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15.1 Eigenfunction Expansions 515

1

0.8

0.6

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
x
FIGURE 15.2 Eigenfunction expansion in
Example 15.4.

15.1.1 Bessel’s Inequality and Parseval’s Theorem

Given a Sturm-Liouville problem on [a,b], we have seen that the eigenfunctions are
orthogonal with respect to the weight function p. To carry the analogy with vectors a little
further, define the weighted dot product of functions f and g to be
b
f · g = p(x) f (x)g(x)dx.
a

We may also denote
b
2
2
f · f = p(x) f (x) dx = f .
a
As with vectors, define f and g to be orthogonal when f · g = 0.

We call f the length or norm of f , and we interpret

f − g = ( f − g) · ( f − g)

b
2
= p(x)( f (x) − g(x)) dx
a
to be the distance between f and g.

Now let ϕ n be eigenfunctions of a Sturm-Liouville problem, with eigenvalues λ n . In the dot
product notation, the coefficients of an expansion of f in a series of these eigenfunctions has
coefficients

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October 14, 2010 15:20 THM/NEIL Page-515 27410_15_ch15_p505-562