Session 5
Characteristic Function Representations
Ernesto Gutierrez-Miravete
Fall 2002
1 Expansion of Arbitrary Functions in Series of Orthogonal Functions
Assume that an arbitrary function f(x) can be represented as a linear
combination of characteristic functions forming an orthogonal set with
respect to a certain weighting function r(x), {fn(x)}
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| f(x) = |
¥
å
n = 0
|
An fn(x) |
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| |
|
Multiplication of the above by r(x) fk(x), followed by integration
from x = a to x = b and the orthogonality of the eigenfunctions leads to
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| An = |
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|
ó
õ
|
b
a
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r(x) f(x) fn(x) dx |
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which determines the formal series representation of f(x).
If the associated Sturm-Liouville problem is proper and p(x), q(x)
and r(x) are regular in (a,b), the formal representation
given above of any piecewise differentiable function f(x) converges
to f(x) inside (a,b) wherever f(x) is continuous and
converges to the mean value whrever finite jumps occur.
Furthermore, if f(x) is continuous and its derivative is
piecewise differentiable, the term by term derivative of the
series converges to f¢(x) wherever the derivative is
continuous.
Note the great generality of series representation in terms
of eigenfunctions in comparison with power series expansions.
While Taylor series representations require the existence of
derivatives of all orders and even in such case the representation may
not exist, this is not the case at all with Fourier series.
The characteristic function representation can be operated upon by the
operator L = [d/dx](p(x) [dy/dx]) + q to give
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| L f(x) » |
¥
å
n = 0
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An L fn(x) = - r(x) |
¥
å
n = 0
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ln An fn(x) |
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and if L f(x)/r(x) is piecewise differentiable
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L f(x)
r(x)
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= |
¥
å
n = 0
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Bn fn(x) |
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with Bn = - ln An. Therefore, in these
conditions the operator can by applied term by term to the
series.
2 Boundary Value Problems involving Nonhomogeneous Equations
Consider the nonhomogeneous differential equation
and its associated homogeneous problem when h(x) = 0 with
corresponding eigenfunctions fn(x) and eigenvalues
ln with the series representation of its solution
y = åan fn
Since L fn = - ln r fn,
substitution into the original ODE gives
r å(L - ln) an fn = h, i.e.
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| |
h(x)
r(x)
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= f(x) = |
å
| (L - ln) anfn(x) = |
å
| An f(x) |
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where h/r is assumed to be piecewise differentiable.
The solution of the nonhomogeneous problem is then
3 Fourier Series
The characteristic functions fn(x) = sin( n px/l) are
eigenfunctions of the problem
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d2 y
dx2
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+ ly = 0, y(0) = 0, y(l) = 0 |
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with the characteristic numbers ln = n2 p2/l2.
If a function f(x) can be represented as
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| f(x) = |
¥
å
n = 1
|
An fn(x) = |
¥
å
n = 1
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An sin( |
n px
l
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) |
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|
in x Î [0,l] then, because of orthogonality,
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| An = |
2
l
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|
ó
õ
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l
0
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f(x) sin( |
n px
l
|
) dx |
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The series is the Fourier sine series representation of f(x)
in terms of individual harmonics
and An are called the Fourier coefficients and are equal
to twice the average value of f sin(n px/l) in [0,l].
Since the sign of each harmonic is reversed when x is replaced
by -x, the Fourier sine series is a representation of an
odd function f(x) (i.e. one for which f(-x) = - f(x))
in the interval [-l,l]. If f(x) is odd and
periodic, the series represents f(x) everywhere.
If instead the following BVP is considered
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d2 y
dx2
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+ ly = 0, y¢(0) = 0, y¢(l) = 0 |
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with characteristic functions
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| fn(x) = cos( |
n px
l
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), (n = 0,1,2,...) |
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and eigenvalues ln = n2 p2/l2 and
this is used to produce a (Fourier cosine series) representation
of the function f(x)
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| f(x) = A0 + |
¥
å
n = 1
|
An cos( |
n px
l
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) |
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the corresponding Fourier coefficients are
and
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| An = |
2
l
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ó
õ
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l
0
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f(x) cos( |
n px
l
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) dx, (n = 1,2,3,...) |
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The Fourier cosine series is a representation of an
even function f(x) (i.e. one for which f(-x) = f(x))
in the interval [-l,l]. If f(x) is even and
periodic, the series represents f(x) everywhere.
Since all harmonics are even and periodic, they constitute
convergent representations to piecewise differentiable even
functions f(x) everywhere.
Since any function f(x) can be expressed as the sum of an even
fe(x) and an odd function fo(x) with
|
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| fe(x) = A0 + |
¥
å
n = 1
|
An cos( |
n px
l
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) |
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| |
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|
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| fo(x) = |
¥
å
n = 1
|
Bn sin( |
n px
l
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) |
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such that , over the interval [-l,l],
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| f(x) = A0 + |
¥
å
n = 1
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( An cos( |
n px
l
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) + Bn sin( |
n px
l
|
) ) |
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where
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| An = |
1
l
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|
ó
õ
|
l
-l
|
f(x) cos( |
n px
l
|
) dx |
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and
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| Bn = |
1
l
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|
ó
õ
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l
-l
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f(x) sin( |
n px
l
|
) dx |
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The series representation of f(x) is called the complete
Fourier sine-cosine series representation.
This can be expressed in more compact form by using
Euler's formula:
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| f(x) = |
¥
å
k = -¥
|
Ck ei k x |
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where
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| Ck = |
1
2 p
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|
ó
õ
|
l
-l
|
f(x) e-ikx dx = |
1
2
|
Ak - |
i
2
|
Bk |
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The complete Fourier series representation of f(x) over
any interval [a,a+P] is easily obtained.
Fourier series representations of f(x) can be differentiated
term by term and the result becomes a convergent representation of
f¢(x) as long as f¢ is piecewise differentiable.
4 Fourier-Bessel Series
The BVP
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d
dx
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(x |
dy
dx
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) + (- |
p2
x
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+ m2 x) y = 0 |
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is of Sturm-Liouville form with
p(x) = x, q(x0 = - p2/x, r(x0 = x and l = m2.
Its general solution is of the form y(x) = Zp(mx) where
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| Zp(mx) = c1 Jp(mx) + c2 J-p(mx) |
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when p is not integer and
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| Zp(mx) = c1 Jp(mx) + c2 Yp(mx) |
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when p is integer. If at x = 0 y is finite and x y¢ = 0
and at x = l, y(l) = 0, the eigenvalues m are the
roots of Jp(mn l) = Jp(an) = 0.
If instead at x = l, y¢(l) = 0, the ms are the roots
of Jp(mn l) = Jp(an) = 0.
In any case the characteristic functions are
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| fn(x) = Jp(mn x) = Jp(an |
x
l
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) |
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which are orthogonal on [0,l] with respect to r(x) = x.
So, excluding the exceptional case p = 0, the
representation
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| f(x) = |
¥
å
n = 1
|
An Jp(mn x) |
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with
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| An = |
1
Cn
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ó
õ
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l
0
|
x f(x) Jp(mn x) dx |
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and
if y(l) = 0, and
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| Cn = |
mn2 l2 - p2 + k2
2 mn2
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[Jp(mn l)]2 |
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if y¢(l) = 0.
The integral involved in the calculation of An can be readily
obtained by numerical methods.
5 Legendre Series
Legendre's differential equation
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d
dx
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[(1-x2) |
dy
dx
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] + p (p+1) y = 0 |
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has the form of the differential equation of a Sturm-Liouville
problem with p(x) = 1 - x2, q(x0 = 0, r(x) = 1 and
l = p (p+1). Finite solutions at x = ±1
require p = 0, 1, 2, ... and are
where Pn(x) are the Legendre polynomials. Since these are
orthogonal, any piecewise differentiable function in the
interval (-1,1) can be
represented as
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| f(x) = |
¥
å
n = 0
|
An Pn(x) |
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where
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| An = |
|
= |
ì
ï
ï
ï
ï
í
ï
ï
ï
ï
î
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2n+1
2
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ó
õ
|
1
-1
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f(x) Pn(x) dx |
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2n+1
2n+1 n!
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ó
õ
|
1
-1
|
(1-x2)n |
dn f(x)
dxn
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dx |
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Because of the properties of Pn, if f(x) is even,
An = 0 if n is odd and viceversa.
Any polynomial of degree k can be expressed as a linear combination
of the first k+1 Legendre polynomials.
6 Fourier Integral
Consider the expression ò0¥ A(u) sin(u x) du where
u is any positive number. Could this be a representation
of a well behaved function f(x) for 0 < x < ¥?
Assume that is the case, therefore,
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| f(x) = |
ó
õ
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¥
0
|
A(u) sin( u x) du |
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Multiplying both sides by sin(u0 x), integrating from 0 to l,
taking the limit as l ® ¥ and rearranging gives
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| A(u) = |
2
p
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|
ó
õ
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¥
0
|
f(x) sin(u x) dx |
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Switching the dummy integration variable from x to t and
substituting in the original representation
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| f(x) = |
2
p
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ó
õ
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¥
0
|
sin(ux) |
ó
õ
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¥
0
|
f(t) sin(ut) dt du |
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This is the Fourier sine integral representation of f(x) and is valid
in "x if f is odd, piecewise differentiable and
ò0¥ |f(x)| dx exists. Similarly, the
Fourier cosine integral representation can be defined as
|
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| f(x) = |
2
p
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|
ó
õ
|
¥
0
|
cos(ux) |
ó
õ
|
¥
0
|
f(t) cos(ut) dt du |
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valid "x if f is even. For a function f(x) = fe(x) + fo(x),
the complete Fourier integral representation in -¥ < x < ¥
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| f(x) = |
ó
õ
|
¥
0
|
[A(u) cos(ux) + B(u) sin(ux)]du |
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with
|
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| A(u) = |
1
p
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|
ó
õ
|
¥
0
|
f(t) cos(u t) dt |
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and
|
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| B(u) = |
1
p
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|
ó
õ
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¥
0
|
f(t) sin(u t) dt |
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Substituting and using complex notation
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| f(x) = |
1
2 p
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|
ó
õ
|
¥
-¥
|
eiux |
ó
õ
|
¥
-¥
|
e-iut f(t) dt du |
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The integral ò-¥¥ e-ixt f(x) dx is called
the Fourier transform [`f(u)] of f. The inverse transform is
|
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| f(x) = |
1
2 p
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|
ó
õ
|
¥
-¥
|
eiux |
f(u)
|
du |
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|
Fourier sine and cosine integral transforms can also be defined as
|
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| fS(x) = |
ó
õ
|
¥
0
|
f(x) sin(ux) dx |
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|
with
|
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| f(x) = |
2
p
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|
ó
õ
|
¥
0
|
fS(u) sin(ux) du |
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and
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| fC(x) = |
ó
õ
|
¥
0
|
f(x) cos(ux) dx |
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|
with
|
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| f(x) = |
2
p
|
|
ó
õ
|
¥
0
|
fC(u) cos(ux) du |
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File translated from TEX by TTH, version 2.34.
On 8 Oct 2002, 17:08.