Page 213 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes are called the expansion coefficients of f(x) with respect to the given orthonormal system.
2
n
Obviously f C dx 0 ...(ii)
1

By writing out the square and integrating term by term, we get
n n
2
0 f dx 2 C . f dx C 2
1 1

n n
or 0 (Nf ) 2 2 C 2 C 2 [Nf means norm of f]
1 1

n
or 0 (Nf ) 2 C 2
1
n
or C 2 (Nf) 2 ...(iii)
1
Since the number on right is Independent of n, it follows that

n
C 2 < (Nf) 2.
1
This fundamental inequality is known as Bessel’s inequality and is true for every orthonormal
system. It proves that the sum of the squares of the expansion coefficients always converges.

For systems of functions with complex values the corresponding relation is
n
2 2
f
f
C (Nf ) ( , ) ...(iv)
1
f
holds, where C is the expansion coefficient C ( , ) .
This relation may be obtained from the inequality
2
n n
x
f ( ) C dx = (Nf ) 2 C 2 0
1 1
The significance of the integral in (ii) is that it occurs in the problem of approximating the given
n
function f(x) by a linear combination with as constant coefficient and fixed n, in such
1
a way that the mean square error
2
n
M = f dx
1

is as small as possible.
An approximation of this type is known as an approximation by the method of least squares, or
an approximation in the mean.

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