Hidden variable recurrent fractal interpolation function with four function contractivity factors

Transcription

1 Hddn varabl rcurrnt fractal ntrpolaton functon wth four functon contractvt factor Chol-Hu Yun Facult of Mathmatc Km Il ung Unvrt Pongang Dmocratc Popl Rpublc of Kora Abtract: In th papr w ntroduc a contructon of hddn varabl rcurrnt fractal ntrpolaton functon HVRFIF wth four functon contractvt factor In th fractal ntrpolaton thor t vr mport ant to nur flblt and dvrt of th contructon of ntrpolaton functon Rcurrnt tratd fu ncton tm RIF produc fractal t wth local lf-mlart tructur Thrfor th RIF can d crb th rrgular and complcatd obct n natur bttr than th tratd functon tm IF Hddn varabl fractal ntrpolaton functon HVFIF nthr lf-mlar nor lf- affn on Th HVFIF mor complcatd dvr and rrgular than th fractal ntrpolaton functon FIF Th co ntractvt factor mportant on that dtrmn charactrtc of FIF W prnt a contructon of on varabl HVRFIF and bvarabl HVRFIF ung RIF wth four functon contractvt factor Kword: Rcurrnt tratd functon tm IF Rcurrnt fractal ntrpolaton functon RFIF Hddn varabl fractal ntrpolaton functon HVFIF Hddn varabl rcurrnt fractal ntrpolaton functon HVRFIF Functon contractvt factor AM ubct Clafcaton: 8A80 4A05 97N C45 Introducton In 986 Brnl ntroducd a notaton of fractal ntrpolaton functonfif bad on th thor of th tratd functon tm IF FIF hav mor advantag n modlng phnomna and fucnton wth om lf-mlart n natur than clacal ntrpolaton functon uch a polnomal and pln Thrfor FIF hav bn tudd n man artcl and appld to a lot of ara uch a functon approomaton gnal p roc and computr graphc tc In fractal ntrpolaton on ma gnrall an IF or a rcurrnt tratd functon tm RIF for a gvn datat and thn contruct th Rad-Baratarbc oprator on utabl pac of contnuou functon ung th IF RIF A fd pont of th oprator th FIF RFIF for th gvn datat Contructon drvatv ntgral dmnon moothn and tablt of th FIF hav bn wdl tudd [-6] nc th vrtcal calng factor whch th contracton tranformaton of IF hav dtrmn th charac trtc of FIF th ar vr mportant To obtan FIF wth hgh flblt contructon of FIF wth fun cton vrtcal calng factor and thr analtc proprt hav bn tudd n man papr [ ] RIF a gnralzaton of IF and produc local lf-mlar t whch armor complcatd than lf- mlar t FIF contructd b th RIF calld a rcurrnt fractal ntrpolaton functon RFIF Contruct on of RFIF for a datat n R and 3 R [4 5 5] In [5] gnralzng th contructon of RFIF wth contant vrtcal calng factor RFIF ung th RIF wth functon vrtcal calng factor wr contructd and th fractal dmnon of graph of th contructd ntrpolaton functon wa tmatd and a contructo n of bvarat fractal ntrpolaton functon ung th RFIF wa propod Barnl t al [ 3] ntroducd a concpt of hddn varabl fractal ntrpolaton functon HVFIF wh ch mor complcatd dvr and rrgular than th FIF for th am t of ntrpolaton data 3 Th da of th contructon of th HVFIF to tnd th gvn data t on R nto a data t on R ma a vctor valud fractal ntrpolaton functon for th tndd data t and thn proct th vctor valud functon onto R whch gv th HVFIF It uuall non lf-affn bcau th HVFIF th procton of a vctor valud

2 functon Th HVFIF hav four fr paramtr: fr varabl contrand fr varabl fr hddn varabl and contrand fr hddn varabl th ar calld contractvt factor Ung hddn varabl w can control mor flbl hap and fractal dmnon of th graph of FIF In man papr th contractvt factor ar contant [3 6 7] Thrfor th contructon lac th fl blt whch ncar to modl complcatd and rrgular natural phnomna To olv th problm H VFIF wth on functon contractvt factor n [] and HVFIF wth four functon contractvt factor wr contructd In th papr n ordr to nur th flblt and dvrt of contructon of HVFIF w prnt con tructon of on varabl and bvarabl HVFIF wth four functon contractvt factor ung RIF Th pap r organzd a follow: In cton w contruct on varabl HVFIF wth four functon contractvt factor for th tndd data t Thorm In cton 3 w prnt a contructon of bvarabl HVFIF wth four functon contractvt factor Thorm 3 4 On varabl HVRFIF Contructon of RIF t a data t P 0 n R b gvn b P0 { R ; 0 n} 0 n To contruct th HVFIF for th datat P 0 w tnd th datat a follow: 3 P { R ; 0 n} 0 n whr and z 0 n ar paramtr Morovr w dnot N n { n} I [ ] and I [ 0 n ] whr I { n} calld a rgon t l b ntgr wth l n W ma ubntrval I l of I contng of om rgon and call I a doman Thn two nd pont of th doman I { l} ar contand n th t { 0 n} and hnc dnotng tart p ont and nd pont of I b rpctvl w gt th followng mappng: :{ l} { n} :{ l} { n} and I dnotd b I [ ] W uppo that l whch man that th nt rval I contan at lat I For ach w ta a { l} and dnot t b t mappng Nn :[ ] [ ] Nn b contracton homomorphm that map nd pont of I to nd pont of I { } { } W dfn mappmg F F : I R R n a follow: z q z q = q z q whr : I R ar phtz functon on I who abolut valu l than whch ar call d contractvt factor and q q : I R ar phtz functon uch that f { } a a { } thn F a Thn F obvoul phtz functon t u dnot b F whr F

3 q q z Eampl An ampl of q q atfd abov a follow: h g g q h g g q g z z g h z z h t R D b a uffcntl larg boundd t contanng n W dfn tranformaton R D : I I W n b F W n Thn a w now from th dfnton of and F W map th data pont on nd of th doman I to th data pont of I For a functon f lt u dnot ma f f W dnot } ; ma{ n Th followng thorm gv a uffcnt condton for W to b contracton tranformaton Thorm If thn thr t om dtanc quvalnt to th Eucldan mtrc on R uch that W ; n ar contracton tranformaton wth rpct to th dtanc Proof W ta a potv ral numbr uch that whr } ; ma{ n } ; ma{ n } ; ma{ n q q up D and a norm on R W dfn a dtanc on 3 R a follow: 3 R Thn t clar that quvalnt to th Eucldan mtrc on R For D I w hav F F W W F F =

4 ma{ } = From th hpoth of th thorm th condton on w hav ma{ } < Thrfor W n ar contracton tranformaton W dfn a row-tochatc matr M p t nn b / a I I t pt 0 I I t whr for vr N n th numbr a ndcat th numbr of th doman I contanng th rgon I whch man that p t potv f thr a tranformaton W mappng I to I t Thn w hav RIF { R 3 ; M ; W n} corrpondng to th tndd datat P W dnot an attr actor of th RIF b A Contructon of HVFIF W ma a contnuou functuon that a fd pont of th Rad-Baratarbc oprator dfnd b th RIF ntrpolat th tndd datat P and who graph th attractor of th RIF For th RIF contructd abov w hav th followng thorm Thorm Thr a contnuou functon f ntrpolatng th tndd data t P uch that th graph of f th attractor A of RIF contructd abov Proof t a t C I b a follow: C I { h : I R ; h ntrpolat th tndd data t P and contnuou} W can al now that th t a complt mtrc pac wth rpct to th norm For h C I w dfn a mappng Th on I b Th F h I Thn w gt Th C I In fact for an { 0 n} thr { } uch that Th F h F whr 0 Hnc w can dfn an oprator T : C E C E on C E Th oprator contracton on In fact w hav Th Th F h F h h h h h h h Thrfor th oprator T ha a unqu fd pont f C I and

5 f F f Th gv that for th graph Gr f of f n Gr f W Gr f E I whr I { N; p 0} n Th man that Gr f th attractor of th RIF contructd n Thorm Thrfor from th unqun of attractor w hav A Gr f t u dnot th vctor valud functon f : I R n Thorm b f f f whr f : I R ntrpolat th gvn datat P 0 whch calld a hddn varabl rcurrnt fractal ntrpolaton functon H VRFIF Furthrmor a t { f : I} a procton of A on R A w now from th proof of Thorm w hav f F f I f F f f I Thrfor for all I th HVRFIF f atf f q f f and f atf f f f q Eampl Fgur how th graph of on varabl HVRFIF contructd b RIF wth four functo n contractvt factor for a datat P 0 ={ } Contractvt factor { 3 4 } { 3 4} { 3 4 } { 3 4 } ar a foll ow: { } { } { } { } { } { } { } { } 3 {9 9 } { } {n0 co300 n00 co3} {099- n0 09- co n co3 } 4 {9 9 } { } {n0 co300 n00 co3} {099- n0 09- co n co3 }

6 Fg Graph of HVRFIF 3 Hddn varabl bvarabl rcurrnt FIF 3 Contructon of RIF t a datat P 0 on rctangular grd b gvn a follow: n 0 m P { z R ; 0 n 0 To ma a HVRFIF for th datat w tnd th datat to th followng on: 4 P { z t z R ; 0 n 0 0 n 0 m whr z z t and t 0 n 0 m ar paramtr W dnot N nm I [ ] Nnm { n} { I [ ] E [ 0 n] [ 0 m] E I I whr E calld a rgon t l b an ntgr wth l N Nt w ta rctangular E l contng of om rgon from E E calld a doman Thn w hav E I I whr I I ar clod ntrval on -a and -a rpctvl nc th ndpont of I { l} ar concdd wth om ndpont of I n dnotng tart pont and nd pont of I b rpctv l w can dfn th followng mappng: :{ l} { n} :{ l} { n} mlarl for I w dfn th mappng :{ l} { :{ l} { Thn w hav I [ ] I [ ] whr w aum that l whch man that I I ar ntrval contanng mor than I I For Nnm w ta { l} and dnot t b W dfn mappng :[ ] [ ] :[ ] [ ] Nnm a contracton homomorphm that map nd pont of I I to nd pont of I I { } { } { } { } Nt w dfn tranformaton : E E b Thn th map vrt o f E to on of E for { } { } ab a { } b { } W dfn mappng F : E R R n m b z t q z F z t q = q t q whr : E R ar arbtrar pchtz functon who abolut valu ar l than and q q : E R ar dfnd a mappng atfng th followng condton: for

7 { } { } a b a { } b { } F z zab Thn F ar pchtz mappng W dnot F b F z whr q z q z t In th futur w dnot mpl F b F t D R b a uffcntl larg boundd t contanng z n m Now w dfn tranformaton W : E D E R n m b W F n m W dnot ma{ ; n Th followng thorm gv a uffcnt condton for hat W to b contracton on Thorm 3 If thn thr t om dtanc quvalnt to th Eucldan mtrc uch t W n m ar contracton tranformaton wth rpct to th dtanc Proof W ta a potv numbr uch that c whr c ma{ c c } c ma{ c ; n 3 ma{ c c c c ; n ma{ ; n up z q q and a norm on R z D Now w dfn a dtanc on R 4 b 4 z z R 4 Thn obvoul quvalnt to th Eucldan mtrc on R and for E D w hav W W F F z z z z z z z c z z = c z z ma{ c } z z = From th hpoth of th thorm and condton on w hav ma{ c } < Th man that W n m ar contracton tranformaton

8 Rmar: In th dfnton of vn n th ca whr changd nto w hav th ml ar rult W dfn a row-tochatc matr M p t N N b / a E E t pt 0 E E t whr : N nm { N} an on to on mappng dfnd b n and th numbr a ndcat th numbr of th doman E l contanng th rgon E whch man that p t potv f thr a tranformaton W mappng E to E t Thn RIF { R 4 ; M ; W n a RIF corrpondng to th tndd datat P An at tractor of th RIF calld a rcurrnt fractal t and dnotd b A Contructon of bvarabl HVRFIF W prnt a uffcnt condton for a fd pont of th Rad-Baratarbc oprator dfnd b th R IF to ntrpolat th tndd datat P and hav a graph whch th attractor of th RIF In th mappng F z dfnd abov w dfn a on atfng condton that for om contnuou functon g ntrpolatng th datat P g z 0 n 0 m F g g { } 3 F g g { } 4 On ampl a follow: l r whr r g E l g E Thn w hav F z l r Nt ampl on of l and r atf ng th condton Eampl 3 A mappng g an ntrpolaton functon of th gvn datat l and r con cd wth g E g E on th doman E th rgon E rpctvl For ntanc w dfn l : g ul g ul g vl g vl u v z u v z u v z u v z l l l l l l l l ul : vl : [ ] [ ] r : g ur g ur g vr g vr ur vr z ur vz r ur vr z uvz r r ur : vr : [ ] [ ] Thn w hav l g l g { } { } r a g a r b g b a { } b { } t th attractor of th RIF atfng condton 3 and 4 b B w hav th followng thorm Thorm 4 Thr a contnuou functon f whch ntrpolat th data t P and who graph th attractor B Proof W dfn a t C E b C E { h : E R C; h ntrpolat P and atf condton 3 4} W can al prov that th a complt mtrc pac wth rpct to th norm nc g C E whch contand n th dfnton of F w hav C E Morovr th functon concdng wth g

9 on E ar contand n C E For h C E w dfn a mappng Th on E b Th F h E Thn w hav Th C E In fact for { } Th F h h Th F h h Thrfor Th h on { : [ 0 m ]} { : [ 0 n ]} 0 n 0 m and F Th F h h Th Th atf 3 mlarl w can prov that Th atf 4 Thrfor w can dfn an oprator T : C E C E on C E It obvou that th oprator co ntracton on In fact w hav Th Th F h F h h h h h h h Hnc T ha a unqu fd pont f C E f F f Furthrmor for th graph Gr f of f N w gt Gr f W Gr f E I Th man that Gr f th attractor of RIF contructd abov From th unqun of th attractor of RIF w hav A Gr f Th vctor valud functon f : E R n Thorm 4 dnotd b f f f whr f : E R ntrpolat th gvn datat P 0 and calld hddn varabl bvarabl rcurrnt fractal ntrpolaton functon 3 HVBRFIF for th datat P 0 Morovr f: E a procton of B on R nc th procton not alwa lf-affn th HVBRFIF not gnrall lf-affn FIF f ntrpolat th t 3 { t t R ; 0 n 0 From th proof of Thorm 4 w gt f F f E f F f f E Thrfor for all E HVBRFIF f atf f f f q and f atf f f f q Eampl Fgur how th graph of HVBRFIF contructd b RIF wth om contractvt fac tor and datat P 0 gvn n th followng tabl:

10 Contractvt factor { } { } { } and { } ar a follow: 3

11 4

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