Orthogonal Function Expansion 正交函數展開 .Introduction of the Eigenfunction Expansion .Abstract Space .Function Sapce .Linear Operator and Orthogonal Function
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 y"+d(x)y'+d(x)y=O,a<x<b With the b.c's: y(a)=yb)=0 The g.s.is y(x)=cu,(x)+czu,(x)where ul,u2 are linearly index.Fucs.And C1,C2 are arb consts. Forb.c's: C4(a）+c242(a）=0 C4(b)+C242(b)=0 The condition of nontrivial sol.of cc2 to be existed if 4(a)4,(a) =0 42(b)4,(b)
Introduction -The Eigenfunction Expansion Consider the equation : y + d1 (x)y + d2 (x)y = 0,a x b y(a) = y(b) = 0 With the b.c’s : The g.s. is where u1,u2 are linearly index. Fucs. And C1 ,C2 are arb consts. y(x) c u (x) c u (x) = 1 1 + 2 2 For b.c’s : c1 u1 (a)+c2 u2 (a) = 0 c1 u1 (b) +c2 u2 (b) = 0 The condition of nontrivial sol. of c1 ,c2 to be existed if ➔ : 0 ( ) ( ) ( ) ( ) 2 2 1 1 = u b u b u a u a
The Euler Column 0,where. d'v EI The g.s. v(x)=c sin Ax+c cos Ax For the b.c.'s v(0)=v(I)=0→C2=0 And c sin Ax=0 for nontrivial solution c0 →sinl=0 i.e.2l=nm,n=1,2,3,4. Sthatwe obin the vaues元，-"T, n=1,2,3 And the correspondingeigenfucs(nontrivial sols)are v (x)=sin=(x) According the analysis,we will have v(x)=0 unless the end force P such that:2=0 ()(x)?Funetion Space
The Euler Column EI P v where dx d v + = = 2 2 2 2 0, , The g.s. v(x) c sin x c cosx = 1 + 2 For the b.c.’s : v(0) = v(l) = 0 ➔ c2 = 0 And for nontrivial solution c1 sin x = 0 c1 0 ➔ sin l = 0 i.e. l = n,n =1,2,3,4..... So that we obtain the Eigen values = ,n =1,2,3...... l n n And the corresponding eigenfucs (nontrivial sols) are v (x) sin x (x) n = n =n According the analysis, we will have unless the end force P such that: v(x) = 0 0 2 n = ( ) ( ) ? b a f x g x = ➔ Function Space
Abstract Space (N,+)為一良序牛群(N,)為良序可交換單子 (亿，+)為一良序可交換群（亿，）為良序可交換單子 Topological Space (亿，+，)為一良序可交换單子環 Metric Space (但，+，)為一有序域 (R+,)為一完備的有序域 Normed Space 連續的有序域 Inner Product Space 有序m>n,m=n,m1+n台m>n m.l>ln台m>n Hilbert Space →由R→R(有序性喪失) Banach Space 代數的原型：+·，(，÷) 完備性：每一Cauchy系列均收斂
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 m l l n m n m l l n m n m n, m n, m n + + = 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 familyX=(UiiI) of subsets of E satisfying the following axioms: (1)E∈X,0∈X (2)The union of any number of sets in X belongs to X i.e. Jmie,JcI→yU,∈X (3)The intersection of any finite number of sets in X belongs to X i.e. JcI→∩U∈X 1∈1
Topological Space Definition A topological space is a non-empty set E together with a family of subsets of E satisfying the following axioms: E X X , 0 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. i i j J I U X J J I Ui X i j finite
Metric Space Definition A metric space is a 2-tuple(,d)where X is a set and d is a metric on X,that is, a functiond: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,3)(triangle inequality)
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(X):in a metric space X=(X,d)is said to be Cauchy if for every there is an N=N(e)such that form,n>N→ d(xm,x)0 or,simply,xn→x Definition:Completeness Any Cauchy Sequence in X is convergence
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 n d x x e x is called the limit of (Xn ) and we write x x n n = → lim x x or, simply, n → for m,n>N ➔ Any Cauchy Sequence in X is convergence ; d ➔0 Definition: Completeness
Ball and Sphere Definition: Given a point oX and a real numberr>0,we define three of sets: (a)B(xo;r)=Xd(x.x)<r (Open ball) (b)B(xo;r)={x∈Xld(x,xo)≤r} (Closed ball) (c)S(xo;r）={x∈Xd(x,o)=r}(Sphere) In all three case,xo is called the center and r the radius. Furthermore,the definition immediately implies that S(xo;r)=B(xo;r)-B(xo;r)
Ball and Sphere Definition: Given a point and a real number r>0, we define three of sets: x0 X (a) (Open ball) B(x0 ;r) = x X d(x, x0 ) r (b) (Closed ball) (c) (Sphere) B(x ;r) = x X d(x, x ) r ~ 0 0 S(x0 ;r) = x X d(x, x0 ) = r ( ; ) ( ; ) ~ ( ; ) 0 0 0 S x r = B x r − B x r In all three case, x0 is called the center and r the radius. Furthermore, the definition immediately implies that
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.Asubset K of X is said to be closed if its complement(in X)is open,that is,Ke=X-K is open. 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 ofall bounded linear functionalson X becomes a normed space, which is called the dual space X'of X
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 Here a norm on a vector space X is a real-value function on X whose value at an xeX is denotedby 20 x=0台x=0 lcal lallbyl x+y≤x+y Here x and y are arbitrary vector in X and is any scalar Definition of Banach Space A Banach space is a complete normed space
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 by x X x x y x y x y x x x + + = = = 0 0 0 Here x and y are arbitrary vector in X and is any scalar A Banach space is a complete normed space.