Page 214 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 13: Orthogonality of Solutions

If, for a given orthonormal system , ..., any piecewise continuous function f, can be Notes
1 2
approximated in the mean to any desired degree of accuracy by choosing n large enough, i.e., if
n may be so chosen that the mean square error.
2
n
f C dx
1

is less than a given arbitrary small positive number, then the system of functions , ..., is said
1 2
to be complete.
For a complete or orthonormal system of functions Bessel’s inequality becomes an equality for
every function f

n
i.e. C 2 = (Nf) 2
1

n
or ( , ) 2 = f 2
f
1
The relation is known as the completeness relation or Parseval’s equation.

Definitions

Closed Set: The set { } is closed in the sense of mean convergence if for each function f of the
n
function space

f
( , n ) 2 = f 2
n 1
Complete Set: An orthonormal set { } is complete in the function space if there is no function in
n
that space, with positive norm which is to orthogonal to each of the functions.
Theorem: If an orthonormal set { (x)} is closed it is complete.
n
If an orthonormal set is closed then for each function f of the function space

( , n ) 2 = f 2 ...(i)
f
n 1
Now, let us suppose a function (x) in the space which is orthogonal to each function { (x)} of
n
the closed orthonormal set such that
0

(f, ) 0,
n
Therefore from (i), we have f 0, which is a contradiction.

Therefore there is no function in space, with positive norm which is orthogonal to each of the
functions (x).
n
Hence the closed orthonormal set { (x)} is complete also.
n

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