On the Levy-Baxter theorems for shot-noise fields. II
We establish sufficient conditions under which shot-noise fields with a response function of a certain form possess the Levy-Baxter property on an increasing parametric set.
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4579 |
| Теги: |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510721684537344 |
|---|---|
| author | Buldygin, V. V. Mel'nik, V. M. Булдигін, В. В. Мельник, В. М. |
| author_facet | Buldygin, V. V. Mel'nik, V. M. Булдигін, В. В. Мельник, В. М. |
| author_sort | Buldygin, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | We establish sufficient conditions under which shot-noise fields with a response function of a certain form possess the Levy-Baxter property on an increasing parametric set. |
| first_indexed | 2026-03-24T03:01:30Z |
| format | Article |
| fulltext |
Y,/IK 519.21
B. B. ByJzanriu, B, M. Me, JIbHHK, B. r . IIInopTIOK
(Hart. TeXH. yH-r YKpaiHtl ,M.III", K~in)
IIPO TEOPEMH J IEBI - BAKCTEPA
tsta 3 'O ;OBHX no,rnB. II
We establish sufficient conditions under which shot noise fields with response function of certain form
possess the Levy-Bakster property on an increasing parametric set.
BCTaHOi~ieHO ]IOCTaTHi yMOBH, npH .,qKHX /]pO~OBi IIOJLq 3 dpyHKtti~lMH Bi/II'yKy neBltOrO BHr.q~l~y
Ma~oTb JIeBi - l~aKc'repoBy ~.nacTtmic'rb Ha apoc'rmoqi~ napaMeTpHqHi~ MHO3KHI-fi.
~Iana pO6OTa e rtpo]~oBYtteHH~iM pO6OTH [29]. l'IpH U~oMy a6epiraloa~cz BBeZ(CHi B ttikI
noaHaqenns, Hy~epau~ qbop~ya Ta ncpmmzjKcpen.
4. Teopem4 JIeni - BaKeTepa/XJtn O/IHopi/InrlX yaaraJmHeHnx/Xpo6onnx no-
Jria ai cne~iaJmHnm4 (l)YnKlfimvm ei/IryKy. Poar~HcMo TCOpC~4y gleai--BaXCTCpa
Ha 3pOCTmOqOMy napa~4cTpnqHoMy inTepsa.ni ~JLq 0/~Hopi~maX yaara.r~HCnnX ~po60-
BHX no.niB, nepeTnOpeHH~ Oyp'e dpyHKI~ii si~ryKy ~tKnX ~ae cTeneHeanl~ xapaKTep
MaJIOCTi Ha rlcCKiHqeHHOCTi. Taxi 3a~aqi ~J~s rayccosvtx npottccia poar~a~ncs
E. rna/xmuean~t [18].
Teope~,ia 5. Hexa~ ~c(?), ? e R m, m > 1 ~ o O n o p i O n e ysamabnene 8po6oee
noae S dpynr~ie~o eiO~yKy f ( ~ ), nepemeopem~ tl>yp'e .~i'sa0oeo,~bn.o.e y~,oey :
I) = gl lXll + o([l ll ,,pu 0e M > 0 , ~>m.
Hexaa p = p ( ~ ) = k , k e N, ascu4o 2 k m - m < 21~ < 2kin+m, ff = (p . . . . . p) e
e N m,
{(~nXn) ~m-~lt, a r g o 2~3~2km+m;
Cn = (O~n,~n)2m_2~[ln(O~n,r.n)[_l ' arv4o 29 = 2 k m + m .
ToOi, xKu4o euronytombc~ ys~oeu
2) lira (z n = ~o,
/l--)~o
3) lira (Zn'C n = O,
?l---)oo
a y 6unaOry 29 = 2k in + m OoOamroao u4e i
2 4) icnye mare 0 > O, u~o 8A~ O06iabl-tOgO n ~ N (Xn'~ n )-- O,
mo
Oe
R
1.i.m. c n tZn m ~ ( A ~ : K(Ck'- T)tzn'Cn)) 2 = c,
n--->~ k -- 1
R,.X. r=l
3 a y s a ~ e n n ~ 4. flKmO xoqa 60/IHH a napa~cTpis 02 a6o 0 w He ~opisHIOe
HymO, TO oqeBtI~HO, 111o c ~ 0.
~ o e e a e n ~ 3 yMosn 1 snnamaae, ulo icnye Taxa qbymud~ g (Z), mo
[)~(z)l = (M +g(~))~-I~, (13)
='0.
P-t-'
�9 B.B. BYJI]II4FII-I, 13. M. MEJIbI-IHI(, B, F. RIIIOPTIOK, 1999
12 ISSN 0041-6053. YKp. ~m..,~'ypn., 1999, m. M. N e I
IIPO TEOPEMH JIEBI - BAKCTEPA }]~l~ }IPOBOBHX HOJIIB. II 13
BH6el~eMO~lOBi.rmHr e > O iwaKe B=B(I~), ulo, axulo I1~11>~, ~ o I~(~)l<~-
CHO~aTKy po3ra~me~o B~ma~o~r KOZm 2kin - m < 2~ < 2km + m.
Ilepeaip~Mo n~xoHa~Ha nepmoi y~OBH n a c a i ~ y 4 3 [29]. 1-/03HauHBmH Bapa3 i3
nepmoi y~osH ~epea I~,, nona~o l~oro y BHr~xai
/In = Irlnl -I'- 4n2, (14)
/Ic
' ln2=( ( i2+(~2)Cn(~n'~n) -m S ( n (2si.~'r<~:")~') IY(~)l~ ,~.
Iq>B,, ,=~
OuimtMo l i n 1. HeBa,'~o no~ec'm, iUO
4,,, _< (,:,: + <~,) <,, (,~,:,,)-" J" ( ~ (2~'"~"'~".)~") I?(~)l= a~ _<
II~II<:B r=l
((~2+~2W) Cn(O~n'~n)-m(~nT'n)2mpB2mp S lY( )l :
= Ki I (CXn,~n)2mp+m-21], (15)
Iq~a
Orate, ia crd~ei/momeHaa (15) Ta y~on~ 3 Te0peMtt 5, mmsmsar lira ll~l = 0.
#l"--)m
Oigmi~to Tenep lln 2. 3a ~IonoMorom 306pa~eHHa (13) oTpitMyeMo
/In2 <-- g2((y2"i'i~2)r ~ I I - [ (^ " ~l'r(gnT'n'~2Plt 2Sln 2 J [[ []ll~li-2~d~-- +
I~l> ~ : B ,-:l
+ KI2E + KI3 ~2 = /ln2! + KI2t; + /('13 ~2, (16)
~e K12 i KI3 - - cTa.qi, mo He 3 a . n c x a ~ si/I n.
PO3FJLqHeMO
lln21 M2 (l~2 ~2w) Cn(l~'n'~n)-m ~ f I-[ f" ~i'rO~nT'n'2P ) = + L2s~,, ~ J ~11 -~"<~=
!~1>~" '=~
= M2(ff2 +(y2) C'i(O~nZn)-mfR.(~=! (2sin~'r%'i'~n]2P)ll~ll-:" < ~ -
- M2(62+(~i') c"(c~n'n)-m I f~.. (2sin~"~] lp ][Iz~ -21~ dZ. (17)
iii<: ~,, r:,
Jpyritlt/Io/IatiOK y (17) oiliHHl~O Hacryrmn'l~ ,.n,moM:
M2(f2+f2)r -m f ( f i {2sin~'r~2nZn)2P ] l l~ l1-2l Ids<
I$SN 0041-6053. Yxp. :itam. ~'vpn., 1999, m. 51, N e I
14 B.B. BYJUIIdFIH. B, M. MEdlbHHK. B. F. IIIFIOPTIOK
<__ m2(o2+(y2)r n)2mp-m(2~) m-I ; p2mp+m-l-2~dp <
9,:11
_< Kl4(anX.)2mp+m-2[~, (18)
~e K~4 < ** i He 3 a ~ e ~ r r ~ si~ n. Tar~M aanoH, i3 cn iss ignotaen~ (14) - (18)
Brln~a~
lim~Iln-M2(ff2 +C;~)C.(an'n)--mjm ( ~ (2sin~'r2"X")2P IH[IF 2~ d[I <
<- ( x n + x,2 e) ~.
3siacg, 3aaaa~r~/~OBinhI-IOMy Br~6opy e, Ma~vIO
lira/in = lim (~2+02) Cn(an'n)-rajm( mrI~= 1 (2sinXr2nxn)2P)ifc(L)12d~ =
= M 1 ( O 2 + O 2 ) 2 m-21~ ;fI~'l[ (2sinur)Ze)[l~ll-2f~d~=c.
R,.k r=l
Ov~e, y~o~a 1 Hacai~y 4 3 [29] B~KOHa~a.
l-Iepe~ip~o s~o~amm Rpyroi y~oeu Hacsti/~y 4. P o 3 r ~ e ~ o
J " j " ( [ sin20tn(kr+~tr)
12" = (0 2 +OW)2 22c. IXn 2rn f i .. 2 ".(sinkr,Ot. "~"~'t' X
sin 2 T, nGn(,tgr + I-tr)\ 2 )
r=l 2
2 ~n2m
R m R "
(~e ~xepe3 H 2 (~,~) ,~na 3pyqH0CTi noaHa~n4Mo ni~i~Terpa.nbHy dp3a-mttho). Toni
12. = I2.1 + / 2 . 2 + t2.3,
.~e
/2nl = (02 • ~2 ̂ 2 ~-2m
U~I~B H~ll-<S
I[[>B Iga>S
Cr,.opHc'r~umcb Hepim-loc'rst~t.
1xl <-Isinx[ ~ Ixl,
i TItM, IIIO
X G
(19)
ISSN 0041-6053. YKp, ~ . acy. px., 1999. m. M. l~ 1
HPO TEOPEMH ./IEBI - BAKCTEPA JJ, rl$I ~[]POBOBHX rlOJIIB. II 15
x2 sin 2 (x:' x) ~2
sup sup < ~ ,
n>l xen sin 2 (x) 4
OUiHH~O ~o~taHXH y (19). Kaa nepmoro ~O~aHxy ~ a ~ o HepieHicrb
de
/2nl < Kl2((Xn'~n)2(m(2p+l)-21~),
(20)
(21)
+O2 '~2('/f'~2m('B'~4mP I [' I~/.~.~l 2 12
~ apyroro ~o~auKy MaeMo
I2n2 r ( 2 ) 2m21-4mp [lnl [In2" (22)
Ia cniani~HomeHb (21), (22) i yMOBH 3 reopeMH BMnmmar mo
lim 12hi = 0 i lim I2n 2 = 0. n...)r •---)e*
OttimiMo TpeTitt ~o~anoK 12n3- IIpn ttboMy 6y~et, io poapianm'~ BHHa~KH, KO~m 2[~--<
<_ 2rap i 2pro < 2[3 < (2p + 1)m. CnoqaTKy OUiHm~O 12n3, npHnycKazowa, mo
2~ <- 2pro:
12n3 _~ (M+E)4(G2-1-ff2) 2 c2 ,:2m(~-~.~-)4~-2m X
X ~ f ( r I s i n 2 ( ' f n l ( U r + V r ) ) ( s i n V r ) 2 P ( s i n u r ) 2 P l •
. . . . . Bznvt n ~>B,cnan rffi! sin 2 (U r +Or)
~ vU >--'-~ 2
X II~[1-21~ [~-213 d~d~.
h-rrerpa.a y npaBitt qac'rm~i oujmoe'rbca aaepxy cKin~xennos3 cyMoIO O~HOT~nHHX ill-
Tel'pallia nO Bcix Ko0p/IHHaTHtlX KyTax. OUJHHMO, tiarlpHx~ia~, r0fl i3 HHX, y ~lxoMy
iHTeFpyBaHH~I rlpOBOIIHTI~I rio nepmoMy xoop~maTHo~y Kyry. POa6HBmH rlepumtl
KOOp~HHaTHH~ KyT Ha 2m-nHMipHi Ky0H 3 ~OB3KHHOIO pe6pa g, OTpHMafMo
/2n3 < (M +g)4(O2 +r C2 0~:2m(~-)4[~-2m X
• ~ ~(~=lsin2(~nl(ur+Vr))(sinVr)2P(sinur)2Pl •
[0,~V" [0,xl m sin (U r +Or) J
( Om :1 X ~ ~f~(l)r~'kr~)2/ II E ~f~(UrJt'qr~) 2 dud~. (23)
iffil k i=0\r=l / J ~ j=l qjffi0kr~l
POaFJI~HeMO cl~YHKl~lO r f i 'e[O,n] m, b > m > 1:
I~lsin ur l ~
JIer~o ~ r m , mo np~ b > m ~ a ~ e l I ~ o ~ a
I$SN 0041-6053. YKp. ~ n . ~]pu.. 1999. m. 51. !r a I
16 B.B. BYJ'Ij]BFIH. B. M. MEJ'IbHHK, B. F. UlrlOPTIOK
m
1-[ lurl"
%,b(~) < ~=I m b/2" + Ko, (25)
k r-~l
~Ie K o m ~e~Ka CTa.~a. T ~ M ~ o M , i3 (23) i (25), o T p ~ y e M o
,i:<u,+~,) J/r,: , :+=o •
<o, : , . , o . : , . , , . , tt~<.: j
2 _2mffXn,~n "~4[~-2m • 2,~,+KO d~dV--" (M+~)4("2+"~)2c"~" t,--y-) x
tt <o,= j.
="~ ('<;'<" +"")] +,<o) ><
[0dt]m [0,~]m r=l
= + = o ) = = < < = + = : I ~ =
• (l=12p"-'l I + Ko) = f i ] ] =in= ('~II('" +"))<t,,. <t,.,,.. (26)
r=l 0 0 s in l (Ur+Vr)
OcKLrn:,Kit ~ l~r iac~ic"r Io .mlpa ,cIietle'pa
sin 2 nx
( in ) -l f 2.dx = I ,
s in 2 _x
2
TO i3 ( 2 6 ) BHII,rIHBae
12n31 --< K23 %,,m, (27)
~ e
K= = (M+e)4(c~2 +o'~v)2gn(n 2pm-213+ Ko) 22-4Lli+3=fi i l " s in (* i lx r )dx < - .
" " " r=l 0 s in ( X r ) r
TaKm4 tmHOM, lim 12,31 = 0. Orxe , aKmo 2[3 < 2 r a p , TO lim 12, 3 = O.
n...r . - r
Y mma/IKy, KO~m 2 r a p < 2 ~ < 2 r a p + m , Aose~emt~ Bi~iamle-'rbc~ mm.le "raM,
mo y npotleci ~oBo~emr po3r~i.~alOTbC~l cl3YHKifii (Path (U) 3 il~UlIMI~ napaMeTpaMH.
A caue: qhp,2~(u), q~2p,2~(vO. TaXHM ,m.OM,/Ipyra y~osa ltacaia~y 4 3 [29]
TaKO~K BI~KOI-~I'I~C~L
rIepesip~I~O s~xoeam~ ~ n z o i y~ose ~aani~y 4. rlos~a~m~o s~pas, mo CTOiT~
ni~ s~axo~ rpammi y TpeTi~ y~osi q~pe~ 1~, i noaa~o ~toro y e~r~.~i
I3n - [3.1 + 13n2 + 13n3' ( 2 8 )
ISSN 0041-605s YKp. .,,,~m. ~. I~., 1999, m. M , N e I
HPO TEOPEMH ~IEBI - BAKCTEPA fl,]Ifl s rlOJIIB. II 17
~c
13n 1 = CnG4(o~nl'Cn) m f f H 3 ( ' ~ , ~ ) d ' ~ d ~ ,
I1~11~ I lgl l~S
I3n2 = 2CnCJ4(Rnl'~n) m [ f H3(~,~'ld~'d~,
IlPll <B IliIl>~
I3n 3 = CnG4(O~nlT, n) m ~ ~ H3(~,~)d '~dX,
I1~>~ ngn>~
a qepe3 H 3 (~, ~) noaHaqeao ni/iitrrerpa.rmny qbynrIIilo.
Ana.aoriqHo, aK i nprt poaraa~i Z~pyroi yMoan, OtfiHnMO nepmnla ~o/xaHo~ y (28).
OTpHMaCMO
Ia.~ _< g3~ (%a~)m(2~+l)-2~,
~e
X~l
OT~Ke,
= r f ld'~ f ? ( ~ ) 1 2 d ~ ] < ~"
k 2 ) L[l~< B tr
lira 13n I = 0. (29)
n--->m
Ottim~MO/~pyrHla ~o~aaoK y (28):
. I3n2 <- 2r -n , - , , -n , ~,-~j ~.-'~-'j •
X ~ ~ ( I ~ sin2 (Xnl(gr +Or))lsin Or[ p ) I f ( 2U ]1
• II~11 -~ d~d~. (30)
IHTerpan no 3MiaHit~ F y npasitt qaCT~Hi cniBBi~HomemIa (30) OtfinmeTt,C.a cgimxe-
HHOXO cyMom O~HOTVInHHX inTerpa.aiB no Bcix Koop~nnaTanx KyTax. ToMy ~OCHTb
6yae otIiHHTa O~HH i3 HHX. HanpaKaa/I TOI~, J/KHI~ 6epeTbca no aeptuoMy Koop-
~aaHaTaOMy KyTy (13n21). Poai6'eMo nepmntt Koop~aaTHn~l KyT na m -BHMipni Ky-
6H 3 ~OBZCanom pe6pa n. OTpnMaeMo
13n2t = ff 4 (M + E ) Cn anm "fn 22m-~ ('gnO~n )mP+~-2m (B] mp - m X
• f ~(Hsin2(x:~(Ur+Vr)))l)(~)[%,,V(w)ldWd~'(3t)
Ba.% [0.x],.\ r=l sin2 (Ur +Or)
Hexafl n HacTim~KH seJ~IKe, UlO BrtKOHy~rBc~ nepi~nieTb BCXn'~ n -< 2g. Togi
irrrerpa~ no ~iHHii~ ~ y rrpaBitt qacT~ni cniBBi~ometma (3 t) ouimOeTSCa cKinqea-
nora cy~om oiaaoTannnx inTerpania no Bcix m-aa~ipHax Ky6ax, pe6pa axax ~e-
~aa~, Ha KOOp/~HHaTItHX OCJIX i o/ma a neptuna cnisna~ar a noqa~xoM KOOp/IrlHaT.
IIpn t~oml iaoa~v.nna pe6pa nopiBnmr ~. ToMy ~aOCHa~ 6ytte oujmn~ omm ia nnx.
Haapn~za~ ro~, aXata 6eper~ca no ~no~aHi [0, ~]m. Oczin~Ka I%,~(~)1 --<
<_ Hm=llV r [p-~/m + KO ' TO Ilel~ irrrerpa~ y cnom qepry poana/la~'TbCJt Ha cy~y ~BOX
ISSN 0041-6053. Yrp. ~tam. ~'vpn.. 1999, m. 51, A te I
18 B.B. BYJI~IHFIH, B. M. MEJIbHHK, B. F. IIIHOPTIOK
/3n31
X
irrrerpa~iB. HaBe/IeMo ol2iltKy ~ntue TOrO 3 HHX, $tKHI~I ai/moBi~ar nepmoMy/Io/IaHKy
B 0ttiHui qbynmaii ~p,~ (~):
/3nil 22m-l~+I ($4 (M + E ) c n ~-m _m . . . . . mp+~-2m (B'~ mp = ~n l'ntl'n~n) tTJ x
x f f ( ~ sin2 ('lTnl(Ur+Vr))l[j~r 2U /{eft VrlP-~Jim)d~d~.
to.:1" to,.1" "=~ sin2(u'+v') 71 ta.~--'~;Itr=~ I
BtIKOprICTOSytOqrI BaacrrtBocri aapa Oeitepa i iaTerpoBHiCTt> t f (7) 2, orprtMaeMo
13n211 -< K32 O~nm('CnCtn) mp-~+m/2,
/ I e K32 < oo OcKi:n, Kn 2rap + m > [~, lira Xna_ = 0 i lira (~n m = 0, TO
�9 n - - . ) o o u 1 l " - ) ~
lim/3n211 = 0 i TOMy
/1 --.~oo
lira 13n 2 = 0. (32) n-.>oo
ORiHrlMO Tenep TpeTi~/~o/IarloK I3n 3 y (28).
$IK i rlpH ROBel~eHHi yMOBH 2 nacai~Ky 4 a [29], nicaa 3a~iHn 3MiHHHX
U r = )bnO~nT';l ~nO~nT, n ~ , 1) r = ~ , r = 1, m,
2 2
iaTerpa~ ottin~eTbCa aaepxy CKiHqeHHO~ cy~o~ O]IHOTnnHaX iHTerpa.nia no acix
Koop~aHaTHHX KyTax. 5[K npnK0aa~, aaae~aeMo OtfiHKy TOrO i3 HaX, y aKOMy inTe-
rpyaaHna npoao~rrr~cz no neptuoMy KOOpZamaTnOMy KyTy (I3,31). Poai6'e~o nep-
tuH~ Koop~naaTnma KyT na 2m-aa~ipai Ky6n 3 ;aoa~nH0~ pe6pa ~t i 306paaI4MO
I3n31 Y aar~zai CyMri o/xaoTnnHnx iHTerpaaia no eaeMeHTaX poa6na-ra. Ylicsta roro,
aK BHeceMo cyMy ni/I 3rlaK inTerpa~ , OTprlMaeMO
0~ 2[~-2m
2 - m r a nT, n sin2('Ur +Or) ) = (M'I'E) ($4Cn~n "lPn ( T ) f ~ fflsi"2('~ll(Ur-I'Vr) )~
[0,g] m [0,g] m\r=l
x 9p.l~(~) 9p, f~(F) d~dF .
CKOpttcTaBIIIHCb OUiHKO~ qbyHxttii tpp, l ~ (F) <_. l"~m=iiVr I p-f~/'' + K o, ao6paaHMo
/3n31 Y Baraa~ti CyMH HOTHpBox ~o~anxia. HaBeaeMo ortinxy TOrO 3 HHX, aKUlt
MicT~v,, ao6yroK I-[r%ilVr IP-~/mlur I " -~ / ' :
-m m (O~n'gn'~ 2~-2m
I3n31! = (M+F')2G4CnfZn "gn t T J X
!] ( ~ T s i n 2 ( ' C n l ( u r + V r ) ) l r ,p-[~/mlv ,p-~/m~
x ~ x x . 2 j t .X~, . . r , , . , )d~d~.
[0,~]' n [0, m r=l Slrl (U r + O r ) =
BI4KOpHc'roByIO'-IH BJIaL~I'HBOC'Ii a/Ipa <l~flepa, HeBa~r ~OBe, CTI4, I.HO
13n311 ~ K33 ~n m, (33)
2m-2l~ | ~, Xn
K33 (M+ ~)20~4 [2P -2-~+1 sin2 Xr < o o .
ISSN 0041-6053. Yrp. ,u~n. ~..TP,.. 1999. m. 51. N ~ I
FIPO TEOPEMH fIEBI - BAKCTEPA }MI~I ~POBOBHX FIO~IIB. II 19
I3 cniBBi/iaomenHa (32) i yMo~a 2 TeopeMn 5 mm.nnBae lim I3n311 = 0, a OTT~e, i
n--->o~
lim 13n3 = 0. (34)
l | -" ' ) ~
TaKrtM qrmoM, i3 cniB~i~anomenb (28), (29), (32) i (34) Bnn.nnnar mo lim I3n = 0,
TO6TO TpeTJ~ yMo~a Hac.ni~agy 4 3 [29] Bngonana. OT~e , y 8nna~Ky, Ko.nri 213 <
< 2 m p + m , TeOpeMy 5 tloe, e~eHo.
Hexa~ Tenep 2 [3 = 2 m p + m. 1-lepeBipnMo BrXX~onanHa neptuoi yMOBa Hacstiz~Ky
4. Bn6epe~o e i B(e) = B TaKi :~, , a g i y Brma,agy 2[3 < 2rap + m , Ta cgopncTa-
eMOCb 3o6paz~erIn.ma (14):
l ln = 11nl+Ilv2"
BaKopHCTOBy~OqH CniBai/IHOmeHHa (15) i Baraa/~ nopMy~OqOi noc~i/IOBHOCTi
(C n, n > 1 ), aerKo ~OBeCTn, mO
I1,,1 -< Kiilln(x,,a,,)l -l-
OcKi:xbKa aa yMOBO~O reopeMn lim (r = 0, TO
cniBBi~HotUeHHa (35) Bnn.nnBae
lim Iln(.~nO~n)1-1
#l--,a#,~
(35)
= 0. ToMy i3
l i m / l n l = 0. (36)
n - .~ oo
PoarslaneMo/xpyrrltt ~o~arIoK 1 i n2- BrmopncTaBtun criiBBiSlttOlSaemi.a (13), no, aa-
Mo uett Z~o~anoK y BrWS~i cyMa
/ In2 = /1,21 +/1n22 + 11n23, (37)
/ ln21 = M2(0.2+~2)Cn(O~,lZn)-m f Ifl(2sin~r2"'cnl2PlH~li-2fSd~,
ii]>Bkr=l
' , . 2 = = f
I i>Bt':l t 2 J j
]1n23 = g2(~)(l~2"t-(Y~)r n)-m f 1fl(2$ill~'rO~2nT'n)2P)il~ii-2[id~.
~i]>B\r=l
OI.IiI-IHMO /ln22" HeBa3KKO 6aqHTH, mO nicJl~l aaMiHH aMianoi u r = 2 , r =
1,'---~, nepexo~y ~o no0-mpm~x KOOp/~nHaT i oqeBHJIHHX npOCT~X O~iHOK o'rpuMaeMO
2~
11n22 ~ 2(0.2"l-0.21])'r ! do f dq)l x
Betn'r <p<l 0
2
lg 1r Ig lg
2 2 1 2~ 2 2 ~
f dcP2 ""fx P dCPm-I + ~ dp jr dCPl I dq~2 "'" f p-2mp-I d~m_ 1J g KI2 2 E,
l ~p o __~ __~
2
~e
•
2 2 2
/~e KI22 = 4MTgm-l(o 2 + 0 "2) < oo . OT•e,
ISSN 004 1-6053. Yrp. ~iam. ~.'vpn., 1999, m. 51, N e I
20 B.B. BYJ-I/1HFIH, B. M. ME.rlbHHK, B. F. IIIIIOPTIOK
lim I1n22 < K122 8.
n---b~
AHa~oriqH0 ~OBOJ~HTbC~[ , m o
-lim Iln23 ~ K123 e2,
,/~e KI23 = 2((Y2+(y2)g m-1 < '~.
O~iH~O l t n2~" ~aXa n~oro no~a~o i~oro y naraza]~i cy~n
/Ie
/In21 = lln211+Iln212,
(38)
(39)
(40)
- - •
2 <|~<~
~u-~ > 1 r = l
CK0pncTaBturlCb nepexoRoM ,rio n o ~ p H n x zooplxrIHaT, 0TpaMacMo nacTynHy
oninzy:
,r,.~_~2 _< M2(,:~2 +,s2w)ltnCc~.'~.)l -! ~'-' F"" ZIh~O~,,.~.)l-'. (41)
mp
OcKiJmza lira [ In (ct n x n) [-~ = 0, TO i3 (4 I) BnnaaBae, mo ~ ~losi~noro k icnyr
rpamma
tim 11,,212 = 0. (42)
IJ.[o60~am'a [l n211, c~opx4c'rae~docL cni~Bi~omenHnM
To~ti ne~a~zxo ~oBec-ra, too, 3 o~aaoro 6oKy,
/ l'~2mp
n -'--) ~
a, s imnoro 6oKy,
lim[ln211 <; M2(tYZ+G 2) X
x j'(eosqhsin~,)2Pd~l]I'I I (eOS%)2~+r-'sin2'~ra~r �9
o
(43)
ISSN 0041-6053. Yrp. ~nn. ~'ypN., 1999, m. 51,1~ ]
FIPO TEOPEMH J"IEBI - t;AKCTEPA fl, fl.q ~POBOBHX FIOJIIB. II 21
IIicaa nepexony y (43) no rpaHHt~i npr! k --->.0 ia cniBainHOmeHb (43) i (44) orpH-
WteMO
2~
lim lln21! = M2(ff2 +ff~V) f (cosg, s ingl)2pdgl •
n,-.+~ 0
m-l 2
• ~ f (cos~Pr)2rp+r-lsin2PcPrdtPr < oo. (45)
2
TaKm~ ~-mHOM, iZ (14), (36)-- (40), (42), (45)sHn.ra, mar
0 < lira I l l , , - lln211[ < (Kll+K12e)e.
n--.>**
OcKi.,mKa e > 0 6y.no ~n6pane noai.rmnaM, TO
2re
lim Iln = M2((~2 + 6 2) ~ (coscPl sin(Pl)2pdCPl •
n--4~ 0
m-I 2
X H ~ (COSq)r)2rp+r-1 sin2Ptpr dtPr < oo.
r=2
2
OT~e, yMoaa 1 Hacai/~Ky 4 BI4KOHy~TbC~I.
IlepeniprrMo ariKOHarma npyroi yMOBH Hac~i/xKy 4. J;I~ i y nonepe~m, oMy arman-
Ky 306paaHMO 12n y BHFJI~I/Ii 12n = 12n I + 12,,2 + 12, 3 (]]JIB. (19)). l-lepmHil ~oga-
HOK omanMo anaaorinHo nepisnocTi (21): 12n I < K21 [In(a,, Xn)[ -2, /Io K21 < oo i
lira Iln(~.Xn)l-l= 0. ToMy
I|--@oo
lira I2n I = O. (46)
...@*o
3 r i c o 3 HepiBHiCaaO (22) 12n 2 < K2211 n III n2" Today
lim [2n2 = O. (47)
.-.~oo
OIIiHHM0 TpeTifl nonaHoK 12n 3 . OCKiJIbKH y K0)KHOMy 3 m-KpaTHHX iHTerpaJIiB,
~Ki BxonJrrs y 12n 3, iuTerpyBarma npoBo/Irrrsca no 3OBHimHOCTi Kyai 3 paniycoM
B, TO irrrerpa.q 12n 3 Mo~z~a omHm-~ aaepxy, poaturIprlamri MaOa<aHy irrrerpyaanHz
no aoaHimHOCTi anHcanoro y Ky~o m-aHMipnoro zy6a ai CTOp0HOIO 2B/' f-m. OT-
pHMaHa OUiHKa JIHIIIO HOCHJIHTI.C~I. JIKIHO 30snimuicTb Ky6a no/~aTH $IK O6'r
nimrpocropin, rpaHHam~H ~tZHX 6ynyT~ n~o~mm rpaHe~t Ky6a. Ta~HM qHHOM, I2. 3
M o n a oI~iHHTH 3~epxy CKiHqeHHOIO cybtoIo 0nHOTHIIHHX nonaHKiB, y JtKHX iHTerpy-
aaan,a rrpo~oarrn, c:a no azazaaax nisnpoc'ropax. ~.,"m npHK~a~y raascnc~o oRiaKy
O/U-IOFO 3 HHX:
12n31 ( 0 2 . 2 X2 2 -2m
- " 4"(~W) On~n f f H 2 ( ~ , ~ l d ~ d ~ . (48)
x I > Bl'J'mm l.t 2 > Bl'~m
Xi>0 ~ > 0
i=2,'-~ i=3,'~,1
CK0pHCTaBmHCb yMOB010 (13) Ta 0rlHCaHHM paHime ~eTO~OM poa6HTTa nepmoro
Koop~HHaTnor0 ZyTa Ha Ky6H a pe6poM noaac~mom ~, ouim~o l~n3~ cy~o~o:
ISSN 0041-6053. Yxp. ~am.~y.pn.., 1999,m.51,N f 1
22 B.B. BYJ'I~HVIH, B. M. MEJIBHHK, B. F. LI.IHOPTIOK
/Ie
/ Ie
I2n31 < 12n311 +12n312 +12n313'
12n311 =
• I
[0,~1" [ore] '~
<s,.,:, 1 sin2 (Ur + Vr) (sin Ur) 2p
l 1 s ( s (Or't'krlt) 2 - - ' -7" X
X t i=2 .= kl=l \r=l
2mp+m~
X j~=l,3q~i=Oq~2=l(r=~ l ( U r + q r ~ ) 2 ) - ~ J d~d'~'
,'2,,~,2 = (<,2 +oh) 2(,~n'~,,)~",~g"lln(,~,,wl-~( M+~)' 2-4"" •
X ~" ~;~I sinE(Tffll(ur+Dr)) )
D~l [0,~]'\ r=l sinE (Ur +Vr) (sinUr) 2p (sinVr) 2p X
2mp+m
i=2 ki=O kr=2
2mp+m
X j~_l.3q~Oq~2=i("r~_i(Ur+qr=)2)-'--2"-Jdud~,
I2n3t3 = (0"2 +0"2) 2 (~. %)E" c~:Emiin(a,,Z,,)1-2(M +E)4 2-4raP •
(r'[ sinE ('~:l(ur + Vr) ) " Op }
• ~ f (SlnUr) ~ (SinVr) 2p •
O.l D~2 k,r=l sin2 (Ur + Or)
X I~ k~,~a=O(~ (13r'l'kr=)2"t'l)2)-~" •
j : l ,3 qi=O r~2
(~ +o~:)2 (~. ~.)E~ ~:ml~n(~,, ~,,)I_2(M +~), 2_,~, •
X
['~.x,,B ~]x[0.~],,,_l. a = D,~ = LE-- '~- ' - D/c2 [0'~]XLT'P~ ~lX[0'~]m-2"
CKopHc'raamncb Hepianic'no (20), .nerKo ~osec'rH, mo
IEn3n < [ln(Ctn Xn)[-2 (aZ + ~2V,')2 ( g +e)4 S~ 2-4mp-Em~ 4m,
~e qepea S l noarmqeHo OUiHKy cy~m p ~ y niR aHaKOM iwrerpaay.
(49)
(50)
(51)
(52)
(53)
I$SN 0041-6053. Yrp. ~tam. ~'vpu.,1999 , m. 51, N ~ 1
OcKi.rlbKH 3a yMOBOIO
lira 12n3~ ~ = O.
?~-...>r
IIo3HaqrlBllIH ~epe3 S 2 cyMy pazty
/ -
i=2 j=2 ki=O k)=l \ r=l
j~i
orprt~acMo ottirmy ~ a a 12n312"
[2n312 ~ 12n3121 + I2n3122,
/Ie
VIPO TEOPEMH dIEBI - BAKCTEPA ~q~ ~DOBOBHX FIOYIIB. II
lim (ct n xn) = O, "ro ia (53) Brln:lmme
n-4,oo
2mp+m
2
23
(54)
OOy (55)
(56)
ISSN 0041-6053. Yscp. atam. x,.'vpn., 1999, m. 51, N e I
12n3i21 = (or 2 +02w) 2 (O~nXn)2m~'~2m[ln(~nXn)[-2(M+e) 4 2 -4rag X
[0,~1 " [0,rq" r=l sin2 (Ur + Vr) dKd~, (57)
12n3122 = ((Y2 + ~ V ) 2 ((~l'nT'n)2ml~n2m[ln(anT'n)[-2Sl 2-4rap X
(58)
o,t [0,~] m r=l sin2 (Ur +vr)
~o/~aHOK I2n3121 .rleFKO ouimoea-bca 3a rtormMorom cniBBi~anomenna (20) i, TaxnM
qHHOM, OqeBH]~HO, mO
lira 12,,3121 = 0. (59)
R .-.9.oo
III06 OUiHnTa 12,,3122 OtfiHaMO [sinVrl--< U r, IsinUrl--< 1, nepe~tgeMo ~0 no-
aqgpHHX KOOpgl, maT, CKOpOTaMO Ha p~mV, otliHHMO ,,3aflBi" ereneid cos q~i i sin cp i
O/~HHHqKOIO, i rloBepTalOqHCb ~IO BHxi/IHHX KOOp/IHHaT, OTpHMaCMO
I2.s,22 <- (cs2 +cr~v)Z (~n %)2" e~:2"lln(~. Xn)l-Z ( M + e) 4S' 2-4mp X
. , _ _
x f f i sinI(ur+Vr) ) I ~ dud~
D= 1 [0,~],n r=l
~Ia.ai, CKOpHCTaBIIIHCB BigoKpeM~eHicTIO HopMrI 11 ~11 Ha MHommd D=I Big Hy.rl~I i
B~aCTrmOC'raMH a/~pa Oeiaepa, orprtMar
z-4m.( B ym
12n3122 = anmlln(an'Cn)l-2(M +e) 4 ~,2---~U-f ) x
• i J~ ~ ~n sin2 (X:I(Ur + 0 r ) )
[ , ] [0,~] sin2 (Ur +Or) durdv r. (60)
I ocldm,ml aa yMoaom reopeMrt lim (XnCXn) = 0, TO ia (60) mm~msar
n-r
lim I2n3122 = 0. (61)
n-..~,oo
24 B.B. BYJ'I/],Hr'IH, B. M. MF_J'IbHHK, B. 1". IIIHOPTK)K
Tax,n~ qrmo~, ia crtiBBi~nomem, (56), (59) i (61) Bl4rl.rlHBar
lira 12n312 = 0. (62)
n-.@eo
Ot~iarI~O Tenep TpeTil~/xo~aHOK y cniBail~HomeHai (49). J"lerKo 6a~irrr~, mo aa-
CTOCOSy~Oqrl CrdBBi/XHOUleHHJ~ (55) /UI.r orlirmn o6ox qbyaKtliona~IbHrIX p.q/~iB niz~
anaxo~ i~TerpasIy, OTpaMar ottitmy, ana.norinrIy (56):
/2n313 ~ 12n3131 + I2n3132 + /2n3133 + I2n3134, (63)
~e
/2n3m = (oh + cr 2)2 ((zn xD 2'n cz~ 2m [In (cx~ zn)l-2 (M + e)4 2-4rap •
x s
D=I 0.2 k r=i sin2 (Ur +Vr)
I2n3132 = (G2 +0~I/)2 (~n Xn) 2m aZ2mlln(a, z.)l-2(M + E)' 2 -4m" •
sin2 (x~l(ur +vr)~ . . . . ~ $2
x S S I~I '2 "(sinur) "~ (sinvr)'e/ dgdV,
Ox i Ox 2 r=l sin (Ur+Or) JIi'VH 2mp+m
12n3~33 = (~2 + ~ ) 2 (an~n)2,.c~:2.,lln(an~n)l_2(M +~)4 2-4mp •
X f S(~=lsin2(xnl(ur+Vr))(sinur)2P(sinvr)2P 1 S2p+mdUdg,
D~I D~2 sin 2 (u~ +v~) I1~11
= 2 2 /2n3134 (0"2 +GW) (IXn'r',,)2"~ g +E)4 2-4rap •
r -'[sin2(x'~i(ur+ur)) :sinv ,2p~ 1 1 dgd~.
x S f t ir~ 7in-'~'~Ur+."~ ) (sinur)2P' "/ Jn~lpm"+'itvilZm"+"
Dnl D~2
Bi~iTaMO, mo /2n3131 olIi/-IIOe"I'bC$/aHaJIoFiqHo ]2n3121, a ]2n3132 i /2n3133
aitaJloFiqlto /2n3122. TaKHM qHHOM,
lim /2n3131 = lim 12n3132 = lim /2n3133 = 0. (64)
/1--.)o~ n..--)oo .-*l.-~oo
OraimiMO 12n3134" }~.rla l.lbOFO OIJjHHMO Isinvrl--. Iorl, [sinu~l--- lurl i, ~aKop.-
CTOSymmI riepexi~ ]Xo no.napmIx KooplxHnar, OTprIMar HacTynHy ottiHKy:
I2.3134 -< ((r2 +C~2)2 (O~n Xn)2" O~:2mlln(an X.)l-2 ( M + E)4 X
( f i sin2 ('c~l(ur +ur))~,,_
0~1 f D~2 ~'r=l ~ sin 2 (u~ +Or) ],u 11 - m I1~11 - ~ d~d~.
x
~a.rli, cKopHCTaBmHCb o6Me:h~tenicTIO aopbl [[ U'II i I1~ Ha ~r iI-ITerpyBaHH.~
amoy i B a a ~ c T m 4 n attpa (l>ellepa, OTpm~aeMo
[2n3134 ~ ('in (x2)-m I1 n (~ . xn)l-2K2.3134, (65)
ISSN 0041-6053. Yrp. uam. mypn., 1999, m. 51, N ~- 1
HPO TEOPEMH IIEBI - BAKCTEPA Ll, rIfl s HOIIIB. II 25
g2n3134 = ( M + E ) 4 (O2 + O 2 ) 2 (B)-2m(,2ml/2) x
X ~= 1 ~ ~ "~nSin2('~nl(ur+vr))
[0,~] [0,~] sin~ (Ur + Vr) dur dvr < ~"
OcKi.rlbKn aa yMOBOIO icnye O > 0 TaKe, mo "~n 0~n 2 >- 0 > 0 /~.rlsI /~OBia-IbHOFO n,
TO iCHyr rpaHnt~a
lim I2n3134 = 0. (66)
n -at oo
TaKHM q-'~InOM, i3 (64) i (66) BrllIJIHBae
lim 12n313 = 0. (67)
t/---)oo
Orxe , a~ Bnnan~ae i3 (40), (54), (62) i (67)
lim [2n31 = 0. (68)
A / 2 e ~OBO~HTb, I.I~0
lim 12,,3 = 0. (69)
n - - )o o
13 cniBBi~aHomeHb (45)--(47) i (69) BnnJmBae, mo lim 12. = 0. TO6TO ~apyra yMoBa
?1---r
Hac~Ky 4 BaKOHyr
IlepeBipaMo ~KOHaHHa TpeT~oi yMOBa Hacni~tKy 4. YloaaaaaMo sapa3 i3 TpeT~oi
yMOm~ aepea 13.. 1-lo3Ha~nBm~ ni~iHTerpan~Hy qbyHKl~im y rpeTiI1 yMoBi ~epe3
H3(~, ~) , noaaMo 13. y B.rnzai
(70) 13n = 13nl + I3n2 + 13.3,
Re
, -1 ,,,m
13 n I = c. 0 4 ['~n (Xn )
- 1 m
13,,9 = 2C nO 4(T, nan )
13n3 2Cn(I4(T, n -1 m
; f H3(~ , '~ )d~d~ ,
[~l>n ign>B
OHjHHMO nepmai~/~o~anoK 13 n l" CKoprtcTaBttmCI. cniBni/momemt•M (20), HepiB-
HiCaaO I sin x I < I x I, HepisHiCTm Fesm/~epa Ta iHTerpoBaicaao y cepe~n~oMy KBaz~pa-
�9 . .oMy o p. a o
l~e
13n t < K3tlln(anXn)[ -1,
m
__ ( ~ ) 2 m ( n ~ 2rap B m
(71)
ISSN 0041-6053. Yxp. ~tam. ~. p.., 1999, m. 51. bl e I
26 B.B. BYJI~IHFIH, B. M. MF_..IISHHK, B. F. IIIIIOPTIOK
OcKi.rlbKlt aayMoao~oTeopeMn lim x n a n = O, TO i3 (71) ann~nBar
n- ->~
lira 13,,1 = 0. (72)
I,--o#~
~'r IXn ~n g r IZ,l Z,1
O I / j H H M O 13,,2. IlicaJ{aaMiHn 3MiHHHX V r = , u r = , r = 1, m,
2 2
i r r rerpaa no aMiHHiit F OtfiHmeTSCa aBepxy CKiHqeHHOm cyMo~o O~HOTHnnHX iHTer-
pav ia no niBnpocTopax, r~e~ami aKriX e n~omrm~ rpanett ry6a , Bnncaaoro B Ky~m:
[[Fi[ _< B a n x.____.~ 5/~ npaH~a/I HaBeAeMo OlljHHy OAHOrO 3 HI, IX:
2
3m
O~ "C nip Utl"~, l mp---~ 13n21 = 2(M+E) t~4Cn('CnO~nl)m ( n ,,) ( 2 ) (~)mP X
( f i Sirl2"tn(Or+Ur) 1
2 (sin vr)e x
X I I sin2 (Or +Ur)
B IXn "~n B ~n "~n r=l
I1~'11~ 2 - v l > ~
vi>O. i=2.m
I ^ ( 2~ "~l _ _2mp+m -- _ x fk ---7; ,Jliloli dodu.
AHa.nori~Ho, ~K i IIpH nepeBip~i Apyroi y~OBtl, poai6'eMo MHO~KHHy { v I > 7t,
v i > 0, i = 2, m } Ha m-BriM'pHi xy6r~ a ~OB~nnO~ pe6pa ft. Flicsm TOrO, aK ane-
CeMO cyMy ni/l 3HaK iHTerpa~y, olliHrlMO 13n21 cyMolo
I3,,21 = (S 3 +$4) 13,ml + 13,,212, (73)
fxe
S 3 = (kr~) 2
i=2 ki=O 1 r=l
2mp+m
9
2mp+m
1-- $4 = f (Or +krl"C) 2 4 ,
j=2 i=2 ki=O kj= 1 k. r = 2
j~i
3m
13n211 = 2 ( M + E ) ( ~ 4 Cn ('Cn O~nl) m (O~n'Cn)mP ( ~ ) mp-'~ (2)mP x
x I I ' sin l (ftr +v r) JIZL --XJI
U g l l < B e t . 'On [O.~l m
2
3m
13n212 -" 2 ( M + E ) ff4 Cn('gnO~nl) m (O~n'g,i)mP(~2-~) mp--~ (~)mP X
( ) _ __-'m'-mi:(. 2.-)1 I I f l sin2('~nl(Ur +Vr))(slnVr)2P I1 11 2 d~d~.
illll:aB~ll.'c. D,I r=l s in2 ("r -l-Or)
2
(74)
(75)
ISSN 0041-6053. Yrp. ,~lam. ~.'vpn., 1999, m. 51, bl ~ 1
riP(3 TEOPEMH JIEBI - BAKCTEPA fl, flfl ~POBOBHX HOJIIB. II 27
OI~iHHMO 13n211" IIpHnyCTttMO, M o n ~aOCnT~ Be~IHKe,/IJIA TOrO, 1Iio6 BHKOHyBa-
~ac~ HepiBnicT~ B (~nTn < 2 g. To~i, BICKOpHCTOBylOqH B~aCTnaOCTi ~apa Oeflepa,
HeBa,'KKO ]~OBeCTH, mo 13 n212 ORiH~OCTbCA CKiHqeHHOIO cyMo~ O~HOTHIrIHHX inTer-
pasha:
13n211 = Z let,
~t
lxe la < Kalln(cx,,'Cn)1-1 ~,-m i
TaKHM qHHOM,
K a = 2-2mp+m+Iigml2(M+E) (y4($3 +S 4) •
I
xffiiZnsin2('~nlxr)dxrl(~[f(z)12d'zl2\r=-! 0 sin2Xr < o o
lim 13n211 = 0. (76)
n---)~
OUiHHMO 13 n212" CKOpHCTaBMHCh OHHCamCM p~ ime MerO~OM nepexo~y ~o n0-
.nApHHx Koop/IHHaT, OTpHMaeMO
3m
/3n212 -- 2 ( M + 8 ) ~ 4 Cn('CnC~nl) m (CCnqs mp-'-~ (B) rap>(
( sin2 ('Cnl(Ur +Vr)) l •
B O[ n '~n m- I r= l
• v 11-7 d5 (77)
BHpaa y npaBiA qaCTHHi cniaBilanotueHHa (77) ottiH~eTbCa aHaJ~oriqHo 13n21t, .aKI~O
nonepeaHbO cKOpHCTaTHCb Bi~oKpeMSxeHiCT~ HOpMH [I FII air Hyna Ha Dnt. TaKHM
q H H O M ,
lim /3n212 = 0. (78)
OTme, a x san:maar is (73), (76), (78), MaeMO lim 13n21 = 0, a ToMy i
r/-.-)oo
lira I3n 2 = 0. (79)
n--->oo
OtfiHrIMO TpeTii~/1o/lanoK 13 n3 Y CniBBi//~-Iomenai (70). CKOpHCTaBMHCr~ cniBBi/I-
HOMeHH~M (13) R;aa nepeTBOpeHuA tl)yp'e dpyHKI~ii si/IryKy i 3aMiHOtO.aMiHHHX U r =
~,r(Xn~ n ~rOLn'Cn = , v = , r = 1, m, aHOBy OtfiHaMO OTpnMaHi ~ana m-xpaTHi
2 r 2
iwrerpa-rmnoaoBnituHOCTicclaep II ll < Ba 'c. i II ll-< B(Xn'gn, CKiHqeHI-IOIO
2 2
cyMo~ O~OTHmmX i w r e r p a ~ no n i snpocTopax , MezcaMH ~XHX e n ~ o u m ~ 6iqm4x
rpaHe~i, Brmca~x y cqbepH x y 6 i s . Fix rrpmc~a~, Hase~eMo OU~nCy OnHOrO a m~x:
[3n31 ( M + e)2 Cn('~nCtnl)m ( - ~ ) 2mp-m = X
ISSN 0041-6053. Yrp. ~un. ~.'vpn., 1999, m. 51,1~ 1
28 B.B. BYJI,/2HFIH, B. M. MEJIbHHK, B. F. IMI'/OPTIOK
X f ~ {~=1 $'ill2('~nl(ur+13r))isinuriPi$illUriP) X
Bcc. "c,, Boo,, % sin 2 (u r +vr)
t41 > ~ ' ~ ' ~ - Vl > 2rail2'
ur>O, r=l,m Or>O, r = l , m
m m
x II ~ II-"P-s II ~ II-m"-~" d~ a~.
~qxnlo ao6pa31rra ncpmlal Koop~imlaTmlll Kyr y Bnrala~i o6'r Ky6i~ a ~IOSXa-
liOIO pc6pa g, TO Mona OiliHnTil I3n31 cyMoIO:
I3n3l < I3n311 + I3n312 + /3n313, (80)
x S ~ - - - |dgd'~,
[o, nl" [o,n]"~, ~= sin" (u r +v~) )
13n312 = ( M + E ) 2 ~ 4 Cn ('tn lx,-~ i ) m ( Ot-'2"r'n-) 2mv-m 2 ( s3 + s4 ) •
• I I r I sin2(z:l(u'+v'))lsinu'lelsinv'l" li~ll-"~-Te~'
D~ 1 [O,l~],n r=l sin2 (ur +Vr)
13,,313 (M+E)2 0"* Cn(T',,(Xnl)m(Ot'~X ) 2mp-m = n X
X ~ ~ (~ $iI12(T'~I(Ur+Vr!')i$inUriPisinVriP} X
/>~i /)~l r=l sin2 (ur + lSr)
#n /11
• II ~ 11-~-7 II ~ II- 'P-7 d'~ a~.
CKOpHCTaBmHCb BdlaCTHBOCT~.MH a~pa Oeilepa, JIeFKO/IOBeCTH, I~O
13 .3 . -< K.H l ln (z , ,c< , , ) l - I a;,m,
T o ~
K3311 =
lira 13n31! = 0. (81)
B--.)m
OnmH~O ~pyr~n~ ~O~aHOK I3n312" CKOpV~cTa~mHCb nepexo~oM ~0 no.~apm~x KO-
o p ~ a T , OTpHMaCMo
1SSN 0041-6053. Yrp. ~m. ar 1999, m. 51, IW 1
FIPO TEOPEMH flEBI - BAKCTEPA fl~Ifl/]POBOBHX HOs II 29
13n312 -< ( M + E )2 ~4 Cn(,.~nO~nl)m (~t2, ~ )2mp-m 2 ($3 + $4) •
sin 2 (Xnl(Ur + Vr ))~ m
x ~ ~ I~ sin 2 (u r +Dr) )]]ull-2dFd~"
D~ I [0,/r r=l
~a~i, cKopncTaBmacs Bi/IoKpeMJ'IeHiCTIO HopMH II ~ 11 Bi/~ nynJt Ha Mnoacrmi Dn~
BaaCTHBOCT~tMa a~apa r OTpHMaCMO
m _m
Me
13n312 -< K3312(CXn 2 ~;n) -'~ ~n 2 I ln(%an)] -~, (82)
711
"Cn sin2 ('c~lxr) ~ 0 sin2 Xr
OcKiasxn za yMOnOrO TeopeMa lira (x .a n) = 0 i icayr TaKe 0 > 0, mO MaJt
]lOBia~noro n _> 1 anXn>- O, TO i3 (82) s~naHBar
lira 13n312 = 0. (83)
OtfiI-IHMO TpeTifl MOMaHOK /3n313" BHKOpHCTOByIOqH oI~iHKH [sin v r I <- IVr [,
I sin ur [ < l Ur I i nepexiM ~o noa~panx Koop~HHaT, OTpHMaeMo
13n313 -< ( g + E ) 2 if4 en(,[nC~,~l)m (0~2. ~ )2mp-m t.......~z •
sin2(,~nt(Ur+Vr))~ m m
• f f f-I ," -Y"
D~ 1 Dx I r=l )
CKopHCTaBmHCb Bi~OKpeMaeHiCTIO HOpM II ~11 i II ~]1 Bi~ ayaa Ha Mno~<aHi D~I i
BaaCTHBOCTaMa ~t~pa Oettepa, OTpHMaeMo
/3n313 -< /('3313 "l:n m r 2m Iln(z,, ani1-1, (84)
Me
( B )-m~m(~i'CnSin2('~nlXr)dxr) < oo.
K3313 = (M'{'E) 2 (~4 2m-2mp ~ sin2xr
r=l 0
OcKknbK~3ayMosmoTeope~m lim ('CnO~n) = 0 i icnye TaKe 0 >0 , m o / l a x ~O-
uiabHOrO n > 1 a2n'~n >- O, TO i3 (84) BHrlJBIBae
lim 13n313 = 0. (85)
n--->eo
TaxaM ~ o M , ia cniBaiMnomerm (80), (81), (83) i (85) B~n~sar
lim 13n 3 = 0, (86)
TO6T0 TpeT~ y~osa aac~i~cy 4 B~XOnyC'mC~. Teope~y ~ao~e/Ieno.
ISSN 0041-6053. Yxp. ~tam. ~.'ypn., 1999, m. 51. A te I
30 B.B. BYJ'ILIMFIH, B. M. MEJIBHHK, B. F. IIIHOPTIOK
3ayea~enna 5. IIopiBHmtOnH npx4 m = I Teope~y 5 a sillrloBillHItM pe3yJlbTa-
WON C. F. FJm/lameBa [18], cJfi/l aaepHym ynary, mo reopeMy C. F. FJmlmmeBa 3a/Io-
~OabnmOT~ rayccoBi npoueca, ~Jm aratx [3 > 1 / 2, a TeopeMy 5 ~ yaaraJmneui/Ipo-
6oni IlpoRecH 3 ~ > 1. YMoBa ~ > 1 BrlnrlKae npr~ nepeBiptfi yMomI 3, a~a Bi~o6pa-
mac cnetmqbi~y yzara~bneHnx JIpO6OBHX npouecia. Oc~i~KH cne~Tpan~na m i ~ -
sica~ yaaranmHeHoro ~tpO6OBOrO npottecy ~opiaHrOr x TO . p . > 1
/Izta~aocTaTm,o Mazmx ~ > 0 BHKOHyerl, y oBa --JR d X < 0~. A Re,
ari/~Ho a TeopeMom Ko~MoropoBa npo HenepepBHiCTb Marline HaneBHe peaJfiaatfifl
cTatlioHapaax mma/IKosax npotlecia [30], rrpaao~aT~/Io roro , tao cenapa6eJmai MO-
wadpi~atlii npoReci~, a~i poarJutaamTbCa y Teopezi 5, Mara~yr~ nenepepnHi Ma~me
nanesHe pea~iaaRii. Flpn [3 < 1 6ea/Io~laTKOBnX yMOB Ha cne~rpaJmHy Mipy H R~o-
ro Mome i He 6yra. /IZa rayccoBHX npoReciB MaeMo iHmy cr~Tyatdm, i TyT ~lJ~a Hene-
pepBHocTi Maltxe HarmBHe pea~iaatlha npoRecy ~OC~T~ y~om4 13 > 1 / 2.
Aaa~ori~Ho TeopeMi 5 /lo~ollrrr~ca HaCTynaa TeopeMa.
Teope~a 6. Hexaa ~(?) , ~ ~ R m, ~ oOnopiOne Opo6oee none s dpynr~ie~o
eiaeyny f ( ? ), nepemeopenna tDyp' e aroi'saOooonbnae y~toe)'
r=l" l<;r~m
1, aruto 2<13r<-3,
r = 1, m, k ~ N ,
Hexaa p , =P,([3A = k, ~ru~o 2k- l<[Sr<-2k+l , k>_2,
m
: (Pl . . . . . Pro), C,,= HCnr, Cnr = ~ , , ( ~ r ) ( t X , , * , , ) 2 - ~ ,
r=l
8e
1, ~nu4o ~r ~ : 2 k + t ;
5n(~r) = [In(~nXn)[ -1, aru40 ~r=2k+l.
ToOi, aru4o euronytombca ys~toeu: 2) lim cr n = **, 3) lim ~nX,~ = 0, 4) icnye
m-I 3m-i
mare 0 > 0 , u4oOnnOoeimmoeo n >- 1 (xn) 2 (Ctn) 2 > 0 >0, mo
1.i.m. Cn a,-~ m _~_ (AP ~ ~ ( ( k - T ) % % ) ) 2 = c,
n'l'~** k : l
ae
r=l R
//pwc~aa2. Poara~HcMoyaara~HeHe~po6oBeno~e ~(~), ~eR ~, m >_ I, 3
qbymcuie~o sinryzy HacrynHoro s~rn~y: f (?) - exp{ - b II ~ II }, ~ e R m, ncpc~o-
perm~ q~fp'r ~KOi ~ae n.rz~/X
2-m l-2m l+m
.~(~) = 2 2 n 2 r(m+l~b(}['~]12+b2)--s ".
\ 2 ]
Or ,, I)(X)] = MII II-'-', o(ll lr'-') npn I:11--,=. = m + 1.
To/fi, axmo m = 1, TO [3 = 2 i TOSV/S yMoBax Teope~H 5 no'rpi6Ho Bn6paTH p = 2 i
I$SN 0041-6053. Ylcp. ~tam. ~. pu., 1999, m. 51, I~ I
FIPO TEOPEMH JIEBI - BAKCTEPA fl~.q/2POBOBHX I'IOJIIB. II 31
Cn= (~n'Cn) -2 , npHqOMy CX n i "~n BH6HpalOTbC~I TaK, IJ.[06 BHKOHyBaJIHCb yMOBrl
l im r n = ~ , lim (x n an) = 0. Tolxi
n---~** I1-'@ oo
N
l ' i 'm" S~2)(~:)n-,-->** = l ' i 'm" cn 0cnl Z n . - * * * k=l (APntz"lc'((k-1) T'nt~n))2 = 8b23 (if2 + i f 2 ) "
flKmo ~ m = 2, To [3 = 3, i TOMy n y~oaax Teope~ri 5 noTpi6no ari6pa'rn p = 1,
c n = (anXn)-2l ln(anxn)[ -1, npHqoMy a n i X n Br~6apamTI,CaTaK, tUO6 aHKOHyaa-
~rtcb yr~onn lim a,z = o~, lim (X, a n) = 0 i too6 icnyBa~o TaKe 0 > 0, mo ~ a a
n - - - ~ I1---> *~
]~OBi~xoro n -> 1 a2n %, > 0. To~ti
1.i.m. S~2)(~) = 1.i.m. a; , 4 %S2 [ln(c%%,)[-1 _~_ (AP a ~: ( (k - T) xn a,, )) 2 =
n ---> 0- n--->~, k = 1
= ~ ( r -4 (2sinur) 2 11~ll-6d~ # 0.
R \ r=l
3ayBaZCHMO, tUO nopiBHanO 3 Bana~KOM m = 1, nopa/xoK piaHnaeBoro onepaTopa
3MeHmHBC~q Ha O]~HHHt.I, IO, a y Hop~ym'~ila nOC;qiIIOBHOCTi 3"~IBHBC$[ ~aO~aT~OBH~ MHO-
~KHaK ]In (a,, %,)1-1.
~IKtuO ar m > 3, TO [3 = m + 1, i TOMy B y~oBax TeOpeMH 5 noTpi6ao na6paTa
p = I i % = (a,, x . ) - 2 , nprlnoMy a n i x n aa6rlpazoT~.Ca XaK. too6 BHKOHyBa.rIHCb
yMOBn l i m a n = -0, iim (%, a, ,) = 0. To;ai
I | .~) c.o n --~oo
1.i.m. S~z2)(~) = l.i.m, c,, a,-~ m _~_ (AP~: ~((k-T)a,,x, ,)) 2 =
I I ' - ) ~ 1 7 6 II '~)e~ k = l
TaKXM qHHOM, IIopiBH~IHO 3 BHIIa~KOM m -- 1,
3MeHLLIHBCJ~ Ha O/~HHHLIIO, a FIOpiBH$1HO 3 BHFIa/I~KOM m = 2,
~OBHOCTi 3HHK MHO3KHHK [In (13~ n "[n)1-1 �9
l (2sinur)2)ll'ffll-2m=2 d'ff ~ O.
r=l
nopa~oK pi3HI4tleBoro onepaTopa
y HopMyIoqilt noczi -
29. BynOuzbt B. B., Men~tu~ B. M., lllnopmtox B. 1". Flpo JIeBi - BaKcTepO~y tmacTnnicT~ ]IpO6OBXX
noJlin. I //YKp. MaT. aCyp. . - 1998.--50, N g I I . - C . 1463- 1476.
30. Kpa~tep F., J7uO6emmep M. P. CTaUaouapmae cJlyaa~hmle npotteccta. - M.: Mnp, 1969. - 398 c.
O~tep~Kauo 26.03.97
ISSN 0041.6053. Yxp. ,~tam. ~.'Ypu., 1999, m. 51, N '~ 1
|
| id | umjimathkievua-article-4579 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:01:30Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/79/cbef74fbdc2fd57134672e71a83a6279.pdf |
| spelling | umjimathkievua-article-45792020-03-18T21:09:14Z On the Levy-Baxter theorems for shot-noise fields. II Про теореми Леві—Бакстера для дробових полів. II Buldygin, V. V. Mel'nik, V. M. Булдигін, В. В. Мельник, В. М. We establish sufficient conditions under which shot-noise fields with a response function of a certain form possess the Levy-Baxter property on an increasing parametric set. Встановлено достатні умови, при яких дробові поля з функціями відгуку певного вигляду мають Леві - Бакстерову властивість на зростаючій параметричній множині. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4579 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 12–31 Український математичний журнал; Том 51 № 1 (1999); 12–31 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4579/5862 https://umj.imath.kiev.ua/index.php/umj/article/view/4579/5863 Copyright (c) 1999 Buldygin V. V.; Mel'nik V. M. |
| spellingShingle | Buldygin, V. V. Mel'nik, V. M. Булдигін, В. В. Мельник, В. М. On the Levy-Baxter theorems for shot-noise fields. II |
| title | On the Levy-Baxter theorems for shot-noise fields. II |
| title_alt | Про теореми Леві—Бакстера для дробових полів. II |
| title_full | On the Levy-Baxter theorems for shot-noise fields. II |
| title_fullStr | On the Levy-Baxter theorems for shot-noise fields. II |
| title_full_unstemmed | On the Levy-Baxter theorems for shot-noise fields. II |
| title_short | On the Levy-Baxter theorems for shot-noise fields. II |
| title_sort | on the levy-baxter theorems for shot-noise fields. ii |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4579 |
| work_keys_str_mv | AT buldyginvv onthelevybaxtertheoremsforshotnoisefieldsii AT mel039nikvm onthelevybaxtertheoremsforshotnoisefieldsii AT buldigínvv onthelevybaxtertheoremsforshotnoisefieldsii AT melʹnikvm onthelevybaxtertheoremsforshotnoisefieldsii AT buldyginvv proteoremilevíbaksteradlâdrobovihpolívii AT mel039nikvm proteoremilevíbaksteradlâdrobovihpolívii AT buldigínvv proteoremilevíbaksteradlâdrobovihpolívii AT melʹnikvm proteoremilevíbaksteradlâdrobovihpolívii |