equation (15.4) with p(x) = 1 and ϕ n (x) = P n (x):
1

f (x)P n (x)dx
−1
c n = .
1 2
−1 P (x)dx
n
∞
The resulting series n=0 c n P n (x) for f (x) is called the Fourier-Legendre expansion of
f (x) on [−1,1], and the c n ’s are the Fourier-Legendre coefficients of f (x) on this interval.

We will look at an example of a Fourier-Legendre expansion shortly. First we will observe
that the Fourier-Legendre expansion of any polynomial q(x) can be achieved purely by algebraic
manipulation. To do this, solve for powers of x in terms of Legendre polynomials and substitute
these into q(x). To illustrate this process, let

2
q(x) =−4 + 2x + 9x .
2
We know that x = P 1 (x).Nextsolve for x in P 2 (x). Since
3 1
2
P 2 (x) = x − ,
2 2
then
2 1 2 1
2
x = P 2 (x) + = P 2 (x) + P 0 (x).
3 3 3 3
Substitute these into q(x) to obtain
q(x) =−4 + 2x + 9x 2
2 1

=−4P 0 (x) + 2P 1 (x) + 9 P 2 (x) + P 0 (x)
3 3
=−P 0 (x) + 2P 1 (x) + 6P 2 (x).
As a simple consequence of this observation, we can show that every Legendre polynomial
is orthogonal on [−1,1] to every polynomial of lower degree.

THEOREM 15.5

If q(x) is a polynomial of degree m, and n > m, then
1
q(x)P n (x)dx = 0.
−1

Proof Suppose q(x) has degree m. Write the Fourier-Legendre representation
m

q(x) = c k P k (x).
k=0

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October 14, 2010 15:20 THM/NEIL Page-525 27410_15_ch15_p505-562