orthogonal function expansion 正交函數展開 introduction of the eigenfunction expansion...
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Orthogonal Function Expansion 正交函數展開
•Introduction of the Eigenfunction Expansion•Abstract Space•Function Sapce•Linear Operator and Orthogonal Function
Introduction -The Eigenfunction Expansion
Consider the equation :
bxayxdyxdy ,0)()( 21
0)()( byayWith the b.c’s :
The g.s. is where u1,u2 are linearly index. Fucs. And C1,C2 are arb consts.
(x)uc(x)ucy(x) 2211
For b.c’s :0)()( 2211 aucauc
0)()( 2211 bucbuc
The condition of nontrivial sol. of c1,c2 to be existed if :
0)()(
)()(
22
11 bubu
auau
The Euler Column
EI
Pwherev
dx
vd 22
2
2
,,0
The g.s.xcxcxv cossin)( 21
For the b.c.’s : 0)()0( lvv 02 c
And for nontrivial solution 0sin1 xc 01 c
0sin l
i.e. .....4,3,2,1, nnl
So that we obtain the Eigen values ......3,2,1, nl
nn
And the corresponding eigenfucs (nontrivial sols) are
)(sin)( xxxv nnn
According the analysis, we will have unless the end force P such that:
0)( xv
02 n
( ) ( ) ?b
af x g x Function Space
Rn
Q
Z
N
Inner Product Space
Hilbert Space
Normed Space
Metric Space
( ) ( )
( , ) ( )
( )
( )
( )
N, N,
Z Z,
Z, ,
Q, ,
R, ,
為一良序半群 為良序可交換單子為一良序可交換群 為良序可交換單子為一良序可交換單子環為一有序域為一完備的有序域
連續的有序域
nmnllm
nmnllm
nn, mn, mm
:有序
由 R→Rn( 有序性喪失 )
) , - ( , : 代數的原型
Abstract Space
完備性 : 每一 Cauchy系列均收斂
Topological Space
Banach Space
Topological Space
Definition
A topological space is a non-empty set E together with a family
of subsets of E satisfying the following axioms:
,0E X X
( )X Ui i I
(3) The intersection of any finite number of sets in X belongs to X i.e.
(1)
(2) The union of any number of sets in X belongs to X i.e.
ii j
J I U X
XUIJJ iji
finite
,
Metric Space
Definition
A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that is,
a function d : X × X → R, such that
d(x, y) ≥ 0 (non-negativity)
d(x, y) = 0 if and only if x = y (identity)
d(x, y) = d(y, x) (symmetry)
d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
Cauchy Sequence Definition: Complete space A sequence (Xn):in a metric space X=(X,d) is said to be Cauchy if for every
there is an N=N(e) such that ( , )m nd x x e
x is called the limit of (Xn) and we write
xxnn
lim
xxn or, simply,
for m,n>N
Any Cauchy Sequence in X is convergence
; d < xn, x >0
Definition: Completeness
Ball and SphereDefinition:
Given a point and a real number r>0, we define three of sets:Xx 0
rxxdXxrxB ),();( 00(a) (Open ball)
(b) (Closed ball)
(c) (Sphere)
rxxdXxrxB ),();(~
00
rxxdXxrxS ),();( 00
);();(~
);( 000 rxBrxBrxS
In all three case, x0 is called the center and r the radius.
Furthermore, the definition immediately implies that
A mapping from a normed space X into a normed space Y is called an
operator. A mapping from X into the scalar filed R or C is called a functional.
The set of all biunded linear operator from a given normed space X into a given
normed space Y can be made into a normed space, which is denoted by B(x,y).
Similarly, the set of all bounded linear functionals on X becomes a normed space,
which is called the dual space X’ of X.
Definition (Open set and closed set):
A subset M of a metric space X is said to be open if is contains a ball about each of
its points. A subset K of X is said to be closed if its complement (in X ) is open, that
is, Kc=X-K is open.
Normed Space
Definition of Normed Space
Definition of Banach Space
Here a norm on a vector space X is a real-value function on X whose value
at an is denoted byXx x
yxyx
yx
xx
x
00
0
Here x and y are arbitrary vector in X and is any scalar
A Banach space is a complete normed space.
A metric d induced by a norm on a normed space X satisfies
Lemma (Translation invariance)
(a) d(x+a,y+a)=d(x,y)
(b) ),(),( yxdyxd
For all x, y and every scalar Xa
Proof. ),()(),( yxdyxayaxayaxd
),(),( yxdyxyxyxd
Let X be the vector space of all ordered pairs of real
numbers. Show norms on X are defined by
211 x
212
22
12)( x 21 ,max
x
ppp
px
1
21 )(
),(),( 2121 yx
The sphere 1,);0( xXxrS
In a normed space X is called the unit sphere.
1
x
14x
12x
11x
Unit Sphere in LP
If a normed space X contains a sequence (en) with the property that every
there is a uniquie sequence of scalars (an) such that
Xx
naseex nn 0).......( 11
Then (en) is called basis for X. series
1kkke
Which has the sum x is then called the expansion of x
1k
kkex
A inner product space is a vector space X with an inner product define on X.
, , ,
, ,
, ,
, 0
, 0 0
x y z x z y z
x y x y
x y y x
x x
x y x
Here, the inner product <x,y> is the mapping of into the scale filed, such
that
Inner Product Space
Hilbert Space
v
Examples of finite-dimensional Hilbert spaces include
1. The real numbers with the vector dot product of and x ),( uv v u
2. The complex numbers with the vector dot product of
and the complex conjugate of .
xC ),( uv
u
,x x x
( , ) ,d x y x y x y x y
Hence inner product spaces are normed spaces, and Hilber spaces are Banach spaces.
A Hilbert space is a complete inner product space.
(Norm)
(Metric)
Euclidean space Rn
The space Rn is a Hilbert space with inner product define by
nnyx ...., 11
Where ),.....()(),.....()( 1111 nn yandx
212
12
1),.....(, nxxx
2122
112
1])(.....)[(,),( nnyxyxyxyxd
Space L2[a,b]
b
adttytxyx )()(,
212
))((b
adttxx
Hilbert sequence space l2
1
,j
jjyx With the inner product
The norm
21
2
1
21
)(,
j
jxxx
Space lp
The space lp with is not inner product space, hence not a Hilbert space 0p
Orthonormal Sets and Sequences
Orthogonality of elements plays a basis role in inner product and Hilbert spaces.
The vectors form a basis for R3, so that every has a unique representation. 3Rx
332211 eeex
13332221111, eeeex
Continuous functions
Let X be the inner product space of all real-valued continuous functions on [0,2π]
with inner product defined by
2
)()(,a
dttytxyx
An orthogonal sequence in X is (un), where
,.......1,0cos)( nntiun
,.......2,1sin)( nntivn
Another orthogonal sequence in X is (vn), where
2
0,
02
,.......2,1
0
coscos
nmif
nmif
nmif
ntdtmtuu nm
Hence an orthonormal sequence sequence is (en)
nt
u
tutete
n
nn
cos)()(
2
1)(0
From (vn) we obtain the orthonormal sequence ( ) wherene
nt
v
tvte
n
nn
sin)()(
~
Homework1. Does d(x,y)=(x-y)2 define a metric on the set of all real numbers?
2. Show that defines a metric on the set of all real
numbers.
yxyxd ),(
3. Let Show that the open interval (a,b) is an
incomplete subspace of R, whereas the closed interval [a,b] is complete.
baandRba ,
4. Prove that the eigenfunction and eigenvalue are orthogonalization and real for
the the Sturm-Lioville System.
6. For the very special case and , the self-adjoint eigenvalue
equation becomes
5. Show the following when linear second-order difference equation is expressed
in self-adjoint form:
(a) The Wronskian is equal to constant divided by the initial coefficient p
(b) A second solution is given by
)(],[ 21 xp
CyyW
x
typ
dtxCyxy
21
12 )]([)()(
0 0)( xq
0])(
)([ dx
xduxp
dx
d
Use this obtain a “second” solution of the following
(a ) Legendre’s equation
(b ) Laguerre’s equation
(c ) Hermite’s equation
Function Space(A) L2 [a,b] space:
Space of real fucs. f(x) which is define on [a,b] and square integrable i.e .
b
adxxffff )(),( 2
2In the language of vector space, we say that
“any n linearly indep vectors form a basis in E ”space”. Similarly, in function space
It is possible to choose a set of basis function such that any function, satisfying
Appropriate condition can be expressed as a linear combination to a basis in L2[a,b]
Certainly, any such set of fucs. Must have infinitely many numbers; that is, such a
L2[a,b] comprises infinitely many dimensions.
(B) Schwarz Inequality:
Given f(x), g(x) in L2[a,b], Define ),)(,(),(,,)()(),( 2 ggffgfthendxxgxfgfb
a
Proof: 0),(),(2),(),(),( 2 ffgfggaagfagfgf
),)(,(),(
0),)(,(),(2
2
ggffgf
ggffgf
(C) Linear Dependence, Independence:
Criterion: A set of fucs. )(),......(1 xx n In L2 [a,b] is linear dep.(indep.) if its
If its Gramian (G) vanishes (does not vanish), where
),.(..........).........,(
................
),().........,)(,(
),().........,)(,(
1
22212
2111
nnn
n
nn
G
The proof is the same as in linear vector space.
(D) The orthogonal System
A set of real fucs. )(),......(1 xx n …….is called an orthogonal set of fucs.
In L2[a,b] if these fucs. are define in L2[a,b] if all the integral ))(),(( xx nm
exist and are zero for all pairs of distinct
Properties of Complete System
Theorem: Let f(x), F(x) be defined on L2 [a,b] for which
n
kkk
n
n
kkk
n
CxF
cxf
1
1
lim)(
.lim)(
Then we have
b
ak
kkCcdxxFxf1
)()(
Proof:
Since f+F, and f-F are square integer able, from the completeness relation
b
ak
kk
b
ak
kk
b
ak
kk
CcdxxFxf
CcdxFf
CcdxFf
1
2
1
2
2
1
2
4)()(4
)(][
)(][
Theorem:
Every square integer able fnc. f(x) is uniquely determined (except for its value at a finite number of points) by its Fourier series.Proof:
Suppose there are two fucs. f(x),g(x) having the identical Fourier series representation
i.e.
b
a
n
kkk
n
b
a
n
kkk
n
dxxcxg
dxxcxf
0])()([lim
0])()([lim
2
1
2
1
Then using we find)(2)( 222
0)]()([(0)]()(2]))([(2
)]()())([()]()([0
222
22
b
a
b
a
b
a kkkk
b
a kkkk
b
a
xfxgdxxfccxg
dxxfccxgdxxfxg
g(x)=f(x) at the pts of continuity of the integrand
g(x) and f(x) coincide everywhere, except possibly at a finite number of pts. of
discontinuity
Proof: Since…
n
kkk
b
a
n
kkk
n
xcxg
dxxcxf
1
2
1
)()(
0])()([lim
And let We can prove f(x)=g(x) at every point.
n
kkk xcxf
1
)()(
Theorem:
An continuous fuc. f(x) which is orthogonal to all the fucs. of the complete system
must be identically zero.
Proof: Since…Assume x2>x1
dxcfdxcfdxcfdxb
a
n
kkk
x
x
n
kkk
x
x
n
k
x
x kk
111
2
1
2
1
2
1
b
a
b
a
n
kkk dxdxcf 1
2
1
Take n 0lim2
1
2
11
x
x
n
k
x
x kkn dxcfdx
Theorem:
The fourier series of every square integer able fuc. f(x) can be integrated term by term. In other words, if
......)(......)(~)( 11 xcxcxf nn
.......)(....)()(2
1
2
1
2
111
x
x nn
x
x
x
xdxxcdxxcdxxf Then
Where x1,x2 are any points on the inteval [a,b]
The Sturm-Liouville Problem
)()()()()()()( 22
2
12
2
0 xuxpxudx
dxPxu
dx
dxpxLu
Self-adjoint Operator
b
a
b
adxupupupudxxLuxuLuuuLu 210)()(
For a linear operator L the analog of a quadratic form for a matrix is the integral
Because of the analogy with the transposed matrix, it is convenient to define the
linear operator
b
a ob
ax udxupupdx
dup
dx
dxuppxuLuuuLu 212
2
01 ][][)]())(([
Comparing the integrands
0)(2)( 1010 uppuuppu
)()( 10 xPxp
upppdx
dupp
dx
udpupup
dx
dup
dx
duL )()2(][][ 210102
2
02102
2
)()(])(
)([ xuxqdx
xduxp
dx
dLuuL
The operator L is said to be self-adjoint.
As the adjoint operator L. The necessary and sufficient condition that LL
The Sturm-Liouville Boundary Value Problem
A differential equation defined on the interval having the form of
and the boundary conditions
is called as Sturm-Liouville boundary value problem or Sturm-Liouville system,
where , ; the weighting function r(x)>0 are given functions; a1 , a2 ,
b1 , b2 are given constants; and the eigenvalue is an unspecified parameter.
0)]()([)(
yxwxqdx
dyxp
dx
d
0)(')(
0)(')(
21
21
bybbyb
ayaaya
bxa
The Regular Sturm-Liouville Equation
It is a special kind of boundary value problem which consists of a second-order
homogeneous linear differential equation and linear homogeneous boundary
conditions of the form
0)()()(
yxwxqdx
dyxp
dx
d
)()()()( xyxqdx
dyxp
dx
dyL
where the p, q and r are real and continuous functions such that p has a
continuous derivative, and p(x) > 0, r(x) > 0 for all x on a real interval a x
b;
and is a parameter independent of x. L is the linear homogeneous differential
operator defined by L(y) = [p(x)y´]´+q(x)y.And two supplementary
boundary conditions
where A1 , A2 , B1 and B2 are real constants such that A1 and A2 not both zero
and B1 and B2 are not both zero.
A1y(a)+A2y´(a) = 0 B1y(b)+B2y´(b) = 0 .
Definition 1.1 : Consider the Sturm-Liouville problem consisting of the differ
entail equation and supplementary conditions. The value of the parameter
in for which there exists nontrivial solution of the problem is called
the eigenvalue of the problem. The corresponding nontrivial solution
is called the eigenfunction of the problem. The Sturm-Liouville problem is also
called an eigenvalue problem.
The Nonhomogeneous Sturm-Liouville Problems
And as in regular Sturm-Liouville problems we assume that p, p, q, and r are
continuous on a x b and p(x) > 0, r(x) > 0 there.We solve the problem
by making use of the eigenfunctions of the corresponding homogeneous problem
consisting of the differential equation
Consider boundary value problem consisting of the nonhomogeneous
differential equation
L[y] = - [p(x)y´]´+q(x)y = w(x)y+f(x),
where is a given constant and f is a given function on a a x b
and the boundary conditions
A1y(a)+A2y´(a)=0 B1y(b)+B2y´(b)=0 .
The Bessel's Differential Equation
In the Sturm-Liouville Boundary Value Problem, there is an important
special case called Bessel's Differential Equation which arises in numerous
problems, especially in polar and cylindrical coordinates. Bessel's Differential
Equation is defined as:
.
where is a non-negative real number. The solutions of this equation are
called Bessel Functions of order n . Although the order n can be any real number,
the scope of this section is limited to non-negative integers, i.e., ,
unless specified otherwise. Since Bessel's differential equation is a second
order ordinary differential equation, two sets of functions, the Bessel function
of the first kind Jn(x) and the Bessel function of the second kind
(also known as the Weber Function) Yn(x) , are needed to form the general solutio
n:
0)( 222 ynxyxyx
)()()( 21 xYcxJcxy nn
Five Approaches
The Bessel functions are introduced here by means of a generating function.
Other approaches are possible. Listing the various possibilities, we have .
1. Gram-Schmidt Orthogonalization
22i
b
a i Nwdx We now demand that each solution be multiplied by i
1iN
ijj
b
a i
b
a i
dxxwxx
dxxwx
)()()(
1)()(2
)()( 00 xux We star with n=0, letting
212
0
00
][
)()(
dxw
xx
Then normalize
The presence of the new un(x) will guarantee linear independence.
Fro n=1, let )()()( 01011 xaxux
This demand of orthogonality leads to
020100101 wdxawdxuwdx
As is normalized to unity, we have 0
wdxua 0110
Fixing the value of a10. Normalizing, we have
212
1
11
)(
)()(
dxw
xx
We demand that be orthogonal to)(1 x )(0 x
212
1
11
))()((
)()(
dxxwx
xx
Where 1111001 ........)( iiiiii aaaux
The equation can be replaced by
212
1 )(
)()(
dxw
xNx i
ii
2j
ii
ijN
wdxua
)(})()()({)( xdttwttuxup jjiij
)(}1{)(1
1
xuPx i
i
jji
And aij becomes
The coefficients aij are given by wdxua jiij
If some order normalization is selected
b
a jj Ndxxwx 22 )()]([
Orthogonal polynomial Generated by Gram-Schmidt Orthogonalization of
........2,1,0,)( nXxu nn
2. Series solution of Bessel’s differential equation
0)( 222 ynxyxyx
Using y’ for dy/dx and for d2y/dx2 . Again, assuming a solution of the form
0
)(
kxaxy
Inserting these coefficients in our assumed series solution, we have
.......])!2(!22
!
)1(!12
!1[)(
4
2
2
2
0
n
xn
n
xnxaxy n
Inserting these coefficients in our assumed series solution, we have
jn
j
jn x
jnjnaxy 2
00 )
2(
)!(!
1)1(!2)(
With the result that…..
)()1()( xJxJ nn
n
y
3. Generating function
)1
)(2
(
),( tt
x
etxg
Expanding this function in a Laurent series, we obtain
n
nn txJe t
tx
)()
1)(
2(
It is instructive to compare.
The coefficient of tn, Jn(x), is defined to be Bessel function of the first kind of
integral order n. Expanding the exponential, we have a product of Maclaurin
series in xt/2 and –x/2t, respectively.
!)
2()1(
!)
2(
00
22
s
tx
r
txee
ss
s
s
rr
r
t
xxt
The coefficient tn is then
......)!1(2!2
)2
()!(!
)1()(
2
22
0
n
x
n
xx
snsxJ
n
n
n
nsn
s
s
n
For a given s we get tn(n>=0) from r=n+s;
!)
2()1(
)!()
2(
s
tx
sn
tx sss
snsn
Bessel function J0(x), J1(x) and J2(x)
4. Contour integral: Some writers prefer to start with contour integral
definitions of the Hankel function, and develop the Bessel function Jv(x) from
the Hankel functions.
The integral representation
1
)1)(2(
2
1)(
v
ttx
v t
dte
ixJ
)1
)(2
(
),( tt
x
etxg
(Schlsefli integral)
may easily be established as a Cauchy integral for v=n, that is , an integer.
[Recognizing that the numerator is the generating function and integrating
around the origin]
Cut line
ie
vt
tx
v t
dte
ixH
0 1
)1)(2()1( 1)(
0
1
)1)(2()2( 1)(
ie vt
tx
v t
dte
ixH
)]()([2
1)( )2()1( xHxHxJ vvv
)]()([2
1)( )2()1( xHxH
ixN vvv
5. Direct solution of physical problems, Fraunhofer diffraction with a circular
aperture illusterates this. Incidentally, can be treated by series expansion if
desired. Feynman develop Bessel function from a consideration of cavity
resonators.
rdrdea ibr
0
2
0cos~
sin2
b
a
rdrbrJ0 0 )(2~
The parameter B us given by
In the theory of diffraction through a circular aperture we encounter the integral
Feynman develop Bessel function
from a consideration of cavity
resonators. (Homework 1)
)sin2
(sin
~)(2
~ 112
a
Ja
abJb
ab
212 }sin
]sin)2[({~
aJ
.....8317.3sin2
a
The intensity of the light in the diffraction pattern is proportional to Ф2 and
Fro green light Hence, if a=0.5 cm cm5105.5
..14)(5107.6sin arcofsradian
(2) Using only the generating function
n
nn txJe t
tx
)()
1)(
2(
Explicit series form Jn(x), shoe that Jn(x) has odd or even parity according
to whether n is odd or even, this
)()1()( xJxJ nn
n
(3) Show by direct differentiation that
s
s
s
v
x
ssxJ 2
0
)2
()!(!
)1()(
Satisfies the two recurrence relations
)(2)()(
)(2
)()(
11
11
xJxJxJ
xJx
xJxJ
vvv
vvv
And bessel‘s differential equation
0)()()()( 222 xJxxJxxJx vvv
Homework
(4) Show that
Thus generating modified Bessel function In(x)
n
nn txIe t
tx
)()
1)(
2(
(5) The chebyshev polynomials (typeII) are generated,
n
nn txU
txt
0
2)(
21
1
Using the techniques for transforming series, develop a series representation of
Un(x)
PS:請參考補充講義