Mathematical Physics Vol 1

5.9 Orthogonal and normalized functions

265

If this norm is equal to one, then the function set is orthonormal with respect to p ( x ) on the observed interval.

5.9.1 Series of orthogonal functions

A significant type of functional series is introduced by means of orthogonal function sets in a simple way. Namely, let g 1 ( x ) , g 2 ( x ) ,... , be a set of orthogonal functions on the interval a ≤ x ≤ b , and let f ( x ) be a given function that can be represented on this interval by a convergent series f ( x )= ∞ ∑ n = 1 a n g n ( x ) . (5.227) Then this series is called a generalized Fourier series 16 of the function f ( x ) , and its coefficients a 1 , a 2 ,... , are called Fourier coefficients of the function f ( x ) with respect to the given orthog onal function set. Given the orthogonality of the functions g i , the Fourier coefficients can be determined relatively easily. Multiplying the left and right side of the equality (5.227) by g m ( x ) , and then integrating from a to b (assuming that element-by-element integration is possible), we obtain

a n  

g n g m dx 

a n g n ! g m dx =

b Z a

b Z a

b Z a

∞ ∑ n = 1

∞ ∑ n = 1

∞ ∑ n = 1

 =

( f , g m )=

fg m dx =

a n ( g n , g m )

For n = m weobtain ( g m , g m )= ∥ g m ∥ g i , ( g n , g m )= 0. Thus, the formula for Fourier coefficients is 2 ,while for n̸

= m , due to the orthogonality of functions

b Z a

( f , g n ) ∥ g n ∥

1 ∥ g n ∥

a n =

f ( x ) g n ( x ) dx ,

n = 1 , 2 ,...

2 =

2

5.9.2 Completeness of orthonormal functions

In practice, orthonormal sets are often used with a "sufficient number" of functions to allow generalized Fourier series of these functions to represent broad classes of functions, for example, all continuous functions on the interval a ≤ x ≤ b . Definition A function sequence f n ( x ) is convergent with respect to norm and converges to function f if lim n → ∞ ∥ f n − f ∥ = 0 , (5.228) that is, if (omitting the square root of the norm)

b Z a

2 dx

lim n → ∞

[ f n ( x ) − f ( x )]

= 0 .

16 Fourier series will be discussed in more detail in the next chapter.