Luận văn Chỉnh hóa một số bài toán moment

ChuangIV :ChinhhoamQtsobaitoaDmoment CHU<JNG IV: CHlNH HOA IVH)TSO BAI TOA.N MOMENT 4.1Chlnhhoabai tminmomentHausdorffmOtchi~u: - Xet bai tmlnmomentHausdortlmQtchi~u: funhamu(x)trongL2(0,1)sao I cho: fu(x).xkdx=,llkVaik=0,1,2...; 0 (4.1) /.l=(/.lk)ladaybi ch~nchotntoc. NguoitachungminhduQcbaito<ln(4.1)lakhongchlnhnghiala khong1uon t6nt~inghi~mvatrongtruonghQpt6nt~inghi~mthlchungkhongphl,lthuQclien tl,lCvaodaki~nchotruoc. M1,lcdichcilachuangnayla chlnhh6abairoanmomentboiphuongphap moment hliu h~n.Ta ap d1,lngphuong phap ch~tCl,ltachuang truoc trong vi~c chQnhamf(t) va fl(E-1/2). Ta sechlnhh6a(4.1)boidaymomenthliuh~n 1U(X).Xkdx=/.lk vdi k =0,1,...,n (4.2). HQtn!cchuin se duQcchob~ngcaeda thuGLegendre.Cho Pnet)la dathuG Legendreb~cn: Pn(t)=:t (n+k)! (t-1t k=o(n-k)!(k!Y' 2k ho~ctuangduangvoi : 1 dn Pn(t)=--(t2 -l ) n 2nn!dtn Ta chungminhk€t quasau: L Pm(t)Pn(t)dt=0 Voi n ~ m Nhan xet: dk ( ) ) " 1 dk ( ) ) " 1 - x- -1 - - x- -1 - o d k x=1- d k x=-I -X . X voi k =0, 1,...,n -1. Khi d6b~ngtichphantungphantaduQc: -.\.() ChuangIV: ChlnhhoamQtsobaitoaDmoment tPo(X)PJX)ctx=~( 1 ,I dm(x' -1)" d'(x' -I)'\2n.n!)2m,m!)11 dxm ' dxn dx -1 i" dm+1 ( d n-l ( 2 ) n = 2 m x-I 2n.n!.2m.m!I dXm+1 x-I). dxn-I dx ' ...... - (y 1 r dm+n - -1 '2n,n!21l1.m!1ldxm+n(X2 -It.(x2 -1)ndx(4.2) Kh6ngma'ttinht6ngqmitgiasli'm2m<ill +n. Do d6 bi~uthucduoi da'utkh phand6ngnha't0 =>Lpn(x)Pill(x)dx=0 vdi n 7=m .:. Chzlngminh: rP;(t)cit= 211 2n +1 Voi n =ill thl tli'ketquatrentac6 : i P2( \r1 = (- 1)" i d211(x2 - 1)" (.2 - 1),dn XJ-lX ( )' J X X1 2n 1- 1 dx-n.n. = (2n).(-1)"i ( 2 - 1)'" \ .( ), x ex2",n!- I ( d'l1- 2- "_;Do ,(x 1) - (_n), daodx-11 (4.3) ham cap 2n coa da thueboc 2nJ Xet : In =L (x2 - l)ndx Ta tinhlnb~ngtkh phantruyh6i. D~t: !u~(X2-1r~ !u'~2nx(xl-1r-' lv =1 lv=x Khi do: J n = ~(X2-1 Y J ~l- 2n L X2(X2-1 )"-1dx =0 - 2n L [(x2-1)+ 11x2- 1Y-I dx =-2nlL (X2-1)"dx+t (X2-1)"-1dxJ =-2n(In+In-I) --1-1 ChuangIV: Chinhh6amots5bairoanmoment"" . =>In =-2nIn -2nIn-1 =>(2n+l)Jn =-2nIn-' 1 =- 2n I n ? I n-I _n+ ? 1 =-::"'11 ,,0 .J I~ =-~l = 2.4 I .- 5 I .., ~ 0J.) 1, =-~l~ =-2.4.61 ~ 7 - 3.5.7 0 .. T6ng quat: ( ) 2.4...2n I In = -1 n 3.5...(20+1) 0 M~t khac:10=2 1 =(- I)" 2.4.6...211 ')=> . ( r-1/ 3.5...2n+11 (- I) " 2" IJ. .n.- - 3.5...(2n+1)' The' (4.4) VaG(4.3) ta duQc: (4.4) I P2(x\r1x=(2n).(-lt. 211,n! .2I "JU 2" I 2" I ..,- 7 (2 1).n. .n. J.) n + - (2n).2 - 3.5...(2n+1)211.n! - [3.5...(2n-1)][2.4.6...2~]2 [3.5...(2n-1)]211.n!2n+1 2.4.6...2n 2= 2",n! 2n+1 2".n! 2-- 2".n!. 2n + 1 2= 2n+1 -1-2 n =1 => n=2 => n=3 => ChuangIV :ChinhhoamQtsobfliroanmoment { Lpn (t).Pm(t)dt=o V~ytaco: r p2(t)dt=~11 n 2n+I ne'un:;t:m .:. Binh nghia: 1n(x)=.J2n+1.Pil(1- 2x ) Tli chungminhtren: Lpn (t)Pm(t)dt=0 voi n =1=m £)~t: t=1- 2x Khi d6tu: tPn(t}Pm(t}it=o=>lpJl-2x)pJI-2x }ix=O => l1n (t}1k(tXlt=0 voi n :;t: k Tuang tl;1'ta co: 11~(t)dt=1 (4.5) (4.6) Tli Ln la dathucb~cn va tli (4.5),(4.6)~ (Ln)la daytn!cchu5nd~ydli ? trongL- (0,1). - '. - n (n+k)! (t-It Thay t - 1- 2x vaG. Pn(t)- L ( ) ( Y'~k=o n - k ! k! 2 Ln(x)=.J2n +1.Pn(1- 2x) =.J2n+1I (n+k) (1- 2x- 1Y k=o(n- k)(k!)2. 2k =.J2n+lI(n+k) (-Iy.xk.2k k=o(n- k)' (k!)2.2' =.J2n+1i (n+k)!(-IY .x' k=o(n- k) (k!)2 => n Ln (x)="C XkL.. nk k=o Cnk=.J2n+1.(-lY. (n+k)! (n- k)!(k!)2 Voi Cho ~=(~k)If!daysotht,I'c. Ta dinhnghladay: i Ai =Ai (~) =L Cjj ~j j=1 k =0, 1,2, ... -+3 ChuangIV : ChlnhhoamQtsobiliroanmoment D~t n pn =pn(}.t)=I Ak(~)Lk(Lk dong vai tra ej C5chuang III) k=O Khi d6pnlelnghi<$mcuabai toanmomenthU'uh~n(4.2). Truongh<;1pdli ki<$nchinhxactacoketquasail : Dinh 1£4.1.. Cho !J.=(!J.k)leldayso thl,l'c.Di~uki<$nc~nvadud~(4.1)conghi<$m1a: ? ~(~ci;~;J <00 (4,7) Voi CiJ' i, j =0,1, ...duQCxac dint boi C[i =(2i +1)112,(-l)J (i +i)!" ( '_' )1( '1) 2 1 )'}', neu j ::::: i va Cij=0 neuj >i. Neu u lelnghi~mduynhatcua(4.1)thlpn(!J.)~ u trongL2(0,1)khin' ~ 00 (pnlanghi~mcuabilltoanmomenthUi.:thi;ln(4.2) iu(x)Xkdx=jLk ) Honnilaneunghi~mulatrongHI(O,1)thlllP" (jL)- ull~ (1 )Il ull HII 2 11+1 . Y nghla: dint 1yneu leu di~uki<$nco nghi<$mcua (4.1) va ChUdgto nghi<$mcua(4.2)hQitl;lv~nghi<$mcua(4.1)trongL2(0,1) khi n ~ 00. Tru'ongh<;1pdffki<$nkh6ngchinhxactacoketquasail: 4.1.2 Dinh Iy 4.2: Cha UnE L 2(0, 1)1anghi<$mcua (4.1) tu'ongling voi !J.() =(!J.~) D~t: f(t)=sz:Js (2t +1"'.3" vdi 0< £ <1; n(£)=[CI (£ ~IJ] ([x] 1elsonguyenIonnha't:::::x). Khi d6 t6nt?i mQthamso Yj(f:);0 <f:< 1 saGehoYj (f:)~ 0 khi f:~ 0 va doivoi nhlingday !J.thoa:ll~- ~oL,=S~pl~k- ~~I< E Taco: IIpn(E)(!J.)-uoll:::::Yj(f:) 44 Chu'angIV :ChlnhhoamQtsobaitoanmoment Han nilane'uUoE HI (0,1)thl : Vdi Il uo l ! H'(a,I) II II? I Ilpn.~E)(Il)-Uo ::;8-+ C(c:) I J -I ( 8.Js JC(E)=2(2ln 3+2'ln2 .Inl n12.* .:. Ynghia: Trongtru'dnghQpdil ki~nkhongchinhxacdinh1)1dachI ra Cl;l th€ hamfer)vason (c:)saochonghi~mcuabairoanmomenthil'uh?n (4.2)xffpXlnghi~mcliabairoan(4.1)NgoairataconcodanhgiaduQc saiso. D€ chungminhdinh1)14.I , 4.2tadt1avaob6d~sail: ., , 4.1.3 Bo de 4.1 Doi vdi m6i v trongHI (0, 1)taco: ? ~( )?' rl "26aI +~ 2k +1- ak ::;1,\'! dx k=2 vdi ak =£V(X)Lk(x):1x Chllng minh: Trudehe'ttaxettru'dngh<..Jpdathuc.Ta co: v =I AjLj; Aj =!Lj(x)v(x}:ixj=o 1\ Lffy d?ohamhaive'taduqc: v' =I AjL~ .1=0 .:. V6i 0 ::;i ::;k I Tli SPm(t).Pn(t)dt=o Ta co: (L'jJLk) =o -I vdi 0 ~j ~k (4.9) Dod6: (V',Lk)=(~Aj Lj,L, J =tAl (L; ,Lk) 1=0 n = L Ai iL~.Lkd'( l=k+1 (4.10) k=0,L n (Do (4.9) Hen t6ng tu k + I den n). .+S ChuangIV : ChlnhhoamQts6baitoanmoment .:. VfIik <i: Tich ph~ntUngph~ntadll<;Sc: (Lj, Lk)~lLjLkdx { U=Lk B~t: Y'=Lj => { U'=Lk y=Lj Khi do: . 1 1 r. (Lj,Lk)=Lk.Lj0-1Lk.Ljdx =L j (l)Lk (1)- Lj (O)Lk(0) Thay t=1-2x ta dll<;Sc: Lj(x) =(2j +1Y/2,Pj(1- 2x) =(2j+1)1/2~~[(1- 2X)2-1p 2J.j!dxJ =(2j+1Y/2.-+-~(4x2 - 4xy"( ~ ) j 2J.j! dxJ 2 (Do difLtt =1- 2x va d(,loham clia ham h<;Sp) . ( ) j 1/2 1 dJ . -1 =(2j+1) .--:-:--,-[-4x(1-x)f.- 2J.]! dxJ. 2 =(2j+1Y/2.(-1Y .4jdj[x(1-x)f (-1Y2J"1 ..-.J. dxJ 2j =(2j +-1Y/2.*. dj[x(x~1)f.J. dxJ ~ Lj(x) =(2j+1Y/2..!.dj[x(1-x)fj! dxj ~ Lj(o) =(2j +q/2 (4.12) (VI [x(1- x)]j=(x- x2yco56 mnnhonha'tla xj ~ d~ohamde'nca'pj co M s6 tl!doHij!, caesf)conl(,lic-ochuaXknenthe-x =0 VaGta chi con l(,lij! ~ L/o) = (2j +1Y/2.\.j!= (2j +1)1/2J. 46 ChuangIV :Chinhh6amQtsobflitoaDmoment TudngWtaco: Lj(I)=(-I)1.(2j+1Y/l Tli (4.10),(4.11),(4.12),(4.13) taauQc: (4.13) n " (v',Lk)= LA/2k+1Y/2.(2j+1Y/2.[(-I}'+k-1] j=k+1 (4.14) Do (4.14)vado(Lk)lahQtnlcchu§nDen: 11-1 Ilvf=I(v',LkY k=o =~[itAi (Zk+1)1/2.(Z j +1)'12(1 - (-1 y")J (kh6ngphl,lthuQCj Denaua 2k + 1fa ngoaiauQc) =%(2k+{t,A.j(zj+l)"'~ - (-It' J]' 11-1 =>llvf=I(2k+l)S; k=o (4.15) Voi: Sk= IA.j(2j+lY/2[1-(-1)J+k] j=k+1 (4.16) Tli (4.15) taco : 0-1 IIvf=I (2k +1:;S~ k=O 0-1 =s~+3S~+I (2k+1:;S~ K=2 0-1 0-3 =S~ +3S~+4I S~+I (2k+1)s~+2 K=2 K=o M~t khac : n-l Ilv'112= I (2k +l)s; k=o 0-3 0-1 =I (2k+1:;S~+ I (2k+1)s~ k=o k=n-l n-3 =:L)2k+l)s; +[2(n-2)+Ils'Ll+[2(n-l)+1ls'LI k=o 0-3 =I (2k+l)s~+(2n- 3)s~-1+(2n-1)s~-1 k=O 47 Chu'ongJY: CrunhhoamQtsO'baitoaDmoment Tli hai ketquatrentadu'QC: 0-3 0-1 211v'II=I (2k+lXS~+s~+J+s~+3S~+4LS~ +(2n- 3}s~-2+(2n-l}s~-l k=o k=2 0-3 =I (2k+3XS~+S~+2)+2(S~+s;)+ 5(S~+sn+ s~+s; +(2n-1}s~-2+(2n+1)s~-1 k=2 . Tli (4.16)taco : S k - Sk+2= 2Ak+l(2k +3)1/2 vdi k =0,1,2,...,n- 2 (4) 7) Th:HV?y : Sk - Sk+2 = i)"j(2j +lY/2[1-(-ltk]- I Aj(2j+1)1/2[1- (-ltk+2] j=k+1 j=k+3 = IAj(2j +1)1/2[1-( lY+k]- IAJ2j +1)1/2[1-( lY+k] j=k+1 j=k+3 k+2 [ ]= LAj(2j +1Y/2l1-(-lY+k j=k+l =Ak+I(2k+3Y/2l1-(-vlt+1J- Ak+2(2k+SY/2l1-(-v1Yk+2J 2 0 ( . ) 1/2 = 2Ak+12k +3 Khi dotU(4.17)taco: 2(S;+SLJ~lsk -Sk+l ~4(2k+3)./L~+I (4.18) M?t khactli (4.16): So-2= IAj(2j+l)'/2[1-(-I)j+0-2] j=o-l =An-I[2(n-1) +1r/211-(-lY-I+n-2j+An(2n+1Y/2[1- (-"l?n-zj 2 0 =2An-1(2n-1)1/2 =>Sn-2=2An-1(2n-lY/2 (4.19) Tu'ongtl,l'ta co : ( )1/2Sn-I = 2An 2n +1 . (4.20) Tli (4.18),(4.19),(4.20)tadu'Qc: 48 ChuangIV :ChinhhoamQtsobfliroanmoment n-l 211v'1122 L2(2k+3y'A ~+I + 12'A ~ + 50'A ~ k-2 '---v--:-' ( ) '---v-:--' ( )- '2 2' 2Vng v(fi 2 S,,+S2 Vng vdi 5 SI +S~ .. n =>Ilvf ~6A~+I (2k+1)2.At k=2 (4.21) Mi[ltkhactUke'tquad ph§nd§u chungminhtac6 : n v=IAJLJ ' Ai = iL,(x)v(x~ J=O => Ak= lLk(x}v(x)ix= lLk(x).IAJLjdx j=O n =I AI iLk (x)Lj{x)ix J=O =0(voi k >n) Do d6 tu(4.21)tac6thSmdrQngfa VOcling. =>Ilvf ~6A~+f (2k+lYA2k k=2 (4.22) V~ytadachungminhdU<;1Cbatc1Jngthuccuab6d~trongtruongh<;1pv la dathuc. .:. Xet trztdnghdpv E Hi (0,1) Pmla daydathucsaochoPm~ VtrongHI (0, 1). Do (Pm)la dayc1athucnenapdl,lng(4.22)tac1U<;1c: '" IIp,~1I~ 6A~m +I(2k +lYA~m k=2 (4.23) Voi AkIn= lPm(x)LJx)ix Ak= lv(x)Lk(x)ix (dod ph§nd§ucuachungrninh) Khi Pm~ v trongHI (0, 1) thl P'~l~ vi. Th~tv~y: doPm~ v trongHI (0, 1)nen: Ilpm-vIIHI(o,l) ~O khi m ~CX). 49 ChuangIV: Chinhh6amQtsf)bairoanmoment Ma c;>Ilpm- vll~I(OI)= !{(Pm- vr +[(Pm- V)']2}dx:2:i[(Pm- v)']2dx = i[(p'm-v'Wdx =IIp':l- vf => Ilp~- v'II~ ° khi m ~ 00 hay P,:,~ v' M~tkhacdo tint lien Wccuarichvohuangtaco : (Pm,Lk)~(v,Lk)khi Pm~V trong H'(O,I} Tucla Akm~Ak khim~oo. Tli' haiketquatrenva tIT(4.23)taduQC: Ilvf :2:6a; +I (2k+lr a~ k~2 Vai ak = iv(x)Lk (xjix =Ak V~ytadffchungmint xongb6d~. .:. Chllng minh dinh IV4.1: 'l: Do (Lk)la h~tn,tcchuinnenu=L AkLk vai u E e(0,1)(Lk dongvai tra k~O nhu ek; (1, X, X2,...)dong vai tra nhuaChuangIII). - Chungminttuangtl;1'nhuaChuangIII tacoulanghi~mcua(4.1). Taco: n pn =pn(I1)=LAkLk k~O (pnla nghi~mcua(4.2)) 00 u =LAkLk k~O (4.24) V~ytaduQcpn~ u trongL2(0,1) khin ~ 00, .:. D€ chung mint Ilpn(11)- till ::; 1 Ilull, n =1,2,... ta dl,l'avao b6 d~4.1.2(n+1) H Ta co: 6a~+I(2k+lY a~::;!Ivfdxk~2 =>I4k2A2k ::; !juf dxk~O 50 ChuangIV: ChlnhhoamQtsobairoanmoment M~tkhac: !Iur dx::;llull~I().I) I 4k2A2k ~ Ilull~I(O,I)' k"O =:> Tli (4.23)va (4.24)tac6: u - P"(~L)=L AkLk k>11 =:> lip11(~) - ul12 =L At k>11 (4.25) (Do (Lk)1aco s6tn,tcchwln). Ma k >n ~ 4k2~4(n+lY (4.26) Tli (4.25)taduQc: (n+ly.llp"(~)- ul12=(n+lYLA"k k>11 - 4(n + 1)2LA2k - 4 bn 1 ~- L4el; 4bl1 (do 4.26) ~~f 4k"At4 bo 1 7 ~ 41Iull~i(O,1) 11 2 1 7 V~y: (n+1)2.llpn(!l)- u ::;41Iull~i().I) => lip" (11)- ul12 :0 4(11 ~I )21Iull~,(nl) =:> IIpn(u)-ull ~ 2(nl+l)'"uIIHi(OII vai n =1 '), -, ... Ta dii chungminhxongdinh194.1 51 ChuangIV: Chinhh6amQtsobairoanmoment .:. Chz?ngminh dinh IV 4.2: Theotinhchatcuachu:1ntaco: Ilpn~)-uoll~Ilpn~)-pn~o)I+llpn~o)-uoll n n , Tli pn =pn~)=:LA, (Il)L,=:L:L C,pllpta co: k=O ,=0p=o pn~)-pn(llo)=fic,p~p -1l~)Lp ,=0p=O (Voi Ckp=0 ne'uk <p) M~tkhac(Lp)Ia cdsatn;1'cchu<1nnen : IIp"0)- P"~"11'=~(~Ck"~lp - ~;)J2~g2.~(~iCkPI)2 (Do giii thuye'tIIIl - Il 0II", = S~plll, - Il~I < E) , 1/2 k (n+k)! TuCnk =(2n+1) (-1), 2 tadude: (n - k)!(k!) , Cm)=(2m+lY/2(-1)j, (m+j) (j!y.(m- j) M~tkhacapdl:mgcongthuc: (a +b +C)n = "" ~,ai, bJ c'L.. "" kli+j+k=n1.J, . Voi a=b=c=1;n=m+j; i =j ;j =j ;k =ill - j , => . \m+j). ~(1+1+It+) U)- .(m- J) (m+j) 3m+J=> ~ (j!Y.(m- J) =>ICijl~.J2i+l,3i+j i i =>:LICijl ~.J2i +1.:L3i+j j=O j=O i =.J2i+l,3i:LY J=O 52 ChuangIV: Chinhh6amQt56bili lOanmoment - ~2 ' 3i (3 i+1 )- '\,ILl +1. - 1 3-1 .J 3 -.2i, ::; 2n +1.-..J . . 2 Tli Ckp=0 voik <ptaco: 00 k I Ckp =I Ckp p=o p=o V?y : (f ICkPI)2 =(i iCkPIJ 2 ~( 3 ) 2.34k(2k +1) p=O p=o 2 Tli ket quatrentaduQC: ~(~lc"lr,; (~)'(2n+If~3" _ ( 3 J 2 I34(n+l)-1] - - .(2n+1). 42 3 -1 (VI k ~n nen2k +1:::;2n+1) (t6ngcuac1psonhann+1soh<;lngyoi: q=34 ,Ul =1). J ~ ( 3 )-.(2n+1)~. 34n2 80 f)~t: f(t)=(~X:~f'(2t +1)'/'3" =~(2t +1)1/2.321 815 Khi do tItchungrninhd dinh19(3.4)taduQc: Ilpn(E)(Il)- pn(e)~0)1~ 81/2 .:. Chungrninhtu'dngt1!d dinh193.4taduQc: IIpn(e)~o)-uoll=( f (fCiJIl~)2)1/2 j=n(e)+li=1 f)~t: T1(E)=EI/2+ ( I ( fCkPIl~ ) 2 ] 1/2 k=n(e)+1P=o Tli (4.29),(4.30)taduQc: (4)7) (4.28) (4.29) (4.30) --.Jj ChuangIV: ChlnhhoamQtsobaitoaDmoment Ilpn(E)(Il)- uoll~11(8) Khi 8 ~ 0 thi8-112~ 00=> [1(8-1/2)~ 00 Ma n (8)=[[1(8-112)]Denn(8)~ 00 M~t kh';C tii' gia fuuy"t ~(~C'~jr <00 1 Ta duQC: f (f CkP~~) - -j> 0 khi 8 ~ 0 tuc la n (8) ~ 00 (chungminhk=~l P=o bangphanchung). .:. Cu6i clingtachoUoE HI (0, 1). Do dinhIy 4.1taduQc: Ilpn(E)(llo)-uoll~2(n(:)+1rlluollHI(u.l) .:. BG-nh giG-n( E): TiY djnhnghlan(E)=[CI(0-~)]=:> n(o)+1>[-I (0-lf2)(dinhnghlaph~n nguyen). Do f Ia hamdondi<%utangDentUketquatIeDsuyfa : r[n(E)+1];'{C'( E.i J] I =>f[n(8)+1]~8-2 ~~~ I 8-15 2t8 +1.321,~ E-2 (4.31) voi tE=n(8)+1 M~t khac to+1~2" '\IteE R Th~tv~y: xethamso'get)=2t - t - 1voi t E R g'(t)=2t -1 g'(t)=0~ 2t -1 =0~ t=0 Bangbienthien (4.32) 54 t '-00 0 +00 - 0 +g'(t) +00 +00 get) I ------.. 0 ChuangIV: ChinhhoamQts~baitoanmoment D1favaobangbienthientac6: 2t -t-1~O V'tER ~2t~t+1 Tli 2tE+1::;2tE+2 =>,j2t[ +1~,j2(to+1)~.fi.-Jy (do4.32) =F2.2 tole IE =>,j2t c +1~.fi.22 (4.33) Tli (4.31)va (4.33)taduQc: 27 t< 8- 0- .J2.22 321< -~ -..jS . >8 2 1 8J5~3 21 "" r ~ 2. - 27-..j2-..jf, t, => eln22 In32t< 8 0- 5 .e > -..jJ - 27J2F£ (do a =elna) 2In3.tE+~ln2 s.J5 2 > =>e - 27J2...Jr. (21n3+~.ln2}c> sJ5 =>e - 27J2'[; L1YIn haivetaduQc: 1 J ( 8-15 J(21n3+21n2 tE~ln 27J2Fs ) -1 ( sJS J=>tc ~(21n3+~In2 .In 27.fi.,J; (4.34) M~t khac tE=neE)+1 ss ChuangIV: Chinhh6amQts6baitoanmoment ) -1 ( 8Fs J::::>2(n(&)+1)=2tc~2(21n3+~ln2 xln 27J2F& The'vaoike'tqua lipn(E) (/l 0) - u0 II ,; [( \ ].lluaIIH' ta dli<!c:2 n 8 + 1 (01) IIpO(&)(IlO)-uoll~ C~8).IIUoIIHI(O'I) ) -1 ( 8-15 JVdi C(E)=2( 2103+~102 .10 n.fiF& J -I ( 8.J5 JM~tkhactir t, ,,( 21n3+~In2 .In 27../2.fo ) -1 ( 8-15 J=> n(o)=t,+1<0(21n3+~In2 .In 27.fi..r. +] Tli bat d~ngthuctrenchopheptau6c1uQngduQcs6 neE). V~ytadachungminhxongdinh194.2 4.2 Chinhhoa bid toon momentcua bh~nd6i Laplacenglidc: Bilitoan TIm ham uEL2(0,oo)saocho: ru(x).e-kxdx=flk k =1,2,.... f)~t t=e-x khidobaitmlntrdthanhiw(t)tkdt =Ilk+1 V6i k =0,1,2.... w(t)=tu(-Int) O~t~l (dachungminhd chuangI) 2 1 Ta co: !lw(t)1dt= !Itu(-In t)1dt. = re-2xlu(xf .e-xdx 7 = [Iu(xf.e -3xdx (do4.34) (4.35) 56 ChuangIV: ChinhhoaffiQtsob:tiloanmoment Do done'uuEL2(O,co)=>flu(xldX <00 Ma x~0=>e-3x ::;1 -, 2 =>[Ju(x)1 .e-3xdx::;r iu(xf dx<-co Tlidotaco:ne'u UEe(O,X) thl \FEL2(OJ) , Ta vie'tl<;1i:llw(tf ell=r:1I(x)12e-;'dx Tli (4.36)vaapdlfngdinh19(4.:2)tadu'QC: (4.36) DinhIi 4.3 Ch L? ( ) I ' -h'~ ., (4 3 ~ ) , ,. n ( n n ) 10 UnE - 0,00 ang l~mcua /..) tu'ongU'ngvol: /1 = /1,,/12"" E OC) £)~tf(/) = 2~(21+l))~.3"/8"5 Voi 0 < £ < I; n(E)=[r-'( E-~J] Khi do t6n t<;1imQthamsC;ll(E) (0 <£<1) saocho 11(£)~ 0 khi £~ 0 vadoivoimQiday ~L=(~L,,~l2,...)th()aII,U-,Li"!/,,:::;E;taco: Ilq"(l:)(,u)-u"t :;//(1;) (4.37) Voi q"(C)(,u)=eXp"(C)(Ji).e-r (4.38) jI=(/12,/13"") IIIII:= [II(x)12e-:'c£'\ (4.39) va pn(E)(/1) nhu'trungl!inh 194.:2 Han nua ne'u UoE HI (0,:1'..)thl: Il qn(&)(J.!)- u II ::; f)< -+-311Uo11H'(0, co) 0 P C(E) Voi C(£) nhu'trongc1inhI)' 4.2 Y nghia: Bai roantim hamu khi bierbien d6i Laplacecuano t<;1icac di€m nguyen du'Qcqui v~bai roanmomentHausdorffmQtchi~u.Tli nghi~mpn(E)(/1)cua bai roanmonenthii'uh<;1ntaxayd~(ngqn(&)(J.!)=ex.pn(&)(jI).(e-X)va qn(E)xapXl nghi~m Uncuabai roanbangeacheh';r~lhamfer)va so neE)Clfth€. 57 ChuangIV: Chlnhh6amQtsobairoanmoment Chungminh Tli dinh1;'4.2tadu'<;jc: lipn(E)(II) - w0 II ::;T\(e) voi II =(1l2'1l3,...) (4.40) Tac6: wo(t)=t.uo(-Int) M~tkhac: Ilpn(E}(II)-wof dn 7 ilpn(E) (lI)(t)- wo(tfdt 7 =flr'(C)(]7)(e-X)-e-x.uo(xfe-Xctc (do co(}(t)=e-x,uo(x)) =flpll(£)(]7)(e-X)ex.e-x-e-x.uo(xf.e-Xctc =fle'p"(E)(}l)(e-X)-uo(xf.e-3xdx fl ole) 1 2 -3x= I q (,u)-uo(x).e dx (dogill thuyetqn(E)(Il)=eXpn(E)(}l)(e-')) = Ilql1(E)(Il)-uoll: =>Ilpn(L)(]7)- woll=Ilq/(C)(,u)- uot Tli (4.40)va (4.41)=>Ilqn(E)(Il)- uot ::;T\(e) (do (4.39)) (4.41) V~ytadachungrninhxang(4.37) Bay giCitaxet Uo E H1(O,oo).Khi d6: w'(t)=u(-Int)- u'(-Int)}.tt =u(-In t)- u'.(-In t) =>W'D =uo(-Int)-u'O(-Int) TheadinhnghlachuffntrongHI (0,1)tac6: IlwoIIHI(O.l)=\!(w~+w~~tr S8 ChuangIV: Chinh h6amQtsf)bfli toanmoment =(1W;dt+ !W ~2dt(' ( r ) x ( r 2 ) x ::; !w~dt +!w ~dt (do~a+b ~.Ja+-Jb voi a;::0,b;::0) ,,;(iII.U" (-In nl' dtr+(i[u"(-In t) - u'" (-1n t)]'dt ) 11 ( , ) 1< ( ) 1/ ? /2 - /2 , /2 ~(l!t.u,,(-,nt)l-dt + iluo(-,nt)!dt + l!u'o(-lntWdt (4.42) M~tkhactli' llw(tfdt= [lu(x)j".e-1xdxva w(t)=t.u(-lnt) =>flLuo(-In nl2dt=[Iuo (x)!2.e-3xdx flu'o(-Int)ldt= [u'o(x).e-xdx Thayvao (4.42)tadu'<;5c: IlwoIIHl(O.I)~(!luo(x)12e-3XdXr +([IUo(x)\2.e-Xdxf +([lu'(x)12e-Xdxf ~31IuoIIHI(O,OO) (do IluI12HJ(O.OO)=[(U2+U'2~X;::llul12va IIUI12Hl(O.oo);::lluf, x;:: 0 =>e-3x~1) V~y : IlwoIIH"(O,I) ::;31IUoIIH'(O.OO) (4.43) M~tkbic tli'dinhIy4.2taco: IlpO(E)(iI)- w011 ~EYz +IlwoIIH'(O,I)C(E) (4.44) (dothay Uobai wJ Tli' (4.41),(4.43),(4.44)tadu'<;5c: II OrE) - II < Yz 3I1uoIIH'(O,oo)q Uo - E + P C(E) V~y ta dff chungminhxongdint1y. 59 ChuangIV: Chlnhh6amQtsobfliroanmoment 4.3 CblnbboabaitoaDnbH~tngu'dctbOigian Tli bai roannguQctimv(x,y)(nhi~tGQt~it =0) ne'ubiet (X-;)2 +(Y-'1) 2 [ [) v(~,l1)e- 4 d~dll=4nu(x,y,l) Ta xettruonghQpv(x,y)=Ov6ix<0,y<O.Khi dod&ngthuctrencoth~ vietthanh: (X-;)2 +(Y-'1) 2 r rv(~,l1)e 4 d~dll=4nu(x,y,l) Voi { X =-2m y =-2n m,n = 1,2... taGuQc: " , C;-+'1- e-(m2+n2) r rv(;,ll)e-~ .e-(I11~_n'1)d;dll= fmn (4.45) voi lOIn=4;ru(-2m,-2n,l) { s=e-; Bang cachd6ibie'n ta GlroC: t =e-'1 rr ( ) ij dd - ""- 01 ;l) J, cos,t S.t s t - /-lij VOl 1,J - , ,-... (4.46) J J ln~ ,+In~ t---- V6i co(s,t)=v(-lns,-lnt).e 4 f-lij = 47Z'.e(i+l)2+U+I)2u(-2i - 2,-2j - 2,1) (vi m=i + 1,n =j + 1) EHiyla bairoanmomentHausdorffhaichi~u,tacoth~dungphuongphapd ChuangIII d~chlnhhoano. Ta nh<1cl~ivaikyhi~u: Cho m,n =0,1,2,...taG~t: v 1 dm 1,11(s)=(2m+1Y2.- - (?(l-sf) m!d? Ll11n(s,t) =Lm(s).Ln(t) Theake'tquadml,lc4.1taco(L I11n) la cosdtn;1'cchucfntrongL2(I) voi I =(0,1)x (0,1) B6i v6i m6i day s6 thvc /-l=(Ili) 1, J =0,1,2,... ta xac dinh day A=A(/-l)=(Ai) nhusau: 60 ChuangIV: Chlnhh6amQtsobairoanmoment i j IIe e I' . ( + ' )1..=.. = .." Y, J mJ, AIJ AI/~) IP' Jq'~pq' VOl Cmj =(2m+1) -(-1), "" 'I p=oq=o (J.) ,(m-j). O::::j::::ri n D;Ittpn =pn(~)=I Ai/~)Lij i.j=O ;2 "HI2 qn(~,l1) =e~.pn(e-;,e-q) GQiL~1akhonggiancachamf saGcho: -7 , c+rr -(c; +'7)--=-- JP.f E L"(R+xRJ voip(C;,17) = e 2 4.3.1Dinh Iy 4.4 Cho (fmn)1adays6th1!c.Neu(4.45)c6 mQtnghi~mV trangL2 (R+,R+) h' n L?t 1q ---) V trang ~ Hon nuaneunghi~mV 1atrangWi,:(R+,R+)thi co mQth~ngsO'C dQcl~pvoi Vvan saocho: II 11 - V II :::: C ,llvllw'Jq L; 2(n+1) voi (theos1,I'xacdinhcuakhonggian L~) II~t,= IIJP ~IL ? v~E L~ ChU'ngminh Tli VEL"'(R+,RJsuyfahamsO' w(s,t) =v(-Ins,-In t).e In: s+ln2t 4 thuQc L 2(1) Do (4.46)w thoabai toanmometnHausdorffhai chieuneBtli'dinh 1"9(4.1) taduQc: Ilpl1 - wllLl(1) ~ 0 khi n ~ CfJ (4.47) (Nghi~mcua bai toanmomentHausdorff hai chieu hUll h~nhQi tl,lden nghi~mcuabai roanmomentHausdorffhai chieuvoh~n), 6) ChuangIV: Chinhh6amQtsobaitoanmoment M~tkhac: Ilpn - wl12 L2(I) =i ilpn(s,t) - w(s,t)12dsdt. ( , , J 2 =rr q"(e-"e-")-v(~.T\)e-'-:"- e-("")d~dT\ (dod6ibie'ns =e-1;, t =e-11va dint nghlap",q") =r r (qn (e-~,e-q)- v(~,TJ)Y p(~,TJ)d~dTJ =IIq"- vl12L~(dodint nghlachuifntrongL~) =>Ilpn- wll\2(I)=Ilqn- v112,L" (4.48) Tli (4.47)va (4.48)tadu<;1C: qn ~ vtrongL~ Ne'uv E W1""(R+,RJ taxet:. In's+ln' t----- w(s,t) =v(-Ins,-Int),e -! Ap dl,lngc6ng thuc d,~l0ham cth ham h<;1ptadU<;1c: ~=-G.~+v~:sJe - '0'';'0' I . aw =>11- :::;Collvllwv as L2(I) Ttiongtl;! !Iaw l! :::; ColivllwlJ at L2(I) Tli dint nghlachuifntrongHI(l) tasuyfa: IlwIIHI(J):::;Cllvllw1.x (4.49) Tli dint ly 4.1taco: lip"(11)- wIIL'I') .,; 2(n1+1)IIwIlH'I'). TU (4.48),(4.49),(4.50)tac6: Ilq" - viiI; ,; 2(nC+I) Ilvllw" (4.50) 62 ChuangN: ChinhhoamQtsobairoanmoment V~ytadachungminhxongdinh1'1. TruonghQpdukil$nkh6ngchinhxactacoke'tquasail: 4.3.2 £>inhIi 4.5: Cho VoELa:>(R+x RJ la nghil$mcua (4.45)tu'angung vdi fO =(f,~n)trongve' phaicua (4.45). Cho F(8)=729(28+1).348va 0 <8 <1;8 ;:::1320 f)~tn(8)=[F-l(8-X)] ([x]1aso nguyenIOnnhat ::';x) Khi do t6n tqi mQt ham so' 11(8)(0 <8 <1) saGcho 11(8) -)00 khi 8 -)00 va vdi mQiday f =(fmn)thoa: suplem2+n2(fmn-f;n)l< E taco: Ilqn(E)-voll!" ~T](E)1l1,n -p (vo la nghil$mchinhxac, qn(E)la nghil$mungvdi bai roanmonenthuuhqn) Hannuane'uV() E wl,a:>(R+, R+)thi: Ilqn(&)-vollz :::;8X+ Cilvollw'x Lp C(8) 1 I ( s-f5 Jvdi C(8)= 2(21n3+2In2t .In 27-fi.Vr. Chungminh: Theotinhcha'tcuachu~ntaco: lipn (J1)- W0" :::;lip n (J1) - p n (J1() )11 + lipn (J10) - w0 II (4.51) n f)~t pn =pn(J1)= I Ak,kz(J1)Lk,k, kl,k2=0 (4.52) n V di Ln(x)=I CnkXk k=O 63 huangIV: Chinhh6amQtsobairoanmoment Cnk =(2n+l)Yz.(-1)k. (n+k)! (n-k)!(k!)2 Lk,k!(Xl'XJ =Lk,(xt).Lk!(x:). kl k! Aklk! =Ak,k!(~)=I ICkIPICk!P!~PIP! p,=o P2=O Tli (4,52) ta c6: 00 YO ( )n - n 0 = C.C - 0 LP (Jl) P (Jl) L I kiP, k2P2 JlPIP2 JlPIP2 P,P2 kl,k2=O P"P2=O Voi C =0 ne'ui <J ',I} Tud6: lip"(~)- p' (~O)11~ .t(J.:"" c,.,.(,..""- ~:",)J' (dogiathuyetI ~- ~l) I <E) Tli ket qua (] ffil;lC4,1 ta c6: C =(2m+1)~.(-1)j.(m+j)!111) (,,)2( _' )1J. m J. Mat khac (m+j)! ~(1+1+1)"1+.1= 3111+).. (,,) 2 ( -' )1J. ,m J, =>ICjjl ~ .J2i +1.311l+j V~yII Cijl~(2i+1)~,:t31+) j=O )=0 , ~ 1 "',"'1=v2n+1. .J ~.J .j=! ~ .,i(",,+1 )=v2n+1 . .J.J - 1 3-1 ~'i,32i.J2n+1. 2 Do Cij =0 v6'ii <j ta co: (4.53) 64 ChuangIV: Chlnhh6amQtsobaitminmoment p,tJCk'P,Ck'P' I =(,~,ICk'P'I) (,t,lCk'" IJ (4.54) Tli (4.53)va(4.54)taco: ( IJCk,p,Ck2PPJIJ 2 :::;; ( ~J -I.(2kl + 1)(2k2+ 1).3-1(kli-k2) P"P2-0 La'y t6ng ke'tqua trenta du'QC: JJJtCk",Ck",I)'~(~r(2n+1)2{~34Jr ( J 2 7 n 00 ",)-1 -,-1(17+1) 1-~I I Ick,p, Ck2P2 I :::;;(~) .(2n+l)2.[ J ",-I- - ]'l,k,=O P"P2=0 - J 1 (t6ngcuaca'ps6nhancon+1s6hang,q=34) ~(%)'C2n +1)2G~]'3"" D~tf(t)=(~rG~ )e2t+1)3"' = ( E-(2t +l)~ .32t J 2 815 Ta co: IIp*\u) - p*\uO)11s;Eli (4.55) (chungminhtu'dngtvd dinh1y3,4) M~tkhactli dinh1y3,4taco: [ J 2 \ 2 00 ex; n(E) 0 - W = C.C 0 lip (1-1) 011 k'.k2~*)+1 p,~=(J KIP, k2P21-1P,P2 (4.56) ( '\~ ~ 00 Y: 0 I D~t: 11(8)=8 2 + ..~ ( LCk,p,Ck~p~1-L P'P2 ] )k"k2-n(E)+1 PI'P:=O Tli (4.51),(4.55),(4.56)taco: Ilpn(E)(Il)- wallS;11(8) (4.57) (,5 ChuangIV: Chlnhh6amQts6bairoanmoment voi wo(s,t)=vo(-Ins,-Int).e In2s+ln2t 4 Chung minh tudngt1,1'd dinhly 4.2taco 11(8)~ 0khi 8~ 0 Tu (4.48)va(4.57)taco:Ilqn(E)(Jl)- vollL2 :::;11(8)" Neu Vo E WI""(R~)thl tu (4.49) ta co: IlwollH'(I) :::;c.llvollwl.~ (4.58) Tu danhgiasai56d dinhly (4.2)taco: ' I 11C. w i I Il n(E) () -w ll :::;s+ IO<lH(I) P Jl 0 L2(I) C(s) J -I ( 815 '\ VOiC(£)=2(21n3+~ln2 .In 27.J2.;J£j Tu (4.48),(4.57),(4.58)tadu'Qc: Il qn(E) -v II :::;s~ + Cilvollw'0 0 L~ C(S) Voi C(8)tadu'Qcxacdinhnhu'tren V~ytadachungminhxongdinhly. * TubairoannguQcHmv(x,y)tavietl~i: (X-~)2 +(Y-l1)2 [) fo v(~,l1).e 4 d~dll=g(x,y) (4.59) voig(x,y)=4nu(x,y,l) tHy laphuongtrlnhtichch~pd6ivoihamchu'abierv(~,11) _(Xl+yl ) . Tu(4.59)taco:h* v=g voih(x,y)=e 4 Laybiend6iFourierhaivetadu'Qc: 1\ , -, ~. ( ) -(w-+~-)- ( )v w,~.e - 2TCg w,~. (4.60) 66 ChuangIV: ChinhhoamQtso'b~lioanmoment voi 1\ ~ i(x(v+.\'.;) h(w,~)= jh(x,y).e dxdy R2 : 1 /\ , h(x,y)=(I )2 fh(w,~).e-'(Xw+Y;)dwd~-'IT 'R- Tli (4,60) ta thay neLll1ghi~mv t(~nt~ithl ~ dllQCchobCii: A , ' ,', v(w,~) = 2TI:,el\-~~-,g(w, ~), Laybiend6iFourierngl()chaivetadUdc: 1 . 1\' ~" A v(x,y) = - je ". g(w, ~),e-l(m+Y~)dwd~/'TT . -IC R- Trang thl,l'cte takh6ngC(')clCi'ki~nchlnhxacma chi co dCi'ki~ndo duQcco sai so,Ta sexay dl,l'ngnghiC'm\.hlnhboa6ndinhd6i voi nhungthayd6icuag, Cho go nhuCi(4,59)cl) nghi~m Vo E e (R 2)Wonglingcivephaisaocho Jig - got:(R ') < E Ta xaydl,l'ngham Y (~ndinh l16ivdi s1/thayd6i trongg, Neu Yola dutrail ta co th~danhgiasai sogiCi'aV(}V:lv. 4.3.3DinhIi 4.6 Ch bK dJ h ' II CT - CJ II <8 O'? ?0 at angtl1c ~ ~lJL2(R2)- . lasu: 1 fe2(W2+~2)lgo(w,~l.(\V2 + ~2)2dwd~ S E2 R2 (4.61) IT ., I II Ii A £)~tv(x,y)=- r re\\'+~'g(w,~).e-i(;m+Y;)dwd~2TI: "- "- d - i'C h voi h2 = { E 2 ( TI'E , f } In- In~.)E v Kh ' d' " E / / I ' K1 0 VOl 8 < - taco ~((jc uQngsal so: e (,7 ChuangIV: ChinhhoamQtsobflitoanmoment E.Ji IIvo -vIIL"; In{ ~{In ~n1 Chum!minh Tudinh1:9Planchereltaco: ,,21 /\ /\ 11 2 -- vo-v IIva- V L2(R2)- 4;r2 IIL2(R2) ~ fe2(aJ2+~2). I go- g2dwd~+ fe2(W'+<'JlgJ dwd; aJ2+~2~R; \112+~2~R; (4.62) /\ 2 ,/\ (do v(w,~)=21t.ew+t;-.g(w,~)) vdiR;~In{~{In~n ~ 2 I 1 2 2 2/\/\ 1 ,,/\ ~ IIVa- vIIL(R2) ~e2Re g- ga dwd~+4 fe2(m-+,;-)ga (w2+~2)2dwd~ R2 R" R2 (do w 2+~2 ~R 2 va w 2 + ~2 ~ R 2 => 1 ~~) E '? E (W2+~2)2 R: Taco: 2R2 ( R2 ) 2. e E =e E ~'em[~ln~r'r ~{~{ln~rr E2 1 =7" In2(~) 68 ChuangIV: ChinhhoamQt56bairoanmoment M~tkhacrheagia thuye'tjig - goIIL'(R') ~ G 1 I "" " ,,1 1\ 1\ - E- 1 " E- ~ e-R& fg- go dwd~ ~ 2""". l ) .8- = - ( ) R1 E In1 E In2 ~ E E The- (4.61), (4.63) VaG(4.62) ta duQC: (4.63) 1 1 11 7 E 7 1 +- h-V;'(R')~ - In2(!) R: (4.64) M - kh' ,. E ,~t ac YOl : 8<- taco e EEl ->e~ In->I~-< 1 £ E In E £ E I E ~-.-<- £ In E E E ) { E( E ) -I } E ~ R~=In-; lln-; < In-; E =>R;<In- E I =>-4>- ( E ) 2 Rg In-; CQngVaG2 ve'batding thuccho ~4 taduQC:g 1 1 2 -+-<- In2(~) R:-R: Tli (4.64)va(4.65)taduQC: 2 2E2 2E2 Ilvo-vIIL2(Rl)~y= { ( ) -I } 8 In2 E. In E G G (4.65) 69