Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis
Pseudodifferential equations of the form $v(D_{\chi})y = f$ (where $v$ is a function holomorphic at zero and $D_{\chi}$ is a pseudodifferential operator) are studied on spaces of test functions of non-Gaussian infinite-dimensional analysis. The results obtained are applied to construct a generalize...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4732 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510896833429504 |
|---|---|
| author | Kachanovskii, N. A. Качановский, Н. А. Качановский, Н. А. |
| author_facet | Kachanovskii, N. A. Качановский, Н. А. Качановский, Н. А. |
| author_sort | Kachanovskii, N. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | Pseudodifferential equations of the form $v(D_{\chi})y = f$ (where $v$ is a function holomorphic at zero and $D_{\chi}$ is a pseudodifferential operator)
are studied on spaces of test functions of non-Gaussian infinite-dimensional analysis. The results obtained are applied to construct a generalized
translation operator $T^{\chi}_y = \chi(\langle y, D_{\chi}\rangle)$ the already mentioned spaces and to study its properties.
In particular, the associativity, the commutativity, and another properties of $T^{\chi}_y$ which are analogs of the classical properties of a generalized translation operator. |
| first_indexed | 2026-03-24T03:04:17Z |
| format | Article |
| fulltext |
Ys 517.9
[-I. A. KaqaliOBCKHi~i (Hn-r ~a'reMa'nmrl HAH ynpamn,1. KHeB)
I I C E B ~ O ~ H c D ~ E P E H I . [ H A J ' I B H B I E Y P A B H E H H ~ I
H OnEPATOP OBOBUIEHHOFO C,~BHI"A
B HEFAYCCOBOM BECKOHEqHOMEPHOM AHA,FIH3E
Pseudodifferential equations of the form v (Dx)y = f (where v is a function holomorphic at zero and
D z is a pscudodifferential operator) are studied on spaces of test functions of non-Gaussian infinite-
dimensional anatys!s. The results obtained are applied to construct a generalized translation operator
~.Z = X((Y, Dz) ) , on the already mentioned spaces and to study its properties. In particular, the
associativity, the commutativity, and another properties of ~z. are proved which are analogs of the
classical properties of generalized translation operator.
Bru)qalor))ca ncen/toitHcl~pemtia~n,ui pilm~nHa Bm'Jlalty v (Dx)y = f (Ire v ~ voJloMopqbtmy ny~fi
~yHK~dX. D x ~ nceBltOlB4~epelll[i~n,ll)lfl onepa'rop) ,~a npocTopax ocuommx tl.~ynKRii, i nevaycci~-
CbKOI'O lleCKil~qelutOBI4MipnoFo allaJtiay. O'cp~Malfi peay.ql;rar~.l aa~:[oco~ym-n,ca ltJ~a no6yJtom~ one-
pa'ropa y3al'tUll,llenovo 3cyl~y ~z := Z((Y. Dz)) Ixa BKa~nnx npocnopax ra mmqemm floro BJmCrU-
BOC'I~[.|" 3OKpeMa, ]tOBe]tellO acoltia'rHBlliCTb, KOM~rI'aTltBII[CTI, Ta illttli HJIa~l'llBOr Tv X , l ifo ~ alla~lo-
I'aMH KJlaCI.IqlIHX BJla~'l'nBOCrFefl o i l epa ropa y3alaJn)l lelIol 'o 3cyBy,
B HaCTO~tRee ~pe~m HeFayCCOBCKH~i 6ecKo~te'aHoMepl-~b~I aHa.n)~3 aKTHBHO paapa6a-
TtaBaerca a o6o6maeTca MHOrm, m aBTOpaMH (C~., HaHptx~tep, [ 1 - 11]). Han6o~lee
tnnpoKHe o6o6merma, rxo.ny,~eHnbte B pa6oTax IO. M. Bepe3aHcKoro H IO. F. KOH-
~pav~,ena [2, 6], tO. M. Bepe3aHcKoro [7, 10], ocuo~am,i Ha aaMeHe ~KCnOHeHT~,~ t~
nponaBO~a~uef,~[ c~yHKttrm nO~HItOMOB DpMnTa rteKoTOpOi.i tte~ofl t~yHKUHe~L a O6b~'a-
HOrO c~mwa ~ o6o6meHm,tM. B pa6o're [12] petuaeTcz o6paTHaa 3a/aa'aa: no r~Be-
CTtmtra xapa~Tepa~ Anneaz n.nn/Ie.n~capTa (aHazoraM nozmto~qo~ Dp~,mTa, cM. [2,
6, 7, 10, 12]) BOCCTaHa~mmaeTca onepaTop o6o6memtoro C,aBHra. l'lo/IO6H~te no-
CTpOeHH~I MO~7.HO npOBCCTH 1~ n c.~lyqae, Kor/~a BMeCTO rlOJIIIHO}4OB DpMr~Ta rtcno.nb-
3ylOTC~ o6o6meamae KBa3Hanrte.neab~ no.nHilo~a c npoH3BO11~ttle~"t tl3yHKRHefi
y ( 0 ) Z ( ( x , a ( 0 ) ) ) [8], rae ~ : C ---> C ~ u e ~ a . a c[~ynKuHa, y n a - - ro:mMopdp-
Hble B HyJle clJyHKRHI, I Ha KOMII.neKcrlcl)tlKaRI-IH N C IteKo'roporo BeRIeCTBeHHOFO
aaepHorO npocTpaHCTBa N. HMettHO, onepaTop o6o6taeHHoro c~aHra ~ ~TO onepa-
Top Tv ~ = Z((Y, DX)), rite D z ~ nceB~aOZmC~cloepeHunaat, tmfl.i onepaTop. CB.,qaaH-
Huff c ynOMaHyTOfl BmtUe qbynKtlnefl Z" TaKnM o6paaolvl. 6a3o|| /~.n,q nOCTpOeHH.a
O606tReHHOFO C/~BHra JtBJI.,qIOTC~I nceBaol~Hqb~epeHuHa~bm,te ypaBHeHrIZ BH,aa
v ( D x ) y = f , r~te v ~ ronoMopqbt!aa B 0 ~ N c t~yHKLIII~I. C~rMeTr~M, qTO TaKne
ypaBHeHH~l Hayqa~HCl, (6e3OTHOClITeJ'IblIO K 0606tReltHoMy C/~Brtry) B O/IHoMepHo~I
cny~ae B [13] (no nOBOdy 6eCKOHeqHOMepllblX O606tl.l.eltH~ CM. [8, 9, I 1] ).
L[e.nl:,lO HaCTOIItRe[.| CTaTbI.I J, IB.n~IeTCJt tl3yqerlHe llCCB/IO/~llqJtl.)epeHtlHaJlbHblX
ypaBHeHafi Brl/.la v(Dx)Y = f ~ a tl3yHKtmt.~ 6eCKoHe~Horo KOJIHqeCTBa nepe~,teH-
Hb~X, a TaKZ, Ke I1OCTpOeHHe H H3ytleHrIe c HOMOILIblO yKa3aHHb[X ypamtenrfla onepaTopa
o6o6meHHoro CaBara ~,Z Ha npocTpaHCTBaX OCHOBm,~X qbytmmtx~ Heraycco~c~oro
6eCKOae'-IHOMepHoro arlanrlaa. (Ylpn :~TOM dpyHKRrla Z ( ( x , 0~ (O))) $1BJlfleTC.,q xa-
paKTepoM/I.na ~TOrO CZBnra: ~,Z X((X ' 0~(0))) = X ((X, a ( 0 ) ) ) X ( ( Y , a ( 0 ) ) ) . ) B
qaCTHOC'rH,/~OKa3alto, qTO BBr162 orIepaTop ~,~ riMeeT aCCORtlaTHBHOC'I-b, KOM-
~tyTaTHBHOCT~ H/xpyrt4c K.rlaccHqCCKI, IC CBO[~[CTBa O606mCHHOrO c~Bt4ra. Pc3y.rlr~TaTrd
CTaT~,a raoa~no ycnonHo paz~e.nwn, Ha ane ,~acTr~. Flepaaa npez~cTa~naeT co6ofi
o6o6memte Ha 6ecxoHeaHor, tepmafl c.ny,~afi COOT~eTCT~yIomax peay.m-TaTon [13 ] B
6onee o6trae~, '~eM n [8, 9, 11 ], nocTm.ionme. Bo mTOpO~t moHxpeTrmnpo~ama ncc:~e-
�9 H. A. KAqAHOBCKH~. 1~)9
1334 ISSN 0041-6053. Yh'l). ,~tam. ~."vpu,, ] 999. m. 51, N e I0
FICEB~O}2HOOEPEHIAHAYlbHbIE YPABHEHH~I H OHEPATOP ... 1335
]~OBaHrl~l [ 12] B cnyqae atla~Haa Ha FIp0cTpaHCTBaX, rIopo.X<~eHl-I~X 0606meHHI:,IMH
KBaarlannencBIaMr! noanHOMaMH,'a I-Ie a6cTpaKTlltaMrl xapaKTcpaMrl Arlrle.rln H.nrI
~e.nbcapTa, T. e. B xaqecTBe xapaKTepa Z (x ; O ) ~l~wypHpyeT Z ( (X, a (e) > ). OTM e-
TI, IM, O/IHaKO, qTO IIpe/~C'I'aBJ~eHHbIe 3~ecs pe3yJqbTaTbl He J~mJIJllOTC~q qaCTI-IbIM C.nyqa-
era [12], llOCKO..qbKy IIOCTpOeHHSIit orlepaTop ~,X OTairlqaeTc~l OT o6o6menHoro
C~Bnra B [12] (~e~CTByeT na ~pyrHx npocTpaHCTBaX), a nponaBo~amaz dpyHKmla
06o6taennr~x KBa3rxanneneaux no.nrxrxoMoB He aB.n-aeTca aacTmaM c.ayqaeM npoHaao-
aame~ dpyHKmta xapaKTepoB An ne~a Hml ~enbcapTa (B qaCTHOCTI4, na-aa Hamtqrla
napaMeTpa a ) . B t~ezora, CTaTba npo~oz~aeT ricc~eaoBamla aaTopa, ony6~qnKoBan-
nsae B [8].
1. FlycTs . q ( ~ Be~ecTBeHHoe cenapa6ezbnoe rn~L6epToBo npo .cTpancTao, N
cenapa6ezbnoe a~epHoe npocTpaHCTBO ~peme, nzoTuo n HenpeptaBnO B~o~enHoe B
.q(, N ' ~ npocTpaacr~o, conpa~KenHoe K N 0TU0CnTeZSHO 2/'. Tor~a (a cn~y
aaepHOCTH) N = prlimY-/'p, N ' = ind l imH_p , roe M'p, p ~ Z , ~ i i e ~ o r o p u e
pen p~l~l
r~t~ab6epTOBb~ npocTpancTaa (XOTOph~e raO~HO a~x6paT~, paayMeeTca, He e/amCTBeH-
roam o6paaora; rata 6yz~era CqriTaTb, qTO TaKo[~ Bu6op c~e:mn ri aadpriKctlpoBaH),
He+ 1 B:~oa<eno B Y'~ onepaTopora Tuna Frmb6epTa--IIIraHZtTa ~ ra~I~oro p a Z,
.q-[_p --HeraTrmtlue npocTpaHcTaa tterto'-ieK H i , ~ .,a/'~ .q.{p. 0603Ha'-Irlra qepea 1. Ip
nopray B ..q-{~, ~epe3 ( - , , > ~ cnaprmam~e raez<ay ~.qeraenTa~H N' r~ N, 5/'_,0 n Mp,
3a]IaBaeraoe pactmipeHnera cxanapHoro rtpor~aBe/ICHHa B .q-/', rt coxpaHnM aTn 060-
3HaqeHH.a/I..r!..a TeH3opHbIX cTeneHefi rlpoc'rpanc'rB H KOrannexcnqbnKai.1;H~.
I'IycT~ N c ~ xoranneKcm~Hxam~z N. 0603na'aHM ,aepea Hol 0 (No) a,nre6py
pOCTKOB ro.noraopqbm,~x B HyneqbyH~tmfl "/: N c --'-'~C, ,aepe3 Holo(N c , N c ) ' - a.n-
re6py pOC'rXOB ro.~OMOpqbH~X B Hyne ~yHKttnia a : N c ----> N c .
~ Z,.~, u n, u e C - - Lie.nan qbyHKUtl.,'t TaKaa, qTO Z (0) = 1, 1-lycT~ Z(u) = =0 n!
Zn~:0 Vn E Z+. PaccraoTpHra tl~ym<tlmo 7(0)X(<Z, a ( 0 ) ) ) , 0 E N c , z E N c ,
rae 7E Holo(Nc) , "},(0) :~ 0; a �9 Hol 0 ( N c , N c ) o~(0) = 0 rt o6paTrwia. Pac-
K.naabmaa ee B a p~t~t Te~'t.nopa no 0, c noraombxo Teoperata o a~pe no.nyqaera c.ne-
,a3qotuee npez~cTaB,nem~e:
r/Ie @ o603naqaeT cmvtraeTpr~qecKoe TeHaopnoe npor~3Be~teHne.
Onpeae.aeHne 1. Ha3o~e~t cucme.atoa (6ecrone,.mo.~tepm, fx) o6o6ucenn~" Kea-
^
3uanneneeb~x nO.aUnOZtOe pV.a := {(p,V.a, q~(,,)>: 9(,,) e N~n, n e Z + } .
1-IycT~ s ~ raHomecTso scex HenpeptaBn~tx I~0mmoraon na N'. /Xsm p, q
e Z+ r~ 13 r [0, 1 ] onpet~eaa~ npocTpancTao ocnosmax qbyHxuH~t (M'p)~q.v. a ~ax
3aMs~xarme ~ (N ' ) no C0OTBeTCTBytomei~t noprae:
Ilq~ll2P:q'l~"t'= := n=0 ~ (n!lt+l~zq" <
FIo~o~HM
ISSN 0041-6053. Yrp. ~tam. ~.'vpu., 1999, m. 51, N ~ 10
1336 H.A. KAt-IAHOBCKHI~t
(N)~, a := prlim(Y'/z,)q~,?,..
p, qcN
Cno~ic'ma cac ' re~u rto.rmao~oa pu n npocTpaac'rs (.q-/'t,){.~,ct , (N)~,c t n p a
= 1 uayqeHta B [8].
3a~e~anue 1. J:lapa Pn v'a , KaK aBeReHrtue rlpOCTpaHCTBa 14 HopMa
U" IIp.q,l~>t,a, aa~ncrr , ~o .equo , OT (I~yHKIIHH ~,. Oaaa~o Mu He 6y~eM non~tepKa-
Barb ~ 3aBrlCrlMOCTb COOTBeTCTBylOmI4M I41-IaeKCOI~I a,rl~q yrlpomeHn~l O6OaHaqeHHlt.
2. 1-lyca~ V ~ H o l o ( N c ) . Onpeae~ma Ha Z ( N ' ) nceaaoa r tdp~epemma~n~ta
orlepaTop v(Dx) := ~]~'=0 ~, <vn'D~ n)' rz~e v n , N'c ~n .z paa~o~erma
V(0) : Z n : 0 l~-l<Vn'O| n o ~ a r a a H a . o n o M a x
I $ ' .
m!Zm_n (x| ~ v n , f~(m)>, (Vn, D~n>(x| (m)) : = l{m?.n} (m_n) !z m
'̂r ~,,, N ' , (p(m)~ " C , x
rx npoz~o.amaz no IIHHeI:tHOCTH (3/~eCb I {m>n} ~ rm/~rlKaTop {m > n }) .
J'le~Ma I. Onepamop V (D z) ~tOa~HO npoSon~um~ Oo ~une~noeo nenpep~tenoeo
onepamopa, Oe~cm~ytou~ezo us (-q/p)~.'t, id ~ ( p)q.v-c.id, zOe p, q ~ N, ~
[0, I ]. Rpfi 9mo~t ecnu 9 ~. (ff'/'p)q~,y.id 3anucana ~ ~uae
Ill : ( l
mo
(v(D z) q,) (x) = ~ (p,~v, id (x), ~C,,0), (2)
m=0
m. e. ae~cm~ue onepamopa v (Dx) ceoaumc a r sastene aOep - mPY'id Ha " inPvu
,q'[ f~ npe,~nee o(k~zna~enue). (~mt coxpanaezt 8ha npoOon~enua V(Dz) na ( p)q,r, ia
,lIoxa3ameat, cmoo. IIycTb 9 ~ ( p)q,7,id rt aanncaua a nnae (1) . Tor~a
0
II~011~.q,~a, id = ~ , = 0 (n!)t+~2q"lq'C")l~ < o. n o ~ o x H M ~0 M(-) : =
~ M (e,Z.r.~d(.),~(,.)> ~ m(~') , ac .o . .~o ~M ' ._,~> ~ ~ro~o~o~. .
: : m : 0 M
(M'p)~q.v.id �9
B~aqrm~_aM (v(Dx)q~ M) (x), xE N'. B [8] ~;oKa~io, wro
<Vn, @n ~,id (m) m[ . u id D~ )(P~ (x ) ,~ ) = ~.,~.) ~ ( e . : , , ( ~ ) ~ , , , ~ ~))
Kpo~e rot.o, ram xe yera~o~aena qbop~yaa
n l
n = 0
I/Icl'IOJlb~j'~ 3TII c0OTHOIIICI-IH~I, I, IMCCM
ISSN 0041-6053. Yrp. ~am, ~'vpu,, 1999, m. 51, N e lO
IqCEB}20~2HtDtDEPEHI2HAflbHbIE YPABHEHH~I H OHEPATOP o,. 1337
n~O l <Vn, D~n> ~_~ y, id q)(m)) = ( v ( D x ) ~ ) ( ~ ) = ,7.' (P,. (~),
= I1! = 0
y~oM ~l ,,~', (m m----L-" 'P. u .)! "&v,, q~C,.)) = __ ~ m : , , t x J , =
I 1 : * : !
n y. id vy, id (p(m)) c;,, p,,,_,, (~) ~ v,,, ,p~"~ ~ = = <P., (x), = ~M (x).
m=0 \ n = 0 m=0
H OTCIO~a BH/IHO, 'aTO cymec'myeT ~ ~ ( p)q, vT.id Taxoe, wro ~M M-~** ) ~ B
Torio~orrirl (ff'/'/,)~,vy, id, rlpatleM [ItPllp, q.13.v$.id -- ][q) [p q,l$ 7.ifl" rIoJIO)KHM
(v (Dz) tp ) (X) : = ~ ( x ) . . q C H O , qTO B aTOM c ~ y q a e V (Dz )
H s id H ~ ' ( p)q.vy, id) rl (v (D;c ) tp ) (x ) HMeev aria (2) (3/Iecb L ( X , Y) - -
r~HO~CeCTBO nriHeHmax Henpep~,mmax onepaTopoB, ~ef~c-rBymmnx ria mmeiaHoro TO-
noJmrnqecKoro npOCTpaHCTBa X B nnnel~Hoe TOnO~orrlqecKoe npocTpaHCTBO Y ),
JIeMMa/toKaaaHa.
HyCTb V(0) r 0. O6oaHaqm~ V(0) := 1,/V(0). ~Ic~io, qTO V H HoI0 (Nc) ,
noaroMy MO~CHO paccMaTpnaaTh onepaTop ~(Dz) . Ha JaeMMu 1 BUTeKaeT TaKOe
cJIeCCCTBHe.
C, ae~cm~ue, l l p u v(0) ;~ 0 0 n e p a m o p v ( D l ) o6pamustt, nputte.~t
v(Dz)-' = (~(Di).
3aste,tauue 2. AHaglor~qHue peay~'raT~a cnpaBe/~mmu ~ ~t~z~ npocTpaHCTB
(Hp)~,v. a , ecJm BMeCTO onepaTopa v ( D I ) ncnoz~aoaaT~ v ( c c - ~ ( D z ) ) , r z e no-
~cne~Hrifl onepaTop CTpOHTCa aHa~orriqHo v ( D z ) no qbyHKRmt V ' (0) : =
: = v ( a - ' ( O ) ) .
3. O]~HO 1.13 KYlaccHqeCKrIX IIpHMeHCrill~l O60~IIIeHHblX KBa3rlartnc1Ieablx no~,InO-
~o~ COCTOHT B pemeHmz ~tnqbqbepeHt~naa~nux ypa~rierint4 cneuaanbnoro aH/la [13]
(w 20). H~,IeHHO, IIyCTb 11 : C --> C ~ rOJIoMopc~Ha~l B 0 H C Ot)yHKRrla, I1 (0)
0, Z = exp. PaCCMOTp~{M/l~qb~epeHmta~bHoe ypaBHeHHe 6eCKOHeqI-IOFO, Boo6me
roBopa, nopa~Ka
11 ( D ) y ( u ) = f (u) , (4)
r/Ie u H C , D ~ onepaTop/IHqbkbepeHu~poBarirIa. ~ a ero petuemia ~O~HO rzpri-
MenriTb c n e a y m m a l t MeTO~: f ( u ) p a c ~ a ~ m a e T c a B pa~t f ( u ) =
= E n = 0 fn p~,id (u), r~te p~,id (U) - -o6o6merms~e IIo,rlrinOMbl A n n e , s c npOHaBO-
a~tueta dpy~KRaef~ rl (0)e "~ 0 H C . Tor~ca peRzeHae (4) I, IMCCT BH/I y (/g) =
= E ~ = 0 f" un" AitaaormlHut~ pe3yllbTaT cnpaBe~nrm ri npa a ~ id ~aaa ypaBHeHH~l
r l ( a -1 (D))y(u) = f(u).
Hay~H~ 6CCKOHeqHoMepHI~ aHan0r aToro MeTo/Ia. PaccMoTpnM nceB]iO/~HdPcl)e-
p~nI.I,H~'Ibn0e ypS~HCHHC
v (D z) y = f, (5)
r;ae v H Hol o(Nc), v(0) ~ 0.
C~c;aym~ee ya~epacaeHHe o6o6maeT tloKaaanH~ati B [ 8 ] I:~3yJlbTaT Ha c~y~all
~.
ISSN 0041-6053. Yrp. ,iwm. ~,,W. lm.. /999. m. 5 I. IW I0
1338 H.A. KAqAHOBCKIdI~I
TeopeMa 1. l lycmb f ~ (Hp)~,u 1~ ~ [0, 1 ]. TozOa ypa~nenue (5) ropper-
mtto paspeutu3to, nputtezt ec.au
f= ~ (~?'id, fn) fn ~ M ~"
' ':C '
n--0
mo peutenue u,~teem 8U(9
p e l f = = Y(') (V(Dz)f)( ' ) ~ (pnV"/'id('),fn> ~ ( P)q.('"t'.id'
n=0
zae 9(0) = z/v(e).
Ec~au Oono,~nume.abno p u q malcosbt, ~mo cyu4ecmsyem Po ~ N , Po < P ma-
roe ,~mo I l ie . l~ , l lns< ~ u -ellip.e,,ll .s < 2 q/2, zc~e ip,l~ , - - o n e p a m o p 8/to~e-
" p "
nu~ H p o H p o , II.llt4S~uop~la Fum, d e p m a - l l l ~ t u O m a , 9 maKooo, ~mo
, , H 1 sup Iv(O) l .< ~ u sup II /v (O) l < ** I ] = I m o y E ( p)q- l ,v , id . O tta-
lelpo =p lelpo =p
cmnocmu, e c n u f ~ (N) l, mo y E (N) 1 ( (N) 1 ~- (N)l,a, c~t. [8]).
t - - n f ~ l ! 3
Ecnu ~ = 1, ~ ~ no~uno,~t u p~, : = max { p ~ N: vn e .~,-t_p.CJ., ,~', . u3 paz-
I (~, , , , o ~ , , > _ , , , (Hp)q.~,,id ~o,.'re,u.R ~(O) = ~'n=O ~ . mo npu p > p~ u f ~ pe tue-
. !
hue (5) y ~ (Hp)q,7,id.
l'Iepaoe yraepTr, JlenHe Henocpe/~craeHno c,qe/IyeT Ha .qeMbi~ 1; ocTa~bm,~e ~;oKa-
z a n u ~ [8].
3 a ~ e q a u u e 3. Ana~orriqnoe y r a e p ~ e H n e cnpaBe/1~nBO B c~'~yqae ct ~ id /I~a
ypaanermt41
V(O~ -I (Dz))y = f,
H I~ (Hp)~q.v.= v~X.,~. na/Io .,rmmb aaMcHn"~ ( p)q.'t.id Ha
4. Ec~H qbyHKtm~ Z ((x, O)), X ~ N', O ~ N c , np~ qbn~cnpo~aHHO~ O ~ -
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cM. ~ [81).
A a T o p 6 s m r o / l a p e H tO . A . B e p e a a a c g o M y H F. (D. Y c y 3a n o ~ e a m a e o 6 c y a < d l e n i a J / n
3aMeqaHHll .
I. ]Ta.~etp~'uii [0. ,,/7. ~Hoprol'Olla.~ll,ln,l~ allaJiov HOJIHIIOMOB :~pMx'['a it o6pato, em4e npeo6paaonalmx
r no Helayccono~ Mepc/ /~yuglmou. anaJm3 a npaJt. - 1991. - 2 5 . N'-' 2. - C. 6 8 - 7 0 .
2. Eepe:~tttctruii I0. M., Konal),on~t~ I0. F. Herayccol~cga~ ana~ma a r n n e p r p y n m ~ / / T a M ~xe. -
1995. - 29. N a 3. - C. 5 1 - 5 5 .
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I 1. Ka~antmcm~ii H. A., Yc" F. O. BHop,oronaJJhtmle cHc'ret, ua Anne~la i~ aHaJm3e na ltyaJmno-altep-
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iSSN 0041-6053. Ytcp: ,~,am. ~.'vpn.. 1999. m. 51. IV a I0
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| id | umjimathkievua-article-4732 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:17Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a0/2fbf3175056afa36e3b4e2985115f2a0.pdf |
| spelling | umjimathkievua-article-47322020-03-18T21:12:54Z Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis Псевдодифференциальные уравнения и оператор обобщенного сдвига в негауссовом бесконечномерном анализе Kachanovskii, N. A. Качановский, Н. А. Качановский, Н. А. Pseudodifferential equations of the form $v(D_{\chi})y = f$ (where $v$ is a function holomorphic at zero and $D_{\chi}$ is a pseudodifferential operator) are studied on spaces of test functions of non-Gaussian infinite-dimensional analysis. The results obtained are applied to construct a generalized translation operator $T^{\chi}_y = \chi(\langle y, D_{\chi}\rangle)$ the already mentioned spaces and to study its properties. In particular, the associativity, the commutativity, and another properties of $T^{\chi}_y$ which are analogs of the classical properties of a generalized translation operator. Вивчаються псевдодифереціиальні рівняния вигляду $v(D_{\chi})y = f$ (де $v$ — голоморфна у нулі функція. $D_{\chi}$ — псевдодиференціальний оператор) на просторах оснонних функцій негауссівського нескінченновимірного аналізу. Отримані результати застосовуються для побудови оператора узагальненого зсуву $T^{\chi}_y = \chi(\langle y, D_{\chi}\rangle)$ на вказаних просторах та вивчення його властивостей. Зокрема, доведено асоціативність, комутативність та інші властивості $T^{\chi}_y$, що є диалогами класичиих власчивостей оператора узагальненого зсуву. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4732 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1334–1341 Український математичний журнал; Том 51 № 10 (1999); 1334–1341 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4732/6165 https://umj.imath.kiev.ua/index.php/umj/article/view/4732/6166 Copyright (c) 1999 Kachanovskii N. A. |
| spellingShingle | Kachanovskii, N. A. Качановский, Н. А. Качановский, Н. А. Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis |
| title | Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis |
| title_alt | Псевдодифференциальные уравнения и оператор обобщенного сдвига в негауссовом бесконечномерном анализе |
| title_full | Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis |
| title_fullStr | Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis |
| title_full_unstemmed | Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis |
| title_short | Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis |
| title_sort | pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4732 |
| work_keys_str_mv | AT kachanovskiina pseudodifferentialequationsandageneralizedtranslationoperatorinnongaussianinfinitedimensionalanalysis AT kačanovskijna pseudodifferentialequationsandageneralizedtranslationoperatorinnongaussianinfinitedimensionalanalysis AT kačanovskijna pseudodifferentialequationsandageneralizedtranslationoperatorinnongaussianinfinitedimensionalanalysis AT kachanovskiina psevdodifferencialʹnyeuravneniâioperatorobobŝennogosdvigavnegaussovombeskonečnomernomanalize AT kačanovskijna psevdodifferencialʹnyeuravneniâioperatorobobŝennogosdvigavnegaussovombeskonečnomernomanalize AT kačanovskijna psevdodifferencialʹnyeuravneniâioperatorobobŝennogosdvigavnegaussovombeskonečnomernomanalize |