Fractal Interpolation

Transcription

1 Uversty of Cetral Florda Electroc Theses ad Dssertatos Masters Thess (Ope Access) Fractal Iterpolato 28 Gayatr Ramesh Uversty of Cetral Florda Fd smlar wors at: Uversty of Cetral Florda Lbrares Part of the Mathematcs Commos STARS Ctato Ramesh, Gayatr, "Fractal Iterpolato" (28). Electroc Theses ad Dssertatos Ths Masters Thess (Ope Access) s brought to you for free ad ope access by STARS. It has bee accepted for cluso Electroc Theses ad Dssertatos by a authorzed admstrator of STARS. For more formato, please cotact lee.dotso@ucf.edu.

2 FRACTAL ITERPOLATIO by GAYATRI RAMESH B.S. Uversty of Teessee at Mart, 26 A thess submtted partal fulfllmet of the requremets for the degree of Master of Scece the Departmet of Mathematcs the College of Sceces at the Uversty of Cetral Florda Orlado, Florda Fall Term 28

3 28 Gayatr Ramesh

4 ABSTRACT Ths thess s devoted to a study about Fractals ad Fractal Polyomal Iterpolato. Fractal Iterpolato s a great topc wth may terestg applcatos, some of whch are used everyday lves such as televso, camera, ad rado. The thess s comprsed of eght chapters. Chapter oe cotas a bref troducto ad a hstorcal accout of fractals. Chapter two s about polyomal terpolato processes such as ewto s, Hermte, ad Lagrage. Chapter three focuses o terated fucto systems. I ths chapter I report results cotaed Barsley s paper, Fractal Fuctos ad Iterpolato. I also meto results o terated fucto system for fractal polyomal terpolato. Chapters four ad fve cover fractal polyomal terpolato ad fractal terpolato of fuctos studed by avascués. Chapter fve ad s are the geeralzato of Hermte ad Lagrage fuctos usg fractal terpolato. As a cocludg chapter we loo at the curret applcatos of fractals varous wals of lfe such as physcs ad face ad ts prospects for the future.

5 Ths thess s dedcated to everyoe who helped me get to where I am today. v

6 ACKOWLEDGMETS I would le to tha Dr. Mohapatra ad Dr. Vajravelu for all ther help ad support. I would also le to tha Dr. L ad Dr. Su for the gudace. v

7 TABLE OF COTETS LIST OF FIGURES... v LIST OF TABLES... v CHAPTER 1: ITRODUCTIO Itroducto to Fractals Iterpolato Process... 3 CHAPTER 2: POLYOMIAL ITERPOLATIO METHODS Polyomal Iterpolato ewto s Iterpolato Polyomal ewto s Dvded Dfferece Iterpolato Lagrage Iterpolato Polyomal Hermte Iterpolato Error Polyomal Iterpolato... 9 CHAPTER 3: ITERATED FUCTIO SYSTEMS... 1 CHAPTER 4: FRACTAL ITERPOLATIO FUCTIOS CHAPTER 5: FRACTAL POLYOMIAL ITERPOLATIO... 2 CHAPTER 6: HETMITE FRACTAL POLYOMIAL ITERPOLATIO CHAPTER 7: LAGRAGE FRACTAL POLYOMIAL ITERPOLATIO CHAPTER 8: USES OF FRACTALS REFERECES v

8 LIST OF FIGURES Fgure 1: Star wars ad Apollo Fgure 2: Pctures of a cloud ad a fer leaf... 2 Fgure 3: I [, ] R Fgure 4: f : I R Fgure 5: Graph of the Pecewse Lear Iterpolato Fucto Fgure 6: Graph of the Fucto h Fgure 7: Graph of the - fractal fucto of h Fgure 8: Graph of the Lagrage Iterpolato polyomal of h Fgure 9: Graph of the fucto f Fgure 1: Graph of the Lagrage terpolato polyomal of f Fgure 11: Graph of the fractal fucto v

9 LIST OF TABLES Table 1: 1 Data Pots (, y )... 4 Table 2: ewto s Dvded Dfferece Table... 5 Table 3: Eample of Lagrage Iterpolato v

10 CHAPTER 1: ITRODUCTIO 1.1 Itroducto to Fractals People are always fascated by terestg shapes ad ofte woder how t s possble to create beautful mages ad scees le those see the moves. There s a lot of terest the pheomeo called Star Wars. George Lucas s a poeer of the move dustry due to the orgal Star Wars moves, whch cotaed specal effects le ever see before. I the flm Retur of the Jed we see a major weapo the sze of a small moo called the Death Star. Fractals were used to create the outle of ths magfcet weapo. Fractals were also used may other otable Hollywood moves such as Apollo 13, The Perfect Storm ad Ttac. The followg are pctures from the moves, The Retur of the Jed ad Apollo 13. Fgure 1: Star wars ad Apollo 13 The feld of fractals s fascatg. Fractals are ot oly ma made but also be see ature. Whe you see a brach a tree, whch s smlar to the tree, the the tree s a eample of a fractal. Other eamples clude moutas, flowers, lghtg stres, rvers, coastles ad seashells. The followg are pctures of a cloud ad a fer leaf whch are eamples of a fractal as the ehbt self-smlarty. 1

11 Fgure 2: Pctures of a cloud ad a fer leaf The mathematcs of fractals bega to tae shape the 17 th cetury whe Lebz cosdered the dea of straght les beg self-smlar. Lebz made the error of thg that oly straght les were self-smlar. It was 1872 that Karl Weerstrass gave a eample of a fucto whch was cotuous everywhere, but owhere dfferetable. I 192, Helge vo Koch gave a geometrc defto of Weerstrass s aalytcal defto of the o-dfferetable fucto ad ths s ow called the Koch sowflae. Such selfsmlar recursve propertes of fuctos the comple plae were further vestgated by Her Poćare, Fel Kle, Perre Fatou ad Gasto Jula. Fally, 1975 Beot Madelbrot coed the term fractals, whose Lat meag stads for broe or fractured. Two of the most mportat propertes of fractals are self-smlarty ad oteger dmeso. Self-smlarty s the property where a small partto of a object s smlar to the whole object. For eample, a bar of a tree s smlar to the tree tself ad a floret of caulflower s smlar to the whole caulflower. Self-smlarty ca be eplaed usg the power law whch s t c d h 2

12 It s called the power law because t chages as f t was a power of h. Tag logarthm of both sdes of the equalty sg we obta log t d log h log c Whch mples that d log( t/ c) log h Fractal dmeso ca be obtaed usg a relatoshp betwee the umber of copes ad the scale factor. For eample, f a le segmet s cut to four equal peces the the fractal dmeso would be d log c log 4, where c s the umber of copes log r log(1/ 4) ad r s the scale factor. 1.2 Iterpolato Process People are ofte search of a good dgtal camera. A camera wth a hgher mega pel level wll have more resoluto ad clarty. It s terpolato that lets the camera obta the mamum level of mega pel, whch maes the mages sharper. Other uses of terpolato clude estmatg a predcted value for the temperature at a grd pot from data from weather statos located ts eghborhood ad predctg the prce of stocs from ts past behavor from a tme seres data. I mathematcs, terpolato process s the computato of values betwee the oes that are ow or tabulated usg the surroudg pots or values. Accordg to Sprger ole referece wors, Iterpolato s a process of obtag a sequece of terpolato fuctos { ( z)} for some deftely growg umber of terpolato f codtos. The am of the terpolato process s to appromate by meas of 3

13 terpolato fuctos f (z) of a tal fucto f (z) about whch there s complete formato or whch s complcated to deal wth drectly. Some of the famous terpolato processes are ewto s dvded dfferece terpolato, Ate s terpolato, Lagrage terpolatg polyomal, Bessel s terpolatg formula ad Gauss s terpolatg formula. CHAPTER 2: POLYOMIAL ITERPOLATIO METHODS 2.1 Polyomal Iterpolato Table 1: 1 Data Pots, y ) ( y y y y If 1data pots such as Table 1 s gve, the our goal s to fd a polyomal p of lowest possble degree for whch p ( ) y ( ) Such a polyomal s called a terpolatg polyomal ad a fudametal result s: Theorem 1: If, 1,..., are ( 1) dstct real umbers, ad y, y 1,, y are ( 1) arbtrary values, the there s a uque polyomal p of degree at most such that p ( ) y ( ) The proof of ths theorem ca be foud [7] 4

14 2.2 ewto s Iterpolato Polyomal Let us assume that the fucto f() s ow at several values of such ad Table 1. It s ot assumed that the s are evely spaced or that the values are arraged a partcular order. Cosder the th degree polyomal P ( ) a ( ) a ( )( ) a ( )( ) ( ) a If the a s are chose such that P ( ) f ( ) at the +1 ow pots, (, f),,,, the P ( ) s a terpolatg polyomal where the a s are determed usg the ewto s dvded dfferece tables [5]. 2.3 ewto s Dvded Dfferece Iterpolato. Let ( ) ( ) the f ( ) f ( ) f [, 1,, ] R 1 1 Here f [, 1,, ] s the dvded dfferece of f at [, 1,, ] ad the remader s R ( ) ( )[,,, ] ( ) 1 f ( 1) ( ) 1! for Accordg Gerald ad Wheately [5], usg the stadard otato, a dvded dfferece table s show stadard form as follows Table 2: ewto s Dvded Dfferece Table f 1 f [, ] f [, 1, 2] f [, 1, 2, 3] f 1 f [, ] f [, 1, 2] f [, 1, 2, 3] f f [, ] f [ 1, 2, 3] f [ 1, 2, 3, 4] 2 f f [, ] f [ 2, 3, 4] 5

15 3 f 3 f 3 4 [, ] 4 f 4 Here, f [ 1, 2,..., ] f [, 1,..., 1] f [, 1,..., ] Hece, P ( ) f [ ] ( ) f [, ] ( )( ) f [,..., ]... ( )( )...( ) f [,,..., ] Lagrage Iterpolato Polyomal. A alteratve method of epressg the terpolatg polyomal P s of the form p( ) y ( ) y ( ) y ( ) y ( ) 1 1 Where, 1,, represet polyomals that deped o the odes, 1,,. Let, p ( ) y ( ) ( ) j j j j Here, s the Delta fucto where = 1 f ad = f. Cosder to be a polyomal of degree, whch taes o the value of at 1, 2,, ad 1 at. ( ) c( )( ) ( ) c ( ) 1 2 j 1 j By puttg we obta 1 c ( ) j 1 j Therefore, 6

16 c ( ) j 1 j 1 ad ( ) c j 1 j j Each of the s, 1,2,...,, are obtaed by a smlar reasog. I geeral j ( ) ( ) j j j For the set of odes, 1,,, these polyomals are called the cardal fucto. The cardal fuctos together wth p( ) y ( ) y ( ) y y ( ) (2.1) 1 1 yelds the Lagrage form of the terpolato polyomals. 2.5 Hermte Iterpolato A loose defto of Hermte Iterpolato s the terpolato of a fucto ad ts dervatves at a set of odes. The smpler terpolato where o dervatves are terpolated s ofte referred to as Lagrage terpolato. Kcad ad Cheey [7] provde a useful eample ther boo umercal Aalyss where they requre a polyomal of least degree that terpolates a fucto f ad ts dervatve f at two dstct pots ad 1. The polyomal whch we see wll satsfy p( ) f ( ) p ( ) f ( ) (,1) 7

17 There are four codtos ad therefore t s reasoable to see a soluto 3, a lear space of all polyomals of degree at most three. A elemet of 3 has four coeffcets. 2 3 Istead of wrtg p() terms of1,,,, we wrte p( ) a b( ) c( ) d( ) ( ) Ths leads to p ( ) b 2 c( ) 2 d( )( ) d( ) 1 2 The four codtos o p ca be wrtte as f ( ) a f ( ) b 2 f a bh ch 1 ( 1) ( h ) f ( 1 ) b 2ch dh 2 j I a Hermte problem t s assumed that wheever a dervatve p ( ) s to be prescrbed (at a ode ), the p ( ), p ( ),, p ( ), ad p ( ) wll also be prescrbed. Let ( j 1) ( j 2), 1, 2,, be the odes ad let the followg terpolato codtos be gve at the ode p j ( ) c ( j 1, ) The total codtos o p s deoted by m+1 ad therefore j m 1 1 Theorem 2 [7]: There ests a uque polyomal p m fulfllg the Hermte terpolato codto p j ( ) c ( j 1, ) j 8

18 Whe there s oly oe ode, we requre a polyomal p of degree for whch p j ( ) c ( j ) j the the soluto s the Taylor polyomal. c p( ) c c1( ) ( )! Hermte terpolatos problems ca also be solved usg ewto s dvded dfferece method ad Lagrage terpolato formula [7]. 2.6 Error Polyomal Iterpolato Theorem 3 [7]: Let f be a fucto C 1 [ a, b ], ad let p be a polyomal of degree that terpolates the fucto at +1 dstct pots, 1, 2,, the terval [ ab., ] To each [ ab, ] there correspods a pot [ ab, ] such that 1 f p f ( ) ( ) ( 1)! 1 ( ) ( ) 9

19 CHAPTER 3: ITERATED FUCTIO SYSTEMS Defto 1: [6] A metrc space s a par ( Md, ) where M s a o-empty set ad d : M M R s a real valued fucto called a metrc o M, wth the followg propertes ) Postve defte,.e., y M, d(, y ) ) Symmetrc,.e., y M, d(, y) d( y, ) ) Tragle equalty,.e., y, z M, d(, y) d(, z) d( z, y ) Defto 2 [6]: Let ( Md, ) be a metrc space ad let a be the famly of all closed subsets of M. For r ad A a, let V ( A) { m : d( m, a) r }, ad defte for r members A ad B of a, d ( A, B) f{ r : A Vr ( B ) ad B Vr ( A )}. Here, d s the wellow Hausdörff metrc. Let M be a compact metrc space ad H be the set of all oempty closed subsets of M. The H s a compact metrc space wth the Hausdörff metrc. ote that A ad B are subsets of M. Let w M M for {1,2,..., } be cotuous. s called a terated fucto system (IFS). { M, w : 1,2,..., } (3.1) Cosder w ( A) { w ( ) : A }. Defe W : H H by W ( A) w ( A) w ( A)... w ( A) w ( A) for A H (3.2) 1 2 Ay set G H such that W( G) G (3.3) 1

20 s called a attractor for the IFS. Accordg to Barsley [1] the IFS always admts at least oe attractor. A IFS s called hyperbolc f, for some s, s 1 ad {1,2,..., }, d( w ( ), w ( y)) s d(, y ),, y M (3.4) I ths case W s a cotracto mappg whch obeys h( W( A), W( B)) s h( A, B ), A, B H Also, W admts a uque attractor. Barsley [1] eplas how to fd ths uque attractor. Gve a set of data pots{(, y) I R :,1,..., }, where I [, ] R s a closed terval, see Fg. 3. Fgure 3: I [, ] R The fuctos that we are cocered wth are fuctos f : I R whch terpolate the data { y :1,1,2,..., } such that as see Fg. 4 f ( ) y,,1,2,.., (3.5) Fgure 4: f : I R 11

21 The graphs of these fuctos G {, f ( ): I} are attractors of the IFS. I other words there ests a compact subset M of I R, ad a collecto of cotuous mappgs w : M M such that the uque attractor of the IFS s G. Barsley [1] refers to such fuctos as the Fractal Iterpolato Fuctos (FIF). Here we are worg wth the compact metrc space M I [ a, b ] for some a b, wth the Eucldea metrc or a equvalet metrc vz. d(( c1, d1),( c2, d2)) Ma{ c1 c2, d1 d 2 } Assg I [ 1, ] ad let L : I Ifor {1,2,..., } be cotractve homeomorphsms such that L ( ), L ( ), (3.6) 1 L ( c ) L ( c ) l c c c1, c2 I, for some l, l 1. Also, let the mappgs F : M [ a, b] be cotuous for some q 1 satsfyg F (, y ) y, F (, y ) y, 1 F ( c, d ) F ( c, d ) q d d, for all c I, d1, d2 [ a, b ], ad {1,2,..., }. We ow defe fuctos w : M M for {1,2,..., } by w (, y) ( L ( ), F (, y )), 1,2,..., (3.7) Here, { M, w : {1, 2,..., } s a IFS, but ths may ot be hyperbolc. 12

22 Theorem 4 [1]: The terated fucto system (IFS) { M, w : 1,2,.., } admts a uque attractor G, whch s the graph of a cotuous fucto f : I [ a, b ], ad that fucto satsfes (3.5) Proof: Let G be ay attractor of the IFS. Let Iˆ { I : (, y) G for some y [ a, b ]}. ote that I [, ] R s a closed terval, see Fg 1. From (3.2) ad (3.3) G w G, ad t follows that Iˆ L ( ˆ ) I. { I, L : 1,2,..., } s a hyperbolc IFS (3.4) whose uque attractor s I. Hece, I ˆ I [, ] To show that G s the graph of the fucto f : I [ a, b ], we start off by provg that there s oly oe y-value correspodg to each -value (defto of a fucto). Cosder the -values{, 1,..., }. Let, ote that from (3.7) S {(, y) G : } for {,1,..., }. w ( s ) ( L ( ), F ( S ) w ( s ) (, y ) 1 1 Therefore, w ( S ) S for 1, we should have w 1 ( S ) S, but w 1 s a strct cotracto o the compact metrc space S, so S (, y) ad smlarly S (, y ). For I {1,2,..., 1} the oly pots whch ca map to S are (uder w ). Therefore, S (uder w 1 ) ad S 13

23 S w ( S ) w ( S ) (, y ) (3.8) 1 Let, Ma{ s t : (, s),(, t) G, I } Due to the compactess of G, the mamum s acheved at some par of pots ( s ˆ, ) ad ( t ˆ, ) G, wth s t. From (3.7) t ca be assumed that ˆ ( 1, ) for some. But, there ests two pots G, ( L ( ), u) ad L 1 ( ˆ, v ) 1 ˆ wth, s F ( L ( ), u) ad 1 ˆ t F ( L ( ), v ) 1 ˆ Hece, s t F L ˆ u F L ˆ v 1 1 ( ( ), ) ( ( ), ) q u v q wth q 1, hece. Therefore, G s the graph of the fucto f : I [ a, b ] whch satsfes f ( ) y. G s uque because the uo of two attractors s stll a attractor. To prove that f() s cotuous, let CIdeote () the Baach space of all cotuous real-valued fucto g : I R. Defe a orm g Ma{ g( ) : I }. Let us defe a cotracto mappgt : C( I) C( I ), where CIcossts () of those g C() I such that g : I [ a, b ), ad whch obeys g( ) y ad g( ) y. Let C ( I ) C( I ). Tg F L g L whe I, 1,2,...,. 1 1 ( )( ) ( ( )), ( ( )) Usg the defto of the orm 14

24 Th Tg Ma F L h L F L g L { ( ( )), ( ( )) ( ( )), ( ( ))} Ma q h L g L q h g 1 1 { ( ( )) ( ( ))} Hece we ow that T has a uque fed pot hˆ C( I) such that ĥ s the attractor of the IFS. Hece the fucto theorem. f h ˆ s cotuous. Ths completes the proof of the 15

25 CHAPTER 4: FRACTAL ITERPOLATIO FUCTIOS I ths chapter we shall cosder the problem of Fractal Iterpolato. Ital results ths drecto were obtaed by Barsley [2]. Before we go to more recet results o ths topc, we cosder ecessary deftos ad some of the ow results whch wll be used subsequet chapters. Defto 3 [2]: A data set s a set of pots of the form 2 {(, f) :,1,2,..., } where, Ths set of data pots has a terpolato fucto correspodg to t whch s a cotuous fucto f :[, ] such that f ( ) F for,1,2,.., Here, the pots (, ) 2 f are called the terpolato pots ad the fucto f terpolates the data pots. The graph of the fucto f passes through the terpolato pots. 2 For eample, let {(, f ) :,1,2,..., } deote a set of data. Let f :[, ] deote the uque cotuous fucto whch passes through the terpolato pots ad whch s lear o each of the subtervals[ 1, ]. That s ( ) f ( ) f ( F F ) for [ 1, ], 1,2,..., ( 1) 16

26 The fucto f() s called the pecewse lear terpolato fucto. The graph of f() s llustrated Fg. 5 Fgure 5: Graph of the Pecewse Lear Iterpolato Fucto The fgure above s the graph of the pecewse lear terpolato fucto f() through the lear terpolato pots{(, F ) :,1,2,3,4}. Ths graph s also the attractor of a Iterated Fucto System of the form { w, 1,2,3,4} where the maps are affe. The term affe stads for everythg that s related to the geometry of affe spaces. A coordate system for the -dmesoal affe spaces s determed by ay bass of vectors, whch are ot essetally orthoormal. Therefore, the resultg aes are ot ecessarly mutually perpedcular or have the same ut measure. Here, w y a c y e f Where, 17

27 a ( 1), ( ) e ( 1 ) ( ) c ( F F 1), ( ) f ( F 1 F ) ( ) for,1,2,..,. Let t t 1... t be real umbers, ad cosder I t, t, the closed terval that cotast t 1... t. Gve the set of data pots t, I :,1,2,...,. Cosder I, t 1 t ad let L : I I, 1,2,..., be cotractve homeomorphsms such that L t t 1, L t t, L c1 L c2 l c1 c2 c1, c2 I ad for some l 1. Also, let the mappgs F : F be cotuous for some 1 1 satsfyg F ( t, ), F ( t, ), F ( t, ) F ( t, y) y, for all 1 c d, 1 1, ad {1,2,..., }. I the precous chapter we defed IFS w ( t, ) ( L ( t), F ( t, )), 1,2,...,. From Theorem 4 we ow that the IFS admts a uque attractor G whch s the graph of a cotuous fucto f : I whch satsfes f t,1,2,...,. Ths fucto s called the fractal terpolato fucto (FIF) correspodg to {( L ( ), (, ))} t F t 1. Let be the set of cotuous fuctos f :[ t, t ] [ c, d ] such that f ( t) ad f ( t). We ow that s a complete metrc space wth respect to the uform orm (, ). Defe T : by Tf t F L t f L t t t t. 1 1 ( ) ( ), ( ) 1,, 1,2,..., T s a cotracto mappg o follows from 18

28 Tf Tg t g, where a ma a, 1,2,..., ad 1 sce 1 1. As a cosequece T has a uque fed pot o from the Baach fed pot theorem. Thus f such that Tf ( t) f ( t) t t, t. Ths fucto f() t s a Fractal Iterpolato Fucto correspodg to the IFS w ( t, ) L ( t), F ( t, ). Also, f : I s a uque fucto that satsfes the followg equatos or f L t F t, f t 1,2,...,, t I f t F L t, f L t 1,2,...,, t I t, t (4.1)

29 CHAPTER 5: FRACTAL POLYOMIAL ITERPOLATIO Utl recetly, terpolato ad appromatos have bee carved out wth the ad of smooth fuctos. These fuctos are sometmes ftely dfferetable. Ufortuately, the real world we deal wth sgals whch do ot posses such smooth qualtes. Sgals recorded wth respect to tme suggest orgal fuctos wth, abrupt chages, whose dervatves posses sharp steps or eve do ot est at all, (avascués [13].) The fractal terpolato fuctos (FIF) are cosdered to be a mportat advace ths feld because the terpolats of the FIF are ot ecessarly dfferetable over a set ad certa cases are ot eve pot-wse dfferetable. They appear deally suted for the appromato of aturally occurrg fuctos whch dsplay some d of geometrc self-smlarty uder magfcato, (Barsley, [1].) avascués [13], her paper, proposes to create a base for fractal terpolats whch are perturbatos of polyomals whose am s to defe a o-smooth fractal verso of covetoal terpolatos. A complete descrpto of the frequecy doma of the fractal fuctos s obtaed by meas of ther Fourer Trasform. Ths fact s partcularly mportat because such fuctos are defed mplctly the tme doma by a fuctoal equato, (avascués [13].) Let us revew the IFS (3.1) whch admts a uque attractor G, whch s the graph of a cotuous fucto f : I R that obeys f ( t ),,1,2,..,. Ths fucto s called a fractal terpolato fucto correspodg to{( L ( ), (, ))} t F t 1. Tll ths day, the most studed fractal terpolato fucto (FIF) has bee defed by the terated fucto system (IFS) [13] 2

30 L ( t) a ( t) b F ( t, ) q ( t) (5.1) where a s a vertcal scalg factor for the trasformato the scale vector of the IFS. w ad ( 1, 2,..., ) s Mchael Barsley, who s a poeer the use of data pots to create fractal fuctos [1], proposes a geeralzato of a cotuous fucto h by meas of a fractal terpolato usg the IFS (5.1) wth a polyomal q ( t) h L ( t) b( t). Here b s a cotuous fucto such thatb( t), b( t). Here, the case that we wll be loog at s b h c, wth the fucto c beg a cotuous ad creasg fucto c( t ) t, c( t ) t. As a eample, avascués her paper [13] cosders the famly t c( t) ( e 1) / ( e 1), o a terval[,1]. Proposto 1 [13]: Let h: I [ a, b] R, be cotuous, : a t t1... t b, be a partto of [ ab,, ] 1, R ad such that 1. The IFS (3.1) where a ( t t 1) / ( t t ), 1 b ( t t t t ) / ( t t ), q ( t) h L ( t) h c( t) ad c a creasg cotuous fucto such that c( t) t ; c( t) t, defes a FIF h ( t ) h( t ) for all,1,...,. Proof: Frst step s to chec ad see whether the codtos for L ad F are satsfed. We loo at (3.6) wth L( t) t 1, L ( t ) t {1,2,..., }, where L : I Iare cotractve homeomorphsms ad h( t ). We have 21

31 ad F ( t, ) q ( t ) h L ( t ) h c( t ) h( t ) h( t ) F ( t, ) q ( t ) h L ( t ) h c( t ) h( t ) h t h t. F ( c, d ) F ( c, d ) d d, the secod varable F s uformly Lpschtz, wth costat 1. Defe T : G G, whereg { g C[ a, b]: g([ a, b]) [ c, d], g( a), g( b) }, 1 1 accordg tot f ( t) F ( L ( t), f L ( t)). Accordg to Theorem 4 T admts a uque attractor, h such that h t F L t h L t, t I. 1 1 ( ) ( ( ), ( )) Usg (5.1) h t h L t q L t 1 1 ( ) ( ) ( ) Substtutg to the above q ( t) h L ( t) h c( t), we obta h t h L t h L t h c t L t 1 1 ( ) ( ) ( ( ) ( )) ( ) h L ( t) h h c( t) L ( t) 1 1 (5.2) h t h h c t L t 1 ( ) ( ( )) ( ) h passes through the pots ( t, ) sce, 22

32 h t F L t h L t 1 1 ( ) ( ( ), ( )) F ( t, ) h t F L t h L t (5.3) 1 1 ( ) ( ( ), ( )) F ( t, ) ote that sce L ( t ) t, we have L 1 ( t) t ad h L 1 ( t ) h ( t ). As a eample let us cosder the graph of the fucto h( t) t cos 2 t. The frst fgure s the orgal graph of the fucto ht (). The secod fgure s the graph of the correspodg fractal fucto wth the partto : Let us cosder the case where c() t 2, a quadratc the terval,1. ote that c(), c (1) 1. Let.2 1,...,8. The thrd fgure s the graph obtaed from the Lagrage Polyomal Iterpolato of the fucto ht (). All of the graphs below are geerated usg Maple 1. Fgure 6: Graph of the Fucto h 23

33 Fgure 7: Graph of the - fractal fucto of h Fgure 8: Graph of the Lagrage Iterpolato polyomal of h The graphs above were geerated usg the followg Maple 1 code. 24

34 Fd a ad b for each. 25

35 Defto 4 [1]: Let h C() I,, c, ad be as Proposto 1. The FIF h c defed ths proposto s termed - fractal fucto of h wth respect to ad c. Defe the -fractal operator (attractor) respect to ad c by O, c : C( I) C( I) h h Defto 5 [1]: A - fractal polyomal s a elemet p ( t) C( I) such that there s a polyomal p P[ a, b] wth O ( p) p. If p has degree m, the p s a - fractal polyomal of degree m. I the defto above, Pm [ a, b ] s a set of polyomals of degree less tha or equal to m o the terval I [ a, b ] ad [, ] [, ] 2 P a b Pm a b. {1, tt,,...} costtutes a bass of P[ a, b ]. For otato purposes avascués ad Sebastá [13] assume that P [ a, b] O ( P [ a, b], P [ a, b]) O ( P[ a, b ]). m m Let us cosder aother eample to emphasze the mportace of fractal polyomal terpolato. I used Maple 1 to plot the graph of a polyomal ad a graph of the fractal terpolato polyomal usg a program created by Ke Mos [1]. The m 1 graph Fgure 6 s the graph of a fucto 1 1 f ( ) s(6 ) o terval[,1]. The

36 graph Fgure 7 s the graph of the same fucto terpolated usg the Lagrage terpolato formula the terval [,1] created usg four parttos ad the oe Fgure 8 s the graph of the fractal fucto o the same terval created usg four parttos. Fgure 9: Graph of the fucto f Fgure 1: Graph of the Lagrage terpolato polyomal of f 27

37 Fgure 11: Graph of the fractal fucto A eample of Lagrage polyomal terpolato s gve chapter 7. The graphs above were draw usg the followg commads Maple 1 respectvely. 28

38 otce that the Lagrage terpolato gave a smooth curve whch dd ot hghlght the fer detals the graph. The stoc maret ad weather data fluctuates by the mute ad the fractal fuctos are deal to capture the mute detals ad predct the future up to a certa etet. The procedure for fttg a real world data utlzg Fractal Iterpolato Fuctos s descrbed the followg paragraph. Frst of all let us cosder the set of data t, y,,1,..., M. The FIF s bult as a * * perturbato of a terpolat g of a subset of the data. Let the subset of t, y,,1,..., M be P t,,,1,..., where * * t, t, ad * * t, t,. Let the terpolat g be a fucto that passes through P. Cosder * * M M the IFS (5.1) wth a ( t t 1) / ( t t ), b ( tt 1 tt ) / ( t t ) ad q ( t) h L ( t) b( t), here b s cotuous ad b( t) ad b( t). Also, let h be the correspodg FIF. 29

39 Let t,, j 1,2,..., m be the termedate pots I t, 1 t ot P: ( ) j j j ( ) t 1 t t j 1,..., m. Accordg to (5.2), h t h t h h c t L t. 1 ( ) ( ) ( ( )) ( ) Addg the codto h t to (5.2) yelds j j h t h b L t 1 j j j Ad h t h b L t. 1 j j j Choosg such that the sum of the square resduals s mmum yelds ( ) 2 m 1 j j j j 1 m E h t h b L t Solvg the above equato for E ' yelds ( ) m j 1 h t h L t b L t ( ) m j j j j j h L t b L t 1 1 j j 2 h t,..., h t ad Let ( ) ( ) j j m m h L t b L t,..., h L t b L t b j j m m ( ) ( ) Substtutg these bac yelds 3

40 b b 2 Accordg to avascués [14], If the terpolat h coverges to the orgal fucto whe the dameter of the partto teds to, we get h t ad. j j Therefore, we ca choose such that 1, 1,2,..,. The fucto h together wth the scale vector determe the fttg curve h. 31

41 CHAPTER 6: HETMITE FRACTAL POLYOMIAL ITERPOLATIO Secto 2.5 gves a bref troducto to Hermte terpolato whch ca also be obtaed usg ewto s dvded dfferece table ad Lagrage terpolato. I [11] M.A. avascués ad M.V. Sebastá gve a troducto to obtag a Hermte terpolato fucto by meas of fractal terpolato. I order to obta the Hermte terpolato fucto let us loo at a couple of theorems by Barsley. Theorem 5 [4]: Let t t 1... t ad L () t, 1,2,...,, the affe fucto L () t at b satsfyg the epressos (3.6). Let a L t t 1 1 () t t ad F ( t, ) q( t ), 1,2,..., satsfyg F( t, ) 1, F ( t, ), t F ( c, d ) F ( c, d ) d d (chec Proposto 1). Suppose for some teger p, q C [ t, t ], 1,2,...,. Let p p a ad q () t F ( t, ), 1,2,..., p a ( ) q ( t ) q ( t),, 1,2,..., p ( ) ( ) 1,, a1 1 a F ( t, ) F ( t, ) wth 2,3,...,, the {( L ( ), (, ))} t F t 1 If 1,,, determes a FIF f C p [ t, t ] ad ( ) f s the FIF determed by {( L ( ), (, ))} t F t 1 for 1,2,..., p. The above theorem leads us to epect the Hermte fractal terpolato problems ca be solved uquely ad assures the estece of a dfferetable FIF. Ths FIF has p 1 dervatves prescrbed at (( t, );,1,..., ;,1,..., p ). 32

42 Theorem 6 [11]: Let 1, p, t t1... t ad ;,1..., ;,1,..., p by p gve. Let 1, 2,..., be real umbers such that a 1,2,...,, wth a t t t t 1. The there ests precsely oe fucto of fractal terpolato f p C defed by a IFS gve by: L ( t) a ( t) b F ( t, ) q ( t ) Where q ( t) 1,2,.., are polyomals of degree at most 2p+1, such that ( f ) ( t ) for,1,..., ;,1,... p. t t 1 Proof: Cosder a t t t t t t 1, b t t ad defe F t, q t a (6.1) for pwth the degree of q, deg q 2 p 1. The polyomal q t s computed as soluto of the system of equatos p q t F t,, 1, a q t F t, a (6.2) The 2p 2uows of the above equato are the coeffcets of q t. Solvg the above equato for q t, we obta 33

43 1 q L t q t 1 1 1, a a, 1 q L t q t 1 a a for p. The fucto q L t s a polyomal wth a degree of at most 2p 1ad whose 1 dervatves up to order p at t 1 ad t are 1, a ad a. Thus t ca be cocluded that q L t s a Hermte terpolatg polyomal the terval t, 1 t. 1 Ths Hermte polyomal ests ad s uque [8] ad thus t ca be deduced that q t ests ad s uque. Let us verfy that the fuctos defed by (6.1) equato referece goes here satsfes the hypothess of Barsley ad Harrgto s Theorem 5. From (6.2) we ow that F t, 1, F 1, t, 2,3,...,. Therefore, we ow that the hypothess s satsfed ad thus from Theorem 5, we ca guaratee the estece of f p C such that {( L ( t), F ( t, ))}. As a f s the FIF defed by the IFS 1 cosequece, we ow that f s the fed pot of a cotracto mappg T : M M. Smlar to Chapter 3, page 15. Let us defe a orm f g Ma{ f t g t : I }. Let us defe a cotracto mappg T : M M, where M cossts of those f M such that f : t, t c, d, ad whch obeys f ( t) ad t( t). Let us defe the mappg T by 34

44 1 1 ( Tf )( t) F L ( t), f L ( t) whe t t, 1 t. Usg the defto of the orm Tf Tg Ma F L f L F L g L { ( ( )), ( ( )) ( ( )), ( ( ))} Ma f L g L 1 1 ( ( )) ( ( )) f g Hece we ow that T has a uque fed pot T M M ad the fucto f of : f s cotuous. T g t F L t, g L t t t, t 1 1 1, f t f t (6.3) From (6.2) ad (6.3) we ca coclude that f t F L t, f L t F t, f t F t, 1 1,1,2,..., ad,1,2,..., p The fucto f geeralzes the Hermte fuctos as 1, 2,...,. f p C ad f t F L t f L t q L t f t t, 1 t. f s a polyomal of degree less 1 1 1, tha or equal to 2p 1 the terval I t, 1 t ad as a cosequece f s a Hermte fucto sce t satsfes all the codtos prescrbed secto 2.5. Ths result mples that the IFS Theorem 6, L ( t) a ( t) b F ( t, ) q( t ), ca be called the Hermte Fractal Iterpolato Fucto. 35

45 CHAPTER 7: LAGRAGE FRACTAL POLYOMIAL ITERPOLATIO As a revew let us loo at the Lagrage polyomals gve by equato (2.1), whch s p( t) ( t) 1 1 ( t), ( t), where, () t ( t t)( t t1) ( t t 1)( t t 1) ( t t ) t t ( t t )( t t ) ( t t )( t t ) ( t t ) t t j j j j. As a eample let us terpolate 4 f ( ) from [1,4]. The Lagrage polyomal ( ) Ls gve below. Table 3: Eample of Lagrage Iterpolato f( ) 1 f ( ) f 1 ( ) f 2 ( ) f 3 ( ) 256 ( )( )( ) ( )( )( ) L( ) f ( ) f ( ) ( )( )( ) ( )( )( ) ( )( )( ) ( )( )( ) f ( ) f ( ) ( )( )( ) ( )( )( ) ( 2)( 3) ( 4) ( 1)( 3)( 4) L ( ) 1 16 (1 2)(1 3)(1 4) (2 1)(2 3)(2 4) ( 1)( 2)( 4) ( 1)( 2)( 3) (3 1)(3 2)(3 4) (4 1)(4 2)(4 3)

46 Defto 6[13]: avascués ad Sebastá defe the - fractal terpolat of the Lagrage polyomal as p ( t) O ( p ) ( t), where, : The partto., : - fractal polyomal of, wth respect to the partto. p () t : A fucto whch passes through the pots ( t, ) as Proposto 1 o page 22. Let L represet a Lagrage operator that assgs a terpolat polyomal to a fucto f wth respect to{( t, ( ))} f t, the p( t) O L( f ). The bass polyomals of the Lagrage operator, () t are orthogoal wth respect to the orm, ( t ) ( t ). The same property s true for,, j,, j,. Therefore,,, j, j, where s the Delta fucto (the defto of the Delta fucto ca be foud page 7). If p P [ a, b ], by the learty of the operator O (see page 46 of [13] Corollary 1) p,. Also, the orthogoalty of, mples lear depedece ad therefore, {, } costtutes a bass for the space of -fractal polyomals for the 37

47 space P [ a, b] o the partto. P [ a, b] has a fte dmeso whch allows us to obta a * p for each h [, ] C a b such that a* a* h p h p p P a b f ; [, ] Utlzg the fractal polyomals the Lagrage terpolato gves us the advatage of obtag a o-smooth verso of the Lagrage polyomals whch wll hghlght the fer detals of the graph. 38

48 CHAPTER 8: USES OF FRACTALS Oe of the may uses of fractal geometry s the geerato of programs that create mages of clouds, trees, ladscapes ad the coastal le o the computer scree, fractals have may other applcatos. We have already wet over the use of fractals scece fcto moves. Fractals together wth owledge of ecosystem are also used to determe the spread of smoe, acd ra, ad other ar bore or water bore tocats. Fractal terpolato also provdes a good represetato of ecoomc tme seres such as the stoc maret fluctuato ad weather data. Wth the curret ecoomc crss we eed some ew models that tae to accout may more varables ad that provde more accurate terpretato of the future behavor ad I th fractal polyomal terpolato ca play a sgfcat part that. The facal marets are churg wth the sub-prme home loa crss the US ad the global bag system. The huge dervatves overhag s a problem that requres global cooperato to maage. He food prces ad crease of poverty. The Govermets together wth the Cetral Bas ad Commercal Bas eed to come up wth a soluto for these global problems. Modelg the problem volves several terdepedet factors mostly o-lear relatoshps- the model, addto to tradtoal quattatve data, has to corporate tagble qualtatve factors such as poltcal polces, chagg food habts, subsdes ad retal supply cha all of whch are dyamcally chagg. We eed to develop holstc models whch are drve by the problem agast the tradtoal approach where the problem s mutlated to sut ow models such as OR resultg short-term o-sustaable solutos. Beot Madelbrot, the father of fractals, has tae o the stoc maret ad he s aalyzg these problems."marets, le oceas, have turbulece," he sad. "Some days the chage 39

49 marets s very small, ad some days t moves a huge leap. Oly fractals ca epla ths d of radom chage." Wth the curret ecoomc crss we eed some ew models that tae to accout may more varables ad that provde more accurate terpretato of the future behavor ad fractal polyomal terpolato ca play a sgfcat part that. 4

50 REFERECES [1] Barsley, Mchael F., Fractal Fuctos ad Iterpolato, Costructve Appromato (1986), o. 2, pp [2] Barsley, Mchael F., Fractals Everywhere, Academc Press, Ic., Sa Dego, CA (1988). [3] Barsley, Mchael F., Fractals Everywhere Secod Edto, Iterated Systems, Ic., Atlata, Georga (1993). [4] Barsley, Mchael F., Harrgto A.., The Calculus of fractal terpolato fuctos, J. Appro. Theory 57 (1989) [5] Gerald, Curts F., Wheatley, Patrc, O., Appled umercal Aalyss Seveth Edto, Pearso Educato, Ic. (24), pp.158. [6] Helems, Alesadr, Lecture ad Eercses o Fuctoal Aalyss, Amerca Mathematcal Socety, (26). [7] J. Stoer, R. Bulrsch. Itroducto to umercal Aalyss, Sprger, ew Yor, 198, pp [8] Kelley, Joh, Geeral Topology, Brhäuser (1975) [9] Kcad, Davd, Cheey, Ward, umercal Aalyss Secod Edto, Broos/Cole Publshg Compay, Pacfc Grove, CA (1996), pp [1] Mos, Ke, Chaos ad Maple Software, Uversty of Scrato, Scrato, PA. [11] M.A. avascués, M.V. Sebastá, Geeralzato of Hermte Fuctos by Fractal Iterpolato, Joural of Appromato Theory (24), o. 131, pp.19-29, Avalable ole at 41

51 [12] M.A. avascués, M.V. Sebastá, Smooth Fractal Iterpolato, Joural of Iequaltes ad Applcatos (26), Artcle Id , pp.1-2. [13] M.A. avascués, M.V. Sebastá, Fractal Polyomal Iterpolato, Joural for Aalyss ad ts Applcatos (25), Vol. 24, o. 2, pp [14] M.A. avascués, M.V. Sebastá, Fttg Curves by Fractal Iterpolato: A Applcato to the Quatfcato of Cogtve Bra Processes, World Scetfc (23). [15] Petge, Hez-Otto, Jürges, Hartmut, Saupe, Detmar, Chaos ad Fractals: ew Froters of Scece, Sprger Verlag ew Yor, Ic., Resselaer, Y (1992). [16] Practcal Fractal Applcatos, retreved o Jauary 17, 28 from ature.bereley.edu/~bgu/uu/geocomp/wee5/rchfractals.ppt. [17] Fractals, retreved o Jauary 27, 28 from mathworld.com [18] Fractals Images, retreved o Jauary 17, 28 from pctures.aol.com. [19] Fractal Dmeso, retreved o Jauary 27, 28 from wpeda.com 42