Page 43 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

ORTHOGONAL SETS OF FUNCTIONS 33
If we now form the inner product of m with both sides of Eq. (3.12) and use the
definition of an orthonormal function set as stated in Eq. (3.11) we see that the inner product
of f (x)and n (x)is c n .
b b

2
c n (ξ)dξ = c n = f (ξ) n (ξ)dξ (3.13)
n
x=a x=a
In particular, consider a set of functions n that are orthogonal on the interval (a, b)sothat

b

n (ξ) m (ξ)dξ = 0, m = n
x=a (3.14)
2
= n , m = n
2 b 2
where n = (ξ)dξ is called the square of the norm of n . The functions
x=a n
n
= n (3.15)
n
then form an orthonormal set. We now show how to form the series representation of the
function f (x) as a series expansion in terms of the orthogonal (but not orthonormal) set of
functions n (x).

b b
∞ ∞
n n (ξ) n (ξ)
f (x) = f (ξ) dξ = n f (ξ) dξ (3.16)
n n n 2
n=0 n=0
ξ=a ξ=a
This is called a Fourier series representation of the function f (x).
As a concrete example, the square of the norm of the sine function on the interval
(0,π)is

π
π

2 2
sin(nx) = sin (nξ)dξ = (3.17)
2
ξ=0
so that the corresponding orthonormal function is

2
= sin(nx) (3.18)
π
A function can be represented by a series of sine functions on the interval (0,π)as
π
∞
sin(nς)
f (x) = sin(nx) f (ς)dς (3.19)
π
n=0 2
ς=0
This is a Fourier sine series.