A Class of Functional Equation and Fractal

7/29/2019 A Class of Functional Equation and Fractal

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A p p l . M a t h . - J C U14B(1999) ,90-98

A C L A S S O F F U N C T I O N A L E Q U A T I O N A N D F R A C T A LI N T E R P O L A T I O N F U N C T I O N S

Sha ZhenAbstract. A new class of functional equation in Co (I) is investigated. It is proved that someclass of FIF satisfies the functional equation. Another functional equation is constructed. Theirsolutions can approximate FIF arbitrarily. And a new approximate estimate between FIF and

e

interpolated function is given.

w 1 I n t r o d u c t i o n

First we introduce some notations. Let 0= x0 ~x ~. .. ~x N= l be a part i t ion of theinterval I= [0 ,1 ] ,I j= [-x~-i ,xj] ,) = 1,2 . . . . . N, and K = I X [a ,b ] , where - -oo


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No. 1 S h a Z h e n F R A C T A L I N T E R P O L A T I O N F U N C T I O N 91 ,

! W j ( x , y ) , j = 1 , 2 . . . . . N ,{ ( x , , y j ) I J 0 , 1 . . . . . N . }.H e r e i s t h e o u t l in e o f t h i s p a p e r . I n w 2 a n d w 3 , w e i n t r o d u c e a n d s t u d y a n e w f u n c -

t i o n a l e q u a t i o n :L ( r = ~ - ] ~ a j (o ~ ) L ( r -4- f ( r ( i . 4 )

i=1I n w 4 , w e w i ll n o t e t h a t a k i n d o f F I F s a t i s f y th e f u n c t i o n a l e q u a t i o n ( ~ . 4 ) , a n d f r o m ( 1 .4 ) w e c o n s t r u c t a n o t h e r f u n c t i o n a l e q u a t i o n s c a ll e d t r u n c a t e d e q u a t i o n s . T h e i r s o l u t i o n sa r e a l so t o a p p r o x i m a t e F I F a r b i t ra r i l y . I n S e c t io n 5 , w e w i l l g iv e a n e w a p p r o x i m a t e e s t i-m a t e b e t w e e n F I F a n d i n te r p o l a t e d f u n c t i o n .

w 2 A C la s s o f F u n c t i o n a l E q u a t i o nF i r s t w e i n tr o d u c e s o m e n o t a t io n s . D e f i ne C o d e s p a c e ~ :

g ] = { ~ = ( i t , i z . . . . . ik . . . . ) [ i, E ( 0 , 1 . . . . . N - 1 ) } .f o r o ~ = ( i l , iz . . . . ) a n d ~ o-- --( ;~ ~2 . . . . ) E g2 , w e i n t r o d u c e t h e d i s t a n c e f u n c t i o n

I o , - ? o l = ~ , , IN *h = lT h e n (2 i s a c o m p a c t m e t r ic s p ac e . F o r w = ( i l , i z . . . . . ik . . . . ) E g 2 , i f i k = O , k ~ n + l , w ew i ll d e n o t e i t b y ( i~ . . . . i . , 0 ) , s i m i l a r l y f o r i ~ = N - - l , k ~ n + l , b y ( i t . . . . . i , , , N - - 1 ) . L e ta '2 " d e n o t e t h e s u b s e t o f I" 2:

O " = { ( it . . . . . i . , 0 ) o r ( i~ . . . . . i . , N - - 1 ) l n ~ 1 , i , E ( 0 , 1 . . . . . N - - 1 )} .T h e S h i f t m a p a : O - - ~ g ' 2 i s d e f i n e d b y

a w = ( i2 , i~ . . . . ) , i f ~o - -- - ( i l , i2 , i s . . . . ) .S o w e h a v e ~ w = ( i t + t , i ; , + z . . . . ) . W e de f in e m a p r

r N .h = lB y t h e e r g o d i c t h e o r y , f o r a l m o s t e v e r y p o i n t x E [ - 0 ,1 ) ( c o r r e s p o n d i n g t o oJ) w e h a v e

n--1

n i=0w h e r e X~ d e n o t e s t h e c h a r a c t e r i s t i c f u n c t i o n o f E a n d r e ( E ) d e n o t e s t h e L e b e s g u e m e a s u r eo f E . T h u s a l m o s t a l l s e t {tb.,~}7~ i s d e n s e e v e r y w h e r e i n l - 0 , 1 ] .

L e t C 0 [ - 0 , 1 ] = { f E C [ - 0 , 1 ] I f ( 0 ) = f ( 1 ) = 0 } , Ilfll= m a x I f ( x ) I . I t i s n o t d i f f ic u l t t oxE[O,l]prDve the following lemma.L e m m a 2 . 1 . L e t

V ( r = ~ , a j L ( r -4- f ( t b ~ . ) ,] = 1

w h e r e L , f E C o ~ O , 1 ] a n d a y i s r e a l c o n s t a n t . I f ~ ] l a i l < o o , t h e n V i s a s i n g l e v a l u e dj= l


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92 A p p l . M a t h . - J C U Vol. 14,S et . Bf u n c t i o n o n [ 0 , 1 - ] , a n d V E C o [ - O , 1 -] .N o w w e c o n s i d e r th e f o l l o w i n g f u n c t io n a l e q u a t i o n s i n C 0 [ 0 ,1 - ] :

L ( ib ~ ) = ~ - ] a j L ( r + f Q b . ) , f E C o l 0 , 1 -] .i = 1S e t

T h e o r e m 2 . 1 .t i o n L .

( 2 . 1 )

f [ i l . . . . . i .+ ~ ] : ?T h i s l e a d s t o ( 2 . 4 ) .S e t

'AI "= a l ,A z = a l A l + a z , (2 . 2 )

9 ~ .. A m - i = a lA m - 2 + a 2A = - ~ - t - . . . + a = - l ,

I f ~ lajl


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No. 1 Sha Zhen FRACT AL INTERPOLATIO N FUNCTION 93By assumption, it is easy to see that I~ -~ 0, j = 1, 2, 3, when m--~c~. Thus (2. 3) is

proved.On the other ha nd, it is easy to verify that t he function L, defined by ( 2. 3) , is really

a solution of equation (2. 1). The theorem has been comple tely shown.Rema rk. Note th at , in the Theorem 2. 1, two conditions ~ [a~ I' ~~ 17 6nd ~ IAj [< oo

j = l j =lare independent with respect to each other in general.Corollary 1. Let aj =( -- 1) J- ~d ~, Id l< l , then equation (2. 1) has a unique solution:

L(~b.) = f(~b.) + df(~ b,. ). (2 .5 )Corol lary 2. Let a i = - - d j , I dl < l, then equation (2.1) has a unique solution:

L(~b,) = f(~b,) -- df(~b,. ). (2 .6)Now we discuss the other case in which the coeff icients {a~} in (2. 1) are relat ive to co.Given {d,}o -a , I & l < l , and o ~ = ( ia , iz . . . . ) E l ] . S e t

d ~ i ) - - dq 9 dq . . . d i .We consider the following functional equation in C0[0, 1]:

L ( ~ b ,) = ~ d , ~ j ~ L ( ~ b g ~ ) + f G b , ) . (2.7)j ~ l

Using the similar method, we have (the details are omitted).T h e o r e m 2 . 2 .

T h e o r e m 2 . 3 .

If d =O ~ i ~ N - - I

k = lIf [d~]%1,i =0,1 . . . . . N- - l , then equation

L ( ~ b. ) = - - ~ d . ( i ) L ( ~ b . , ~ ) + f ( ~ b ~ ),j =l

has a unique solution:

T h e o r e m 2 . 4 . I f d = o


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94 A p p l . M a t h . - J C U Vol. 14,Ser. Ba n d

I L ( r - - L ( r ~ K ~ ? O + K ~ 2 k _ l d ,. N ~"1 k~. + 2IISH 9 2 k - ' d k.k = l k = m + l

N o t i c e t h a t 2 d = N - ~ a n d u s i n g t h e s i m i l a r m e t h o d a s s h o w e d i n [ - 2 ] w e c a n c o m p l e t et h e p r o o f o f th e t h e o r e m .

N e x t w e c o n s i d e r m o r e g e n e r a l f u n c t i o n a l e q u a t i o n a s f o ll o w sL (r = ~ a j ( o a ) f ( r + f ( r ( 2 . 1 2 )

j = lS u p p o s e t h a t f o r a n y j , t h e a j (o a ) is a s i n g l e v a l u e d f u n c t i o n o n ~ , a n d a j ( c o ) = a 1 ( ~ o ) f o ra n y o a = ( ij . . . . . i . , D ) , ~ o = ( i 1 . . . . i . - - 1 , N - - 1 ) E ~ ' . S e t t in g

"A1 (oJ) = al (oD ,~ - 1 ( 2 . 1 3 )

A~(o~) = ~] ai ( o~ )A . ~(o~w) + a . ,(oo) , m > 1 ,i ~l

W e h a v e th e f o l l o w i n g th e o r e m .T h e o r e m 2 . 5 . L e t a i ( w ) E C ( ~ ) , V j a n d f 0 f f C 0 [ - 0 , 1 ] . B o t h s e ri es ~ l a~ ( o~ ) l a n d

/= 1

~2] ] A j ( w ) [ a r e u n i f o r m l y c o n v e r g e n t o n J ~ , t h e n t h e e q u a t i o n ( 2 . 1 2 ) h a s t h e f o l l o w i n g) - - 1

s o l u t i o n i n C 0 [ - 0 ,1 ] :L ( r = f ( r + ~ A i ( w ) f ( r 1 6 2 ( 2 .1 4 )

9 j=l

w 3 E x a m p l eA s a n e x a m p l e w e d i s c u s s a s p e c i a l e q u a t i o n :

L ( ~ & ) = d L ( G ~ ) -+- dZL(r ~ f ( r f E C 0 [ O , l ] , d ~ O,W e a s s e r t t h a t t h e f o l l o w i n g th r e e s t a t e m e n t s a r e t r u e :

( 1 ) I f O ( d ~ " r t h e n t h e e q u a t io n ( 3 . 1 ) h a s a u n i q u e s o l u t i o n :2 '

( 2 ) If O ~ d ~ - -

w h e r e

a n d f E L i p a , t h e n w e h av e2( 0 ( 6 ~ ] lo g 3 1 ) , i f a = fl

~ o ( L ; 3 ) = ~O ( 3 r ) , i : f a :/~ f l, ) ' = m i n ( a , f l ) ,

f l = lo g/f d . 1 -+- 4 ~ - ] ' / / l o g ( 1 / N ) .2( 3 ) S u p p o s e t h a t t h e f u n c t i o n f ( x ) i n ( 3 . 1 ) is a n " h a t " f u n c t io n

( 3 . 1 )

( 3 . 2 )

( 3 . 3 )

( 3 . 4 )


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No. 1 Sha Zhen FR AC T AI . I NT E R POL AT I ON FUNC T I ON 95

I x , 0 ~< x ~< 1/ 2,f ( x ) (3. 5)~ l - - x , 1 / 2 < x - . . < 1 .Simultaneously, set N= 2 and

, /- g - - ! J - g - - 1~ K 9 [( 1 + ~/-g-)d~ m. . . . . . T ' , if m is large enough9It is clear that the nu mbe r of sets of square wit h side ~ which can cover L is larger than

K 9 2 --~ 9 [-(1 3. 4'- 5-) d~ m.Thus

dimag raph L ~ lim log K 9 2 ~-~ 9 [-(1 3" 4' -5- )d] " log(1 3" ~- 5- )d. . . . log2,~+1 ----- 1 3" log2 = 2 - - ft.Combining it with ( 3. 7) , (3) is proved.


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96 A p p l. M a t h . - J C U Vol. 14,Ser. B

w 4 A p p r o x i m a t i o n o f F I FL e m m a 4. 1. If two f j ( x ) ( j = l , 2 ) are the FIF associated with

( x , y ) = ( L . ( x ) , d . y + q . (x) ) , n = 1 ,2 . . . . . N ; j = 1,2,( { ( x , , y ~ j ) ) l i = 0 ,1 . . . . . N ; j = 1 , 2 } ,

respectively, then fl are the FI F associated with.~ " I (x " d -- L ( r + ~(r (4 .3 )

where t$(~b~)=h(~b~) + 1~ d , , , ( j ) h ( rj=l


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98 A p p l . M a t h . - J C U Vol. 14,Ser. BU s i n g ( 4 . 2 ) , w e o b t a i n L ' = f + h . W h e n m ~ n , w e h a v e L ' ( x ' ) = L ' ( x ' ) ,T h e r e f o r e ,

r - - f ( x ) =r - - r ) A - ( r -4- h ( x " ) ) - - ( f ( x " ) + h ( x " ) ) -4- f ( x " ) - - f ( x ) ----( r - - r ) ) A- ( r + h ( x ' ) - - L " ( x " ) ) -4- ( f ( x " ) - - f ( x ) ) .

Us in g Le mma 5 .1 a n d (5 .4 ) , we h a veI ~ ( x ) - f ( x ) , ~ t o ( ~ ; ~ , ) + e ( N ; n ) d - NK----~.

The Theorem is p roved .

R e f e r e n c e s

1 Barnsley, M.F . , Fractal functions and interpolation, Constructive Approximation, 198 6,2: 303~3 29.2 Sha Zhen, Holder property of fractal interpolation function, Approx. Theory Appl. , 19 92, 4:4 5~5 7.3 Sha Zhen ~. Chen Gang. , Haar expansions of a class of fractal interpolatio n function and their logical

derivatives, Approx. Theory Appl. , 1993,4:73~88.

Dept. of Appl. Math. , Zhejiang Univ. , Hangzhou 310027.

A Class of Functional Equation and Fractal

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