Page 196 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 11: Sturm–Liouville’s Boundary Value Problems

Ν Notes
2 2
x
x
R N ( ) f ( ) A f ( ), ( )
x
x

f
i
i
i 0
Ν Ν
*
*
x
A f ( ), ( ) , A A f ( ), ( )
x

x
f
x
f
i i i i i i
i 0 i 0
Ν
2 * * *
f ( ) { A a A a A A }
x
i i i i i i
i 0
Ν
2 2 2
a
x
f ( ) {|A i a i | | | }
i
i 0
The error is therefore smallest when A = a for i = 0, 1, ...., N, so the most accurate approximation
i i
is formed by simply truncating the series (14) after N terms. In addition, since the norm of R (x)
N
is positive,
Ν
2 b 2
x
a i f ( ) dx
a
i 0
As the right side of this is independent of N if follows that
1
b
2 2
x
a f ( ) dx
i ...(16)
i 0 a
which is Bessel’s inequality. This shows that the sum of the squares of the expansion coefficients
converges. Approximations by the method of least squares are often referred to as approximations
in the mean, because of the way the error is minimized.
If, for a given orthonormal system, (x), (x)..., any piecewise continuous function can be
1 2
approximated in the mean to any desired degree of accuracy by choosing N large enough, then
the orthonormal system is said to be complete. For complete orthonormal systems, R (x) 0 as
N
N , so that Bessel’s inequality becomes an equality,
b
2 2
x
a i f ( ) dx
a ...(17)
i 0
for every function f(x).
The completeness of orthonormal systems as expressed by
2
b N
lim f ( ) a ( ) dx 0
x
x
N a i i
i 0
x
does not necessarily imply that f(x) = a i i ( ) , in other words that f(x) has an expansion in
i 0
x
terms of the (x). If however, the series a i i ( ) , is uniformly convergent, then the limit
i i 0
x
and the integral can be interchanged, the expansion is valid, and we say that a i i ( ) ,
i 0
converges in the mean to f(x). The completeness of the systems (x), (x).... , should be seen as
1 2
a necessary condition for the validity of the expansion, but, for an arbitrary function f(x), the
question of convergence requires a more detailed investigation.
The Legendre polynomials P (x), P (x),... on the interval ( 1, 1) and the Bessel functions J ( x),
0 1 t
J ( x),... on the interval [0, a] are both examples of complete orthogonal systems (they can easily
2
be made orthonormal), and the expansions of unit 1 to 5 are special cases of the more general
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