Hàm Suy rộng CoLombeau 5

Luận văn Thạc Sĩ toán học-ngành Toán Giải Tích -Chuyên đề : Hàm Suy rộng CoLombeau

CHUaNG I

VIEHAM COLOMBEAU

I T~P' H<)PtYiq

.Dinh nghla 1.1:

Voi q = 1, 2, ta gQi udq la t~p hQp cac ham cp E @(Rll) thoa cac di~u

ki~n sail:

f <pet)dt = 1

R"

Ita <p(t)dt = 0

R"

vail:::; lal:::; q

.H~ qua 1.2:

n Vai ~(A) = (27r)-Z fe-iAx<p(x)dx, (A E Rll) la anh Fourier cua cp(x) thuQc udq,

ta co:

n

ii) Da ~(O) =0, 1:::; I a I :::;q

Chang minh:

i) ~(O)=(27r)-z f <p(x)dx =(27r)-Z

R"

-ii) Theo [1] ta co «iA)~Da ;)(A) =D~((-ix)a cp(X))(A)

5

ChQn~= (0, 0, , 0) ta co

f

R"

-Bjnh Ii 1.3:

i) ,;4q =I-0 voi mQi q = 1,2,

ii) ,;41 =:;,;42 =:; =:;,;4q =:; ;4q+1=:;

CX)

iii) n ,;4q= 0

q=l

(Vlly do ky hi~u lien chI chung minh tHrong hQp n = 1)

i) GQi <pet)E YJ(R) thoa f<p(t)dt= 1 (trong rdJ(R)hi€n nhien co ham nhu

R

v~y)

Bay giO ta xet day ham sail day:

<P1(t)= <pet)+ al<P'(t)

<Pq+1t = <Pqt + aq+1<P (t)

Ta luon tIm duQc aq thich hQp thuQc C d€ <PqE dq

Th~t v~y:

Voi q = 1 ta co:

f<pj(t)dt = f<p(t)dt+ a} f<p'(t)dt

= 1 + al.O

= 1

Jt<pj(t)dt = Jt<p(t)dt+aj Jt<p'(t)dt

= Jt<p(t)dt - a] J<p(t)dt

= Jt<p(t)dt-a\

R

R

Voi q tuy Y thuQc N: quy n~p theo q ta se tim duQc aq E C d€ CPqE ,Jdq

ii) DuQc suy ra tn!c ti6p tu dinh nghla.

iii) Gia sa

(jJ

q =1

=> t6n t~i cP E ~q, q = 1, 2,

Theo [1], voi dinh ly Paley-Wiener ta co:

A

<p(~),~ E <C

la ham nguyen thoa: yoi mQi N > 0, t6n t~i CN > 0, yoi mQi~E C ta d~u co:

I

I\

(.\:) I < CNexp(Rllm~l) (1) U 1, ?;:: " ?,.:'

d" b/

k/

hR

cP '? - N ' R a qua cali tam (j goc tQa Q, an III ,

/\

chua Stipp cpoKhai tri€n Mclaurin ham nguyen cp(~) t~i g6c tQa dQ va sa d\lng

1\

Do cp(O) = 0 voi mQida chI s6 a (I a I ~ 1) ta thu duQc:

cp(~) = cP(0) = (2;r) -2:, voi mQi ~ E <C.

7

Khi lay ~E R va cho I ~I c1ulOn, ta se co c1U<;1cmall thu~n voi bat c1~ng thuc (1)

V~y

r:JJ

n ~q = 0.

q=l

II s6 PHUC SUYRONG

.Djnh nghla 1.4:

Voi cp(x) E QlJ(Rn), ta c1i;itCPE(X)= ~cp

(

~

J, x ERn, E > O.

Tli c1inhnghla 1.1, 1.4 va dung phudng pha p c16ibie'n s6 ta d~ dang chung minh c1lfdch~ qua sail c1ay

.Hf qua 1.5: Ne'u cP E ,>:iqthl CPEE ~q

Djnh nghla 1.6:

i) $'0 la t~p hQp cac ham R(cp) tli ,->:ilVaGC

ii) $'M la t~p h<;1pcac ham R trong $'0 ,thoa c1i~uki~n: t6n t'.li s6 tlf nhien

dudng C va 11c1€I R( CPE) I ~ ~, '\IE E (0, 11).

E

T6n t'.li s6 tlf nhien N E N va day y E r (N, y phl;l thuQc R) saG cho voi

m6i cP E ,>:iq (q ~ N) c1~u tlm c1u<;1c2 s6 dudng C va 11c1€ I R(CPf;) I ~ CEy(q)-N,

'\IE E (0, 11).

.Djnh ly 1.7:

i) $'M la mQt vanh voi phep cQng va nhan anh X'.l

ii)J la mQtIc1eancua $'M

(Chung minh c1inh ly nay xin trlnh bay (j ph~n phl;l chudng)

.fJjnh nghla 1.8:

T~p hQp cac s6 phuc suy r<)ng la vanh thudng ~M /J, ky hi~u la C.

ZE <Cta co j(z) =R( CPIJ+ J

ddoR(CPE)=Z, Vcp E~Y11,V8>0

Chang minh:

+ j la anh x:;t VI: chc,>llN =1=> V cP=~Y11 ta co:

8

+ j la d6ng c§u vanh: hi~n nhien

+ j la don anh VI:

n€uj(z) = 0 E C.

=> R(cp) EJ

=> T6n t:;ti s6 N E N va day y E r d~ vdi cPE ~~q (q :2:N) d~u co hai s6 du'ong C, 11sao cho

8

va cho 8 ~ 0+

Suy ra z = 0 E C

Tli m~nh d~ tren ta co z EJ q Z=0 (duQc hi~u theo nghla nhung)

ID.HAMSUYRONGCOLOMBEAU

Ky hi~u: ~O[Rll] la t~p hQp cac anh x:;t R tli ~l x Rll vao <Cma khi c6

dinh cPta duQc R( cP,x) khcl vi mQi c§p theo x

9

.EJjnh nghia 1.10:

T~p h<jp $'M[Rll] g6m cac ham R(cp, x) trong $'0[Rll] thoa vdi mQI

t~p compact K c Rll va da chi s6 a d~u t6n tqi s6 h! nhien N EN sao cho vdi

m6i cp E JiN d~u Hm du'<jc2 s6 du'dng C va 11d~ I oCt R( CPf;,x) I ~ E~ dung

Vx E K, VE E (0, 11)(trong do oCtRia dqo ham dip I a I cua R theo bien x).

.EJjnh nghia 1.11:

T~p h<jp A1Rll] g6m cac ham R(cp, x) cua $'0[Rll] thoa vdi mQi t~p

compact K c Rll va da chi s6 a d~u t6n tqi s6 tt! nhien N va day Y E r sao cho

vdi m6i cPE Jig (q ~ N) d~u Hm du'<jc2 s6 du'dng C va 11 d~:

C Er(g)

E

i) $'M[Rll] la mQt vanh vdi phep cQng va nhan anh Xq

ii) QJf/[Rll] la idean cua $'M[Rll]

(xin trlnh bay b~ng chung minh dinh ly nay d phgn phl;!chu'dng)

T~p h<jp cac ham suy rQng Colombeau la vanh thu'dng $'M[Rll]/./11Rll].

Ky hi~u la y[Rll].

Tudinh nghla 1.13 ta co cac di~u sail day:

+ M6i ham suy rQng Colombe au la mQtlOptu'dng du'dng co dqng:

G =R(cp, x) + /11Rll], R(cp, x) E $'M[Rll]la mQt dqi di~n cua G.

+ Phep cQng va phep nhan hai ham suy rQng Colombaeu du'<jcth\fc hi~n nhu' sail:

G1 + G2 =Rl(cp, x) + R2(cp, x) + J1;[Rll]

G1.G2 =Rl(CP, x).R2(cp, x) + J1;[Rll]

Trong d6 R1(cp,x) la d(;lidi~n cua G1, R2(cp,x) la d(;lidi~n cua G2

1.14.-Gia tri cua ham suy rQng Colombeau G t(;li Xo E Rll la so phuc suy rQng

R(cp, xo) +:J E C.

Trong d6 R( cp,x) la mQt d(;lidi~n cua G trong ~M[Rll]

D~ dinh nghla 1.14 hejp l~ c~n phai chung minh hai di~u sail day:

i) R( cp,xo) E ~M

ii) Dinh nghla 1.14 khong phl;!thuQc vao d(;lidi~n

Th~t v~y:

i) La'y t~p compact K c Rll chua Xo,da chI so a, I a I = O.

Do R( cp, x) E ~M[Rll] t5n t(;li so tlf nhien N d~ vdi m6i cp E udN, d~u tIm

du'ejc 2 so du'ong C va 11sao cho:

IDa R(CPE'x) I ~ ~, Vx E K, VE E (0,11)

E

=> R( cp, xo) E ~M

ii) Gia sa R1, R21a hai d(;lidi~n cua G

=> (Rl - R2)E J1;[Rll]

=> Vdi t~p compact K chua Xoda so chI so a, I a I = O T5n t(;liso tlf

nhien N va day so Y E r sao cho vdi m6i cp E Yiq (q ~ N) d~u tIm du'ejc 2 so

du'ong C va 11d~:

E

C.E y(q)

E

rQng trong C

v D~O HAM CUA HAM SUY RQNG COLOMBEAU

.fJjnh nghla 1.15:

Da(G) =DaR( cP,x) + JI1Rll], trong do G E y[Rll] co d~i di~n la R( cP,x), DaR( cP,x) la d~o ham dtp a cua R( cP,x) theo bien x, con a la da ChI s6 tuy y.

£)6 dinh nghIa 1.15 hQp l~ ta c§n chung minh hai di€u sail:

i) DaR(cp, x) E $'M [Rll]

ii) Da(G) khong phl;l thuQc vao d~i di~n.

Th~t v~y:

i) Voi t~p compact tuy yK va da ChIs6 ~=> ~+ a cling la da ChI s6

Do R(cp, x) E $'M[Rll] lien co s6 tl! nhien N sao cho voi m6i cp E ~JiiNd€u

tlm du'Qc 2 s6 du'c5ng C va 11d6

I

a+13

I

C

D R(CPE'x) ~ N' \:Ix E K, \:IE E (0,11).

E

I

I

C

E

=> Da R( cP, X) E $'M[Rll]

ii) Giii su R}, R2 la hai d(;lidi~n cua G => R =Rl - R2 E ut[Rll] Voi t?P

compact K chua Xo,da chi sf)~(a va ~ la da chi sf)lien a + ~ clingla

da chi sf))

Do R E A1Rll] lien voi mQi t?P compact K c Rll va da chi sf) ~ + a co sf) tl! nhien N va day sf)YE r sao cho voi m6i

cpE '~q (q ~ N) d~u tim duQc 2 sf) dudng C va 11d~:

C Ey(q)

I Da+P R( q\;, x) I ~ . N ' '\Ix E K, '\IE E (0, 11).

E

C.E y(q)

= I D~(D<XRl - D<XR2(cpc;,x)) I ~ ~

E

=> (D<XRl - D<XR2)(CP,x) E ut[Rll].

Tli dinh nghla 1.15 va chung minh tren ta co m~nh d~ sail:

Mfnh di 1.16:

i) M6i ham Colombeau d~u co d(;loham mQi ca'p

ii) D(;loham cua Colombeau cling la ham Colombeau

.Mfnh di 1.17:

Cong thuc Leibnitz v~n con dung voi d(;loham cua ham Colombeau

D<X(GIG2) = ICt.Da-fJGJ.DfJG2.

O~fJ~a

Th?t V?y: Voi R1(cp,x) la d(;lidi~n cua Gl, R2(cp,x) la d(;lidi~n cua G2

D<X(GIG2) = D<X (RIR2) + ut[Rll]

= ICt.Da-fJ R1.DfJ R2 + ut[Rll]

O~fJ~a

= ICt.Da-fJGJ.DfJG2

O~fJ~a

13

VI PHEP NHAN, PHEP CONG SO PHUC SUY RONG (HO~C SO

PHUC) VOl HAM COLOMBEAU.

.M~nh di 1.18:

y[Rn] nho ghep nhung

trong do c E C co d(;lidi<%nk( ep)

Chang minh:

Truoc he't ta co nh~n xet dng: Tli cac dinh nghla 1.6, 1.10 va 1.11 d~ dang co duQccac di~u sail day:

e N€u R(ep) E $'M thl R(ep) E $'M[Rn]

eNe'uR(ep) EJthlR(ep) E J1'[Rn]

e Ne'u R(ep) E h[Rn] va R(ep) kh6ng phl;!thuQc x thl R(ep) EJ

Bay giO ta chung minh m<%nhd~ 1.18 theo cac buoc sail:

+ j la anh X(;l,th~t v~y:

do keep) E $'M => keep) E $'M[Rll] => j(c) E y[Rll]

gia sli' k 1(ep) la d(;li di<%nkhac cua k

=> [k 1(ep)- k( ep)] EJ

=> [kl(ep) - keep)] E J1'[Rll] va keep),k1(ep)E $'M[Rll]

=> keep),k1(ep)la hai d(;lidi<%ncua mOt ph~n tli'trong y[Rll]

=> j kh6ng phl;!thuQc vao d(;lidi<%n

V~y j la anh X(;l

+ j la d6ng diu vanh, th~t v~y:

Gia su kl(CP)la d(;lidi~n cua cl E C

k2(cp)la d(;lidi~n cua c2 E C

=>k1(cp)+k2(cp)lad(;lidi~ncuacl +c2 E C

k1(cp).k2(cp)la d(;lidi~n cua cl.c2 E C

Ta co: j(Cl) =k1(cp) + J1;[Rll]

j(Cl).j(C2) =k1(cp).k2(CP)+ /V[Rll]

=(k 1.k2)( cp) + J1;[Rll] =j( Cl.C2)

=> j la d6ng diu vanh.

+ j la ddn anh, th~t v~y:

Gia su j(Cl) =j(C2) => k1(cp) +J1;[Rll] =k2(cp) +J1;[Rll]

=> [k1(cp) - k2(cp)] E J1;[Rll]

ma [k 1(cp) - k2( cp)] khong phl;}thuQc X

=> [k1(cp) - k2(cp)] EJ

=> j la ddn anh

V~y j la ddn ca'u

Tli m~nh d€ tren, ta co h~ qua sail day:

-Hf qua 1.19:

Ne'u: k( cp) la d(;li di~n cua C E C

R( cP,x) la d(;li di~n cua ham Colombeau G E y[Rll]

15

Thi: j(c) + G =(k(ep) + R(ep, x)) + J1I[Rn]

j(c).G =(k(ep).R(ep, x)) + J1I[Rn]

Ta co thti nhung t~p s6 phuc C vao t~p heJpcac ham Colombeau y[Rn]

nho rich cua hai phep nhung da nh~c d€n a m~nh d~ 1.9 va m~nh d~ 1.18.

Voi nhung co sa tren ta di d€n dinh nghla phep nhan, phep cQngmQt s6 phuc suy rQng c (ho~c s6 phuc Z E C) yoi ham Colombeau nhu' sail:

.Dinh nghia 1.20:

Gia sa c E C co d~i di~n la k(ep)

G E y[Rn] co d~i di~n la R(ep, x)

Z la s6 phuc thuQc C

Ta gQi: c.G =k(ep).R(ep, x) + ,/11Rn]

c + G =keep) + R(cp, x) + J1I[Rn]

Z.G =z.R(ep, x) +J1I[Rn]

Z + G =z +R(ep, x) + J1I[Rn]

Tli dinh nghla 1.20 ta tha'y vanh y[Rn] la mQt kh6ng gian vec to tren

tru'ong s6 phuc C.