О кОЁФФИцИЕНтАх тРИг ОНОМЕтРИЧЕскИх НУль-РьДОВ

Analysis Mathematica, 11 (1985), 139--177

O uo3~mlueuTax TpHrOHOMeTpHqecRHx uyab-pgao~

H. B. I IOFOCItH

w 1. B~e~Ienne

B pa6ore ~io~a3~,~aeTca

T e o p e M a 1. ~r ~to6oft nocze~)oeame5bHocmu ,tuce:t {Ok}~=0 marco& ~tmo

(1.1) ~ min 0~, = + 0% r = 1, 2, 3 . . . . n=0nr~k (n+l)r

cyu4ecmeyem mpueotto~empuuectcu~t t~y~b-paa euOa

(1.2) ,:~ 0;, cos 2rt (kx +Ok) k=0

' ' < I eOe Ok = q~ O~a ecex k = 0 , 1 , 2 . . . . .

HanoMr~nM, aTO p~41 (1.2) na3~,maeTca HyJIb-p~aOM, ec~r~ or~ CXO~I~TC~ ~ i~y.um no,~rr~ ~cm)~y ~a oTpe3~e [0, 1] ~ XOT~ 6t,I OJIt'II-I ~ogdpqb~rar~enT 0~ p~aa OTanuen OT HyJIX.

OqeBltAHo, qTO rmcamaoBaTeoa~,uocs~, ae~cea {0k}~=0, )Ia~ KOTOpOI~

(1.3) 0k~0 ( k - + ~ ) r~ ~ 0 ~ = + ~ , k = 0

yaXOBJIeTBopzeT COOT~toineHl~m (1.1), nO3TOMy g3 TeOpeMbI 1 B~,iTeKaeT

T e o p e M a 2. ,gsta npou3eoezbnoft noc~zeOoeame~cbuocmu ~tuce:t {~k}~'=0, yOoeJtem- eopa~o~e~ ycJtoaua~a (1.3), cyutecmeyem mpueoHo~tempu~ecKu~t uyJtb-paO (1.2) C Kog~6-

9'Ju~luenma~vttt " -< IOk[ =Ok, /C=0, 1, 2, . . . .

Cy~ecTBoBanHe TprrrOHOMeTpre~ecir n y ~ b - p ~ a BnepBJ, ie 61,urn ycTauoaaelm M e n ~ m o B ~ i u (CM. [15] n [2, crp. 804]). HenocpeacTBeuuo n3 o n p e a e a e a n , rpu-

r rOlmMeTpn~Iecroro nya~,-paaIa caiea!yeT, ~TO 0k-+0 npa k - ~ I~ qTO CyMMa ~aaa~- paTo~ ~oaqb~nmaenTO~ 0~ pacxoaaTCZ. Paa mawepecn~x uCCSleao~aHn~ 6~t~ noc-

HocTymaao 27 .~aa 1980, nepepa6oram~a~a Bapr~asir- 9 znBapa 1984.

140 H.B. UorocnH

BS~IL~eH B~,IYICtteHI/I~O BO3MO~KHOJ~ cKopocTg CTpeMJIeH~iYl K HyJl~O KOg~)@HIUdeHTOB

rIyas-pz~oB (1.2) (CM. paSOTt,I [t4], [28], [29], [23], [22] rl [12]). B ~acTriOCTrt, C a a e - MoM [22] 6bI~o ~oKa3aHo, qTO ~Yt~ KaT~ofi nocae~oBaTe~t,nOCT~ ~rrcea Akt~ (k-~ ~) cymecTsyeT TpHro~oMeTpn~ecKr~ Hy~s-p~ (1.2), KOg~m~HeHTt, I I~oToporo ~rMemT nop~or : o(k-~/~Ak). B 3TOM ~anpa~e~nr~ H s a m e B L ~ M - - M y c a T o B ~ [12] 6~,ia no~yqen c~eRy-~om~ o6mnfi pe3y~TaT. H y c ~ noc~eRo~aTem,Hocr~, �9 ~rrce~ {@~}~=0 co CgO_~cTBa~H (1.3) y,~os~eTBowte~ cae)IymmH~t ycsm~r~s~ pery- o~tpnocTn y6~,ma~rr~:

k o ~ O ( k - ~ ) , k ~ + ~ ( k ~ )

~J/Jt Bcex ~ > 0 , Ilp~Iq~M )IJIft ~eI;oToporo e0>0 nocJIeJlo~aTea~,HocT~, {k~+~0~k}~=0 MO~OTOIma; Tor)la cymecTByeT Tpnro~o~eTpH~ec~n~ Hy~,-p~Jl ~r~a (1.2), rRe O~ = 0 (Ok) n p ~ k -~ oo.

Bonpoc o6 oc~a6aeHr~tt ycaom~f~ pery:bqpHocTn y6~manr~ n~cea ~k ~ 9TO/~ �9 eopeMe I/I~ame~a--MycaxoBa o6cym~aaca na ceMrmapax (cvt. Ta~me [2, cTp. 840]),

I;a~ naBeCT~O, ero cnpa~e~JlrmOCT~,, ~or~a ~ocaeaosaTe~HOCT~, {~k}~'=0 y)lom- J~eT~opaeT ai~m~, yCJ~OBnZ~ (1.3), ocwaBaacx OTI;pI~IT/,IM (CM. ImnpnMep [26, CTp. 24], [27, cTp. 12]). IIoaomt~Tea~n~I~ OTBeT Im 9TOT Borlpoc cae;~yeT ~3 T e o p e ~ 2.

TeopeMa 1 ~omeT 6~,~, no~y~eHa ~am ~aCTH~N c~y~a~ 6oaee o6mero yTBep- a;)IeHH~t, ~a~ qpop~yJ~rrpozmr~ ~OTOpOro yAo6~m noJ~,aOBaT~cZ onpe~ene~gz~i~ (c~. [24, CTp. !33]).

O n p e j i e ~ e u n e 1. Hycmb {f,, (x) }~=l --npou~eoabnaa cucme~ua u~mepu~btX

96ynmtu~, no~tmu ae~Oe t~one~mbtx na ompe~t~e [0, 1]. Pac)

(1.4) Z, f , (x) n=l

na~bteaemca 6-ynueepcanbnbt~t, eczzu Oaa n~o5o(t ux~tepu~to~ u no~tmu ae3c)e tr

06ytm~luu F(x) cyuteemeytom ~ue~a ,5,=0, 1 (n=>l), c).~a ~omopb~x pac) ~ 6,f~(x) ?t=i

cxoOumca no~tmu ectoOy Ha [0, 1] ~c F(x).

Onpe~ le~en~e 2. PaO (1.4) na~taaemca a-ynueepca,a~nb~g, ec:m O:~a :tto6o~

u~gepu~ao~ ~yn~a4uu F(x) (F(x) ~ao~cem paet tambca + co na ~utto~ecmeax no:to-

~cume~t~no~t ~tep~t) ~uo~cno ma~ nepecmaeum~ ~4/lel4bt pm)a (1.4), ~mo6~t anoe~ no:~y- ,~enn~t~ paO cxoOu:zca no~tmu ecmc)y na [0, 1] ~ F(x).

Cnpa~e)lJ~rma cze~ymlua~

T e o p e M a 3. ,!Ira :tto6ogt noc~eOoeame:tb~ocmu ~tucmt {~k}~'=0, yOoeswmeo- pamule~t coomnoutenwo (1.1), eyu4ecmayem mpueono~tempuuec~cu~ p aO (1.2) c ~og~6- OSuvue~tma.,tu 1o~l<=b,I, k - 0 , 1, 2, . . . , ae.aatou~uftc,,t oc)ttoepe~etmo 5- u ~r-ynueep- Ca./IbHblM.

TparoaoMeTpitqecK~ie i~y~lb-ps~z 141

B gaCTttOCTI4, ec~au 5~ 0)=0, 1, k =>0, j = 1, 2 Taxi, e, qTO pea

5 O), ~," cos 2n(kx+Ot) k=0

CXOanTC~ nOrTh Bcmay Ha [0, 1] X (--1) ~ ( j = l , 2), TO B I{a~lecTBe HCKOMOrO nya~-p~a , cynlecwBoBaHne KOTOpOro yTBep~JlaeTCII B Teope Me 1, MO>KIIO Blbi-

6paT~ pa~

,~ 1 rs(~ ) • o' k J ,~o "~" t k ,k COS 2n(kx +Ok).

OTMeTI4M, qTO eymeewBoBaune cxoa~meroc~ nOUTn Bcmay wpnrouoMevpnuec- ~oro p~aa ~ ~aaanuofi u3~aepuMo~ ~ no~zu ucmay ~ouewao~ qby~r~unu 6~Iao ao~a- 3auo M e u ~ m o B ~ [16], a cymecsBoBan~e 6-yn~Bepca~noro spnrouoMevpmtec- ~oro p~aa - - T a a i a a ~ o ~ [25].

)IoI(aaavea~cT~a TeopeM 1, 2, 3 IIpOgOdI~ITCa MeTO~OM, IIO3BOflRIOIIIIIM ycTa- I~OBrlTb cylllecTBoBarlrle 5 rI a-3qmBepca~Ibm, Ix p~ROB no nmpolmMy KoIaccy OpTO- roi~aJmm, IX (mlH 6a3rlci~,ix) C~CTeM qbyHxtmf~ (no~po6nee CM. ~ w 4).

w 2. BcnoMoraTeabn~ie yTBep~enna

3aec~, M~,~ Aoxaxl). B I(a~ecTBe OCHOBIm~ 06aaCTX~ o~pe~IeaeuH~ nepeMeHHoR x pacCMaTpuBaeTc~ oTpe30~ [0, 1]

YleMMa 1. Ilycmb 05YUmluuf,(x) U3A4epUAIbIe U IIf~(x)[]l~l 0 ~ n = l , 2, 3 . . . . . TozOa cyulecmeytom nobnoc.aec)oeame~bnocm~, {fk(x)},~x, Cynnl4un co(y) {yE[O,+~o)), bee noe.aec)oeamem, nocmu U3.,uepuMblX ~6yH~tfU~t {g,(x))~~ {q,(X)},~=x, yc)oeaemeopatoutue ycnoeuag:

1) A . ( x ) = g,,(x)+,1.(x),

2) z ~ m { r / , , ( x ) # 0 } < + ~ ~ n=J.

3) m{Ig,,(x)l > y} < o)O')y-*

4) c0(y)-,-0 1,I o)(y)y-140

5) f (o (y)y- ldy ~ 2. 1

f[OKaaaTeJH, CTBO. 1-[yCTb

�9 , (y ) = m{lfv(x)[ > y};

g,(x)q, (x) = 0 3.~n ecex xE[0, 1] u n => 1;

Onnecex y > O u n ~ l ;

npu y ~ + ~ ;

4 Analysis Mathematica

142 H.B. IIorocan

~CHO~ tlTO

~ ( o ) ~ , ~(y)r (y -~+~) ~ ?~(y)dy<=~. 0

17o TeopeMe Xeznn (CM. [30, cxp. 223]) nocze~lozaTea~no ~I6epe~ noalcnc- ~ r { ~ d = { e ~ } = { e ~ } = . . . = { e ~ , ~ } = . . . Ta~e, ,To ~p~ .~m6o~ t_->~ {~l(y)}~=a cxo~nTcg zcmay na oTpe3~e [0, 1]. O6paayeM ~ a r o n a a ~ n y m nocae- aO~aTeJH, ImCT~, {~,~}~=a -=t&(~)I~176 Jv=~" IIpeaeJi 3TOI~ IlOCJmao~aTesn, ImCTI~ - - ne- Bo3pacramlImZ na Iio~Iyoc~ [0, + ~ ) q b y m ~ r 7~(y). rio TeopeMe JIe6era

0 o OTCtO~a

(2.~)

H~ CJ~e~OBaTeJIBHO~

(2.2)

(v -~oo, I = I, 2, 3, . . .) .

f ~ e (y)dy < 0

lim yT(y) = O.

L~a oTpeaKa [0,2"] Harlem noMep k, (k,>k,_l, n = 2 , 3 , 4 . . . . ) H3 nocJm-- ~OBaTeJIBtIOCTIr {r~}7= 1 TaKO~, qTO

]~k,(Y)--T(Y)[<8-" npa y = 0 , 1 , 2 . . . . ,2".

Tenept, nocTponM co(y). Hoslox<nM

fy~P(y--1)+y -I, ecJm y--> 1;

co(y)=l t 2, eeJm 0 < = y < l .

Ha (2.2) Bl, ITeI~aeT, ~ITO w(y)+0 (y-* + o~). Ya~ KaK qbynxlmx co(y) y-a MOnO- romeo y6~,maloma, Ha m~Tep~aaax (0, 1), (1, + ~) ~i coo --0) =>co(l) ~co(1 +0) , TO co(y)y-~O IIpI~ y-,+o~.

]~a~ee, ~ c~J~y (2.t) ~MeeM

~ ? i f co(y)y-~dy = 7t(y-1)dy+ y-~dy <= 2. 1 1 1

OnpeReanM qbyHri~rm q. (x) n g. (x). Floaoz~rlM

g.(x) =A.(x) , x~{fA~ ~- 2"};

q.(x) =A.(x), x~{lA~ > 2"}.

B ap3wi~x To~IcaX ~r q,(x) H g,(x) ImaaramTc~ paBnl, IMn nymo. CnpaBeaJI~I- BHOCTI, IIpejIcTaB.rIeHI/R I) HelIocpe~cTBeBtlO Bt,ITeKaeT H3 olIpeae~lelm~ aTnx qby~x- ILm~. JIelxo ~laeT#, ~TO m {~, #0} <=2-", n =>1, owKyaa n B~,ncelmeT 2).

Tp~ro~oMexpaaecgue ~fy.,~b-p~b~ 143

3aMeTrrM a-enepb, ,tTO npH 1 = y 2

nt{lg,,(x)] > y} <~ m {Ig,(x)[ > [y]} < ~ ( [ y ] ) +

+ 8 - " < T ( y - 1 ) + y -z = co(y)y-~.

Ec:m ~ e 0 < y - < l , TO m{Ig,(x)] > y} --<= 1 < co(y)y-L

51eMMa 1 ~;oI(a3a~a.

3 a ~ e , I a n r r e 1. F a n o m I ~ v ~ 6J,~z~o yCTaHO~JIe~to (CM. [6, ~leMMa 1.2.4]),

'l~-O ec~m IIf.(x)lll<= 1, n >= 1, TO cymecT~yeT iioAc~cxeMa {f~ (x)},7=~, yAOB~IeTBop~IO- ma~t yc.rlO13n~r~ 1)---2) ~e~Ma,l 1 H Ta ta r , qTO nOCJIe)~oBaXe~bHOCTb qbynrlmfi

{g,,(x)}~=~ CXO~I~TC~ cna6o B L~[0, 1]. 3aueTrlM x a r x e , qTO N3 yc~IOgrtfi 3) H 5) :~eMMJ,[ 1 gblTeKaeT, qTO HocMe~o-

BaTe~a~,Hocr~, {g,,(x)}~=~ i,IMeew pal3HOCTelienrlO a6co2IrO'ri~o ~elipept,mut,~e HOpMbI,

T . e .

lira f I g. (x) l dx = 0 mEmO E

paBHOMepHo no n, rfO~TOMy rio TeopeMe o cna6ofi roMnaKxHocxH nocsIeaoBaTe:n,-

HOCTH 13 FIpocTpaHcTBe LI[0, 1] (CM. [5, CTp. 317]) nOCSle~o~3aTeslbnOCTb {g,(x)}n= 1 Bcer~a co~epX~T cJm6o cxo~nmytoc~ B La [0, I] nO~nOCJ~eAOBaTe~I, HOCTS

{gk,,(x)},7=~.

.YleMlvla 2. Hyemb ] [ f , , ( x ) ] l~ l , n = l , 2, 3, . . . . ToeOa eyu4ecmeytom noOnoe~e- 3oeame~buocmb {fk,(x)},,~ 1, nocaeOoeamenbuocmu no,w~tcumeJtbubtx mtceA {e,},~__l,

{ 6 , } Z 1 c

maKux, ~tmo naatcOaa noOcucme~ua {fkpn(X)}~=l, 1 <=Pl - i),

eOe Cymcquu h,(x) u ~,(x) yOoe:temeopnmm c:teOytoutu~t yc:toeua~t:

[h,(x)]<=ne, npueeex x6[O, 1] u m{~. (x)~O}-<6, , n = 1 , 2 , 3 , . . . .

] I o l i a 3 a x e J i l , cx13o. HyCTb nocae~oBaTem, HOCT~ qbynrllkIfi' {fk (X)}~=t, {g,(x)}~=l, {r/.(x)}~~ rI qbym~nst co(y) B3a6panl, I TaK, ~ITO 13~momt~iOTCSi ycJIom~a

1)--5) 3IeMM/,I 1. Mo:x~-Jro CqHTaTb, qWO

m { , t , (x ) ~ o} <_- , , -" , ,, = a, 2 , 3 , . . .

4*

144 H.B. Horocuu

Ta~ KaK ueBo3pacTammag ua [1, + ~) ~yHr~u~ co(y)y -1 unTerpHpyeMa~, TO

~ , o~(lO l -~ < + ~o 1=1

~ u JIIO6OrO ~ c ~ a e >0 . H o J m x ~ r

e ~ = 2 -~ npu r , _ - < n < r ~ + 1, v = > 0 ( r o = 1 ) 14

6, = e 2 1 n - l m(ne~) + n -~,

r ae IIOCJIe~IOBaTeJIBHOCTb MOHOTOHHO pacTyILIHX r v B~I6pa~a TalC, r

o)(12 -~) 1-1 < 4 -~, v = 1, 2, 3, . .... /= r v

JlerKo BH~eT~, ~TO ~,~0 H

n=J_ v=O l=r v n = l

~JI~ lIpoI, I3BoJIbnOI~ IIoc.rIeJIOBaTeOlbI-IOCTn {fkpn(X)}~= 1 IIOJIO)KI;I"IV[

h , ( x ) = ~ A v " ( x ) ' e c ~ lay (x)[ <= nen;

t 0, ec.rln l A y ( x ) [ >- ne,,,

I4

~. (x) = f % (x) - h . (x), n = 1, 2, 3, . . . .

5lcao, aTO h , ( x ) ~ , ( x ) = O npH Bcex xC[0, 1]. ~aaee , war raK

{~ . (x) # 0} -- { I g , . ( x ) + ~ , . ( x ) [ > n~,,} c {tg~.(x)[ > ,,~o}U {~ , . (x ) # 0}.

TO c o r Y l a c H o yCJ/OB~IIO 3) JIeMM]bI 1 r r M e e M

m{~.(x) # 0} <- m{[gv.(x)[ > n e , } + m { t l v . ( x ) ~ 0} < e, n l n - ~ ( n e , , ) - k - p n - z ~ a n.

JIeMMa ~ora3aua.

3 a M e ~ a u n e 2. OTMeTRM, qTO g yC2OB~X ~eMM~I 2 ~aU ~ 6 p a u u o ~ noacnc-

TeM~ {f~ (X)}~=~ cnpaaeaarma em~ oKenxa

Z r f IA ,,=~ Vn {i&v,00[>,,~,, } v. (x)[ ~/s d x -< + ,~,, 1 <= p~ -< p~ -< Pa "< . . . .

)Ie~c'r~n'resmno, r~3 npe)xc'raBoIeuriu 1) r, ycJioBI~ 3) aea,im,I 1 ~,rrerae'r , wro

f tA, (x)l. . d x <_- ~> Ig,. (x)l v~ d x + f [tlv.(X)[ ~/s d x -< {Ifkpn(X)l>nen} {[0p (x)l nen} {~tp (x)#0}

< ~.~,+ f o~(y=)y -~ d y + ( m {r/v.(x ) # 0}) v= Iln,,.(x)lll/*, .r

TpnroHoMelp~i~ecKHe nyJ]~,-p~a~,~ i45

n xaK ra t cy~tua nepBoro a Tpexbero caaraeM~,~x ne ~epeaocxo~uT ]/n 6,, (c~. ,RoKa3aTeJII:,CTBO 5IeMMbI 2), TO ,ROCTaTOqHO IIoKa3aTt,, qTO

f +=.

B~n~ly y6~,maHu~ qb3~Klmn m(t) t -~/~ Ha (0, +o~) mueeM:

, ,= a 1/ n ~ ,= ~ V n ,~.

<- - - Z co ( / 2 - ~) l - ~/s < - - 2 =- = rv<_/l<rv+ 1 l = n

~ 5 2 <_ 1 Z 2vl~- Z 1 1 1 -~/~ 2~/~ ~E' 7-7, o ( 1 2 - 0 -<_ 0,(12-,,) 1-~ < +oo. v=O t = r v t n = r v ~//'2J v=O l = r v

I43 3IeMMbi 2 HellOCpe~cTBeHHO BblTeKaeT

C~e,~cTBrte 1. Hycmt, Ilf,,(x)lh<=l, n=>l. Toec)a cyu4ecmayem noOnoc.ae3oea- meabnocmb {f~,,(x)}~=~ matcan, ~tmo c)xa t~aozt)oft noc~te3oeameabnocmu ao~pacmato- ulux no~epoe 1 <=pl <pz <Pa < . . .

lim n- l j~ (x) --- 0 n ~ ~ Pn

no,tmu actoc)y ua [0, 1].

Bo~ee c~a6oe yTaepa~enne co;~epac.uTc~ B pa6oTe [20] (CM. ~eMMa 1). HycTb 11[0= {U} MHoxem~o He~pep~mBo ;mqbqbepeHu~pyeM~ix Bo3paeTamumx

t~yHrRri~ u(y) (yC[0, +oo)) Ta~nx, qTO U(0)=0, u(y)<=y ~v~ acex y > 0 H

(2.3) ? u(y) y -2 dy = + co. 1

O6o3naqnM ~epe3 v=vu qbynKRnm o6paTttym K uEll 0. 5lcuo, ~ITO V ( Z )

(zE[0, + ~ ) ) Heapep~tano aaqbqbepemmpyeMa~ Bo3pacTammaz qbynram~, v(0)=0 v ( z ) ~ ~o ( z ~ ~). KpoMe TOrO, no,aara~ z = u ( y ) n nnTerpnpyn no ~aeT~M, icv~eeM

~ npon3Bo~bnoro a ~ 1

/ z[o * ,

~(~) ~ fib + f u(y) y-Z dY" 1 v(i)

~OaTOMy pacxo~;nMOCTb ~mTerpa:a (2.3) paBnocm:bna pacxo~I~4MOCTH unTerpaaa

(2.4) / dz v(z) - + oo.

146 H.B. l-lorocn~

J I e ~ M a 3. H y e m b U3~tepU~Wble ~ytIICIIUU f~(x), n = 1 , 2 , 3 . . . . matcue, umo

cyutecmaytom nocaeOoeamem, noemu ~fiynxuu~ {ut}~=xCll o u ~uo)*cecma {Et}7= ~ c mE, ~ l (I-~oo), 0 ~ nomop~x

sup [ u , ( t L ( x ) l ) d x < + ~ , ~: 1 , 2 , 3 . . . . . n ~ l Et

ToeOa cyu4ecmeymm noc~eOoeame~b~tocmo Heeo3pacmatoutux noctogtcumectb~tbtx

~4blCe~l

: --, n c fl~ : +oo tl n= l

u noc)nocaec)oaame.ah;qoemb {f~ (x)}~'=a mar, ue, ~tmo c)aa npou~aom, m, tx ttomepoa 1 <= ~a

~P~ < P~ <Pa <- . -

(2.5) i]irn= fl,,fi,, (x) = 0

n o , m y ectoOy , a [0, 1],

) IOKa3aTea],CTBO. He orpaI~n~n~aa 06mHOCTn, MOXHO CqnWaTl,, UTO MHO- XecTBa El pacmI~p~mmnecz (T. e. E~cEt+~, I~1) .

MyCTb V t ( / ~ ] ) - - ~byHKKg~[, o6paTHan I~ 0pyH~m~ U t. TorAa (CM. (2.4))

(2.6) ,,_-Z1 vt(n) = +~ l = 1, 2, 3, . . . .

l-lojl~3y~cb aeMMO_~ 2, noc.~e,aoBaTe.abHo Hafi,a6M no,acncTeMt, I {f,,) m {]),t-} {f~(,, ) m . . . m {A(,)} m . . . . noc~eAoBaTenm{OCTn noaoxnTeabm, m ,~cea {6~)}7= 1,

(~ ' l ~ l Tax,e , UTO np1~ nm6OM l ~ l Ann npola3Bo~,- t = 1 , 2 , 3 . . . . c Z 6,,- < ~ , t l = l

HO~ HO~FIOCJIe~OBaTe3]bHOCTH H o M e p o B ~ (0 {kp , j , ,= lc {k,, },,=1. crlpa~em~Bt, I O~eHKn

(2.7) m{x~E~; u~(Ifkp,(x)!) > n} < 6,(,'~, n = 1, 2, 3 . . . . .

B cHAy IIOAO)KHTe.~bHOCTH ~IAeHOB J1 t060~ IIOCfleAOBaTeJIbHOCTIt n = l

l~>], ~ yc:m~H~ (2.6) ~OXHO ~m6paT~ ~oMepa r~=0, rt+t >r~, l ~1 , ~ ~OTOp~•

Y

I > r~e a l : 1, a l = i=-~//t'~-t(r~)+2vi(ri+ 1) ' = 2 H

(2.8) Z ,~(o 2 1 = 2, ~ 4, - . < _ - t , 3, . . . .

t~ =l'l ,-~ 1

Tp~ronoMeTpa~eca~ae ~yns-ps~Ib~ 147

I-[yCTI3 {k,}~~ ~ t'~n/lr(n)}~'Jn=~ - - ~rlaroHanbHaa nocne,/~oBaTenbrlOCTl, / 4

a,} B . = m i n ' v ~ ) , e c a u r , < n ~ r , + ~ , 1_->1.

~oxa~eM, ~ITO no)IcrtcTeMa {f~ (x)},~=~ H ,~acna ft,, n @1, - - rmKOM~,~e.

l~ at ! "+1 ]~ Jler~o ~mlea~, ~rO at =<2 -t+~ (1~1), a uocJIeao~axem, uocr~ ltvt(n----~ln=,,+~lt=~

y6~,maex i~ nya lo MOaOTOUUO, n o 3 x o u y X~ fly,' 0 ( n - ~ ) . Ta~ xaK

2 k flu,, = m i n 1, v, (2 k) l = + ~'' TO tf ft, = + ~x~. k = 0 I = 1 rt<2 ~_rt+ 1 n = l

~JI~ IlpOrl3BoJIbrlO.~ no/InocJIejIoBaTenhrlOCTrI {Ap,(X)}~=l 1I][ HoMepa

mvleeM npr, rt <n<=r,+ ,, l>=s: s>=2

{x<e.; (.)l > l&(x) t > {x<e,; . , ( l&(x) t ) >

,noaToMy ZBH,ay (2.7) n (2.8)

Z m x<es ; J ~ . A , ( x ) t > < Z a(,~ < = v. 2s-1, l = S / l = r l + 1 1=$ ~r=r/,4-1

oTKyaa BNrexaer cnpaBe)xnnBoca~ cooxnolneun~ (2.5) i io~rn Bcm)Iy ~ia Muox~ecxwe

E,. TaK Ka~ mEs-~l (s -~o) , TO (2.5) cnpaBe)Iarmo uouxu ~cmay na [0, 1]. JIeMMa 3

ycTaitoBJleifa.

PaccMoTpmvt rIocae)IoBaTenbsocrn H3MepHMbLX HpOCTbIX, T.e. yaKgx, '~TO

oTpe30K [0, 1] npeacTaBa~eyc~t B BruIe cyMMN Koneuaoro ~aH cqeYHoro qncna

aenepeceKalomr~XC~ ~an0*eCTB, tm ~OTOp~x 3Ha~enrr~ (l~ynI~RI~ nocroannN, qbyuK-

R ~ {h~(x)},~= 1, yaoBneTBopamm~m ycnomrm (~):

(~) ,g.~n .am6oeo i>=2 ,unoacecmao {hi ( x ) r na.~nemcn ~unoa~cecmao~u noc- monncmea ~ynxqu~ h~(x), 1 <=j<i.

CrlcTeMl~I ~yuKl~Hfi {hi(x)}~=, TeCt-tO cB~3arlbI c MaprrmranaMu (onpeaene~rm

r~ C~OfiCT~a ~apT~uraaou c~., ~mnpm~ep, [18, raaBa 4]). IlycT~ % . = ~ ( h , , h~, ..., hN) a-aare6pa, n o p o * a e r m a x ~HOX~ecTsa~U nOCTO-

N

nHCT~a qbyH~m*R hi(x), h2(x ) . . . . . hN(x) ~ TN(X) = Z~ h,(x), N = 1, 2, 3, . . . . Ycno- = 1

~r~e (e) pa~nocmasHO woMy, ~Wo nprr nm6OM N_-->2 MHoX~eCTUO {hs (x )~0} llea~- ~OM npnuaJIaexvnT HeKOTOpOMy aro~ay u3 aN_,, T. e. TaKOMy MHO*eCTBy ~t3 0-~_~, KOTOpOe HeBO3MO3t~HO rlpeJICTaB~Tb B B~a~Ie o6~,ejInHeHi~ HenycTbIX Mtto~eCTB I13

~N- I �9

148 H.B. Horocnr~

OueBg:~HO, UTO {au}~=l - - MOHOTOH]:I]bI~ I(TIace a-a~lre6p (aNca~.+l, N ~ I ) r~dpymc~gr~ TN(x ) ~N-rr3Mep~Ma. B cay~ae, xor:Ia ~yH~nHH hi(x), i ~ 1 HnTerpr~-

1

f h,(x) dx=O, i=>2, nOCJIe:IoBaTeJIbHOCT;~ {TN(x), as, N=>I} cocra- o

BYI~teT MapTI~Hra~ (Bepo~tTI~OCTHag Mepa coBna:~aeT c Mepo~. Yle6era). C :Ipyro~ CTOpOHbI, ecJ/rr {~N}~=I - - IIpOH3BO~bHI,I'~ MOHOTOrlttI~I_~ I<Jlacc

gorre~1,ix a~re6p ~t {Tu(x)}~= 1 - - noc~e:~o~a~en~,nOCTr, rlpOCTblX qbyn~H~ TaKrlX, 'ITO {TN(X), ~ N , N=>] } ~tBJ/~eTC~I MapTHHFas TO (~yHIfIIHtf TN(X ) ~IorlycKalOT npe~cTa~enHe:

qN

r~(x)= Z h;(x), qo=O,q~=~,qN+~>qN, N>=~, i=qN_l§

] r~e CnCTeMa {h~ (x) }~%1 y~o~:~eT~op~eT yC~O~HW (e); h~ (x), i =>2 HMemT Hy~eB~m

(h" ' , ) c _ ~ cpe~qne n a ~ ~, h h N ~ 1. Hano~m~M (CM. [18, CTp. 144]), qTO ~eao3naaHa~ qbynr~Ha n(x), 0 < n ( x ) < ~ ,

onpe:~e,~ennaJ~ Ha ~enycwoM ~3MepHMoM HO)IMHO~I~ecTBe oTpe3Ka [0, 1], Ha3]bi- BaeTc~/ MOMeHTOM ocTat~oBKII OTHOCHTeJISHO raacca a~re6p {a~}~=x, e c ~ np~ arO6OM N ~ 1 MHo~ecTBO { n ( x ) = N } rlpHHa:ule>KrIT aare6pe aN.

)Iorax~eM c~e~lymmym aeMMy, ~ B ~ m m y m c ~ a n a ~ o r o ~ aeMM~ 2.1 pa6oTS~ F a m a ~ [t0].

J I e M ~ a 4. Hycm~ ~ynKguu hi(x ), i=1 , 2, 3 . . . . yOoaaemeopmom ycaoaum (~), a n(x) - - npou~eostbm, Kt ~tomeHm ocmaHoetcu ( x 6 E c [0, 11). ToeOa ~uoateno ebt6pam~, uuc~a 6~=0, 1, i ~ l mar, ~mo

~' 6iaih~(x) = ",[SN(x) npu 1 ~ U <- n(x); (2.9)

~=~ [S~(.,0(x ) npu N > n(x), N

r .r {ai}'~=~, N ~ I u x~E, eOe S u ( x ) - ~ a i h i ( x ) . B vacm,ocmu, Cyn~guz i=1

S,(x)(x) npeOemae:t~,~emca a auOe cy~tb t paOa

S.(~(x) = ~ 6,a~h~(x), f= l

cxoOauleeoc.~ ecmOy Ha E.

~ o ~ a 3 a r e . ~ b c ' r ~ o . HycT~, Fu, N = I , 2, 3, . . . , - - MHo:~eCTSO H~e~coB i > N , 2~.ng rOTOp~,IX { /h (X) r {n(x )=N} . IIoaoT~r~

[0, ec~H i ~ ) r ~ ;

l , B IIpOTI4BHOM cTIyrqae,

r~oKa~CM, qTO ql~c~la bi I~CI<OMMr

Tpnro~o~eTpx~ec~ne ~;yn~,-p~A~,~ 149

T a r raK ~M'HO)KeCTBO E npeJIcTaBJ~eTc~ g Bn~e cy~rumi Henepece~alom~xcz

~noX~eCTB {n(x)=N}, N = I , 2, 3 . . . . , TO ~OCTaTO~nO Ao~aaaT~, cnpa~e~am~ocTb

pa~encx~a (2.9), ecJIn x6{n(x)=No} ~ qbnrcHpoBanlmro N~=>-I. B y ~ e ~ paa-

n a ~ a ~ a~a cay~aa N<-_No r~ N > N o. HycT~, N<=No. P a c c M o T p ~ HoMepa i<=N T a ~ e , qTO i ( U F~. I I p e a n o -

| < N

, a o x ~ , ~TO i~FN~, r~e N~<N. Ta~ r a~

{hi(x) # 0 } c {n(x) = N~}, TO hi(x) = 0 t lpn x({n(x) = No}.

OTCtoAa BBITegaeT, NTO

N

~ 6ia~hi(x) = S s ( x ) - ~ aih~(x) = SN(x). i = 1 i ~ N

i~ U F~ l<N

EcJm me N > N o, TO B CH.ay CBO!~CTBa (Ct)-H tTN0-I~3MepI4MOCTI~ Mno:~eCTBa

{n(x)=No}, nag 3HanenH~ i, No<i<=N nan {lh(x)r n:i~ o6a

a~vr Mao~ecTBa ne nepece~amTcs, r logToMy 6~h,(x)=O B 060~X cayaazx . C~e-

~OBaTeJIbHO, N No

6 a , h,(x) = X 6 a , h,(x) = S..~(x). i = I i = l

Ae.,vrMa 4 AoKa3a~a.

N 3 a M e q a ~ n e 3. IlyCTb P ( x ) = . ~ ail~i(x) (N>=n ~ 1 ) - - npOH3BO~,n~,~ r toa~-

~o~vt n o c ~ c r e M e { h i ( x ) } ~ = 1 . BBeAe~ c~e)IymmHe o6o3na~ie~H~:

(2.10) q q

Re(x)= m a x ~_,aihi(x), r p ( x ) = min Xa~hi(x). n ~ q ~ N i = n - - - - n ~ q ~ N i ' ~ n

rIyCTf, F~, F2, ..., F , - - npon3BOm, n ~ e Henepecerarotur~xca Mnoz~ecxBa Tai,:r~e,

aXO 0 F k = [ 0, 1] v~ nprI ~ro6oM i, n<=i-<_-N Mao~eCTBO {h~(x)r ~e~nr:oM k = l

r~pm~a~.rle~aT O~ttOMy It3 MIto~KeCTB F k. PaccMoTpnav~ npoI~3~o~at,aym npocTym

qbynr~um f(x), npnHn~amtuyIo rIOCTO~Httoe 3ttaqeHI4e Ha Mtto~ecTBe Fg,

k = 1, 2, . . . , s. I !pe~noaoa~mu, qTO

(2.11) re{Re(x) >= f(x) >= rp(x/} > 1- -e l max sup !aihi(x)l -~ e~, n ~ i ~ = N x~ [0,11

rAe elC(0, 1) l~ % > 0 - - n m 6 ~ i e u~c:m.

I lora~reM, ~tTO npH ~THX yCJtOBH~X M O ~ H O OHpeAeJII~Tb napy no~HnoMoB N

(I) (~)_ O,(x)= Z a7 ~a,h~(x), l = 1, 2, ~ae ~ c ~ q'~ ~ p ~ M a m ~ ~.a~e~H. 0, 1 ~ 6 6 --0,

150 H . B . lqoroc~n

i=n, n + l . . . . . N, ~rl~l KOTOpBIX cnpaBeaauB~ caealylomue oIIeHK~I:

(2.12) O~(x) Qz(x) = 0 (x~[o, 1]);

(2.13) R~, (x) < f + (x) + e.,., re, (x) > f - (x)-- e=,

)Ia~ ~cex x~[0, 1], r;ae f+(x) tteoTp~IIIaTeall, raa~, a f - ( x ) HelioolorgitTea~,Has ~lacrl,

*yI~rII.H f(x) ( f - ( x ) = l [f(x) +_ lf(x),]);

(2.1.4) m{[f(x)-Q~(x)-Q=(x)[ >= e~} < ea.

[lycw~ E0, E~, E~ - - Mrm,ecT~a, na r o r o p ~ x COO+~TCTBeHno f = 0 , f>O, f < 0 . K a * a o e ~3 MHOX<eCTB {hi(x ) #0} npnHaa~ex<nv oanoMy u TOglbKO OaUOMy MnO- Z<eCTBy /7l, 1=0, 1, 2. Hoaox~rnv~

P,(x) = x" /~ a~hi(x), i~rt

rAe F, = {i; n ~ i _--__- N, {h,(x) # 0} c Et};

q

S ~ ( x ) = Z aih~(x), S~(~ Z a, hi(x), i=n n~i~q

is

q = n, n + l , ..., N;

A~ = {xEE1; Re~(x) ~ f ( x ) } , Ae = {xEE2; re2(x) ~ f ( x ) } ,

TOrlla MnoxecTe~a A1, Ae, a(h,, h,,+l .. . . , hN) - - rI3MepivlMble. O~IeBHjIHO, RTO S~l)(x)=.Sq(x) HpPI xEE z r~ S(qt)(x)=O rlpH x(~Et, rmaTOMy

ffMeeT ~eCTO BKJIIOHeHHe

(2.15) {Re(x) >= f(x) >= re(x)} c= EokJA~UA2.

Onpeale~rM tleamBHam~J,m cl~ynrt~rm nt(x) (xE[0, 1], 1=1 ,2 ) c.~e;IyromnM o6pa3o~:

rain{q; n ~ q <=N, ( - l ) 'Sq (0 (x ) ~_ (-1) t f (x)})!JI~ xEAI; nt(x) = N ~sI~ xE[0, 1 ] \A t .

~ICHO, HTO nl(x ) ItB~ISIeTC~ MoMeHTOM OCTaHOBKH. [ I o omM~re 4 cynlecTByeT no:IrmOM Qt (x )= ~fi~~ ) (6}~ 1) Tailor, ~ITO npH nm6ora q, n<=q<=N,

iC F t

ts~2 (x), e ~ H . _~ q < n,(x); X a}'~ ~,,h,(x) = ~=

,,~i~-q l Ql(d), eca~ nt(x ) -~ q ~ N. i{F~

3aMerma Tenepl,, HTO IIO.rlHHOMI, I Ql(X), Qz(x) HCROMble (~t)=O,i~F1) . )~e~crsrr-

Tern, Ira, Rq,(x)=re,(x)=Re,(x)=r1,(x)=O np~ x~El oTry~a, z nacrrmcTm c~eayeT (2.12). OI~enr~r (2.13) ~enoepe~ez~e~a~o s~,rreraror n3 onpe~Ie~eHr~ Ql(X), Q~,(x). )!a~ee, ec~arr x~EoUA~UA~,, TO

f + ( x ) ~ Ql(X) < f + ( x ) + ~ , ~ f - ( x ) - ~ . ~ < Q~(x) <= f - ( x ) ,

Tp~ronoMeTpaqecKne Hyn~-pa~9i 151

n a n cyMMupyz, n~ieeM [f(x)-Ql(x)-Q,,(x)l<e~. 1-[O3TOMy conocTaBo~a (2.15)

c rmpso~ n3 o~4eHot~ (2.11), r l o n y , ~ (2.14). ,/IoI<axeM TaKxe o~Hy .rleMMy 0 CXO/~tlI'VIOCTII pa~loB llO CI4CTeMaM {hi(x)}T= x.

~)Ta oaeNMa MOXeT 6tJITb noslyqei~a •aK cJIe/IcTBHe TeopeMI, I ~ y 6 a o CXO)IHMOCTH

Mapwn~raao~ (CM. [4, TeopeMa 2.1] unn [3, caejicTB~e 2]). C ue~t,m nonaow~t H3aO- xeH~a rlpnBe)I6~ ee AoKaaaTesI~CXBO.

J I eMMa 5. Hycmb ~6ynmtuu h~(x), i = 1 , 2, 3 . . . . . yOoeaemeopmom ycaoauto (~), hi(x)-~O ( i~o) paeno.~epno lm [0, 1] u

1

i = 1 0 ~

~a~tee, nycmb ttoJ~iepa Nk~, (k-~oo) ma~ue, ,tmo

(2.17) l im sup sup ~ hi(x)<= 0 k ~ m ~ N k i = N k

na netcomopo~ ,r E c [0, 1] no,~,,o.~teume:~t, noft ~wpbt. ToeOa pnO

(2.18) Z h,(x) i=1

cxoOumcn noqmu ectoOy tta m~toowecmee E.

~oi~a~aTesI~,CT~O. ]IocTaTOUnO no~a~aT~,, ~ro nOrTh ~cm~y na M u o , e c v e e E cnpaBeanvIUO pa~enCTBO

(2.19) lira inf ~ hi(x) = O, k ~ m>=Nk i=Nk

UTO CO~vleCTHO C (2.17) oana~meT CXO;In~OCTb p ~ a (2.18). Iloo~3y~c~, uepaBeuc+- ~ o ~ (2.17), ~n~eeM

l i m s u p inf ~ h ~ ( x ) ~ l i m s u p [ l i m i n f inf h i ( x ) + s u p h~(x) <= k ~ m ~ N k i=Nk ~ p ~ m~Np i=Np m_~-~_2~" k i r

n~

<= lira inf inf ~ hi(x). p ~ m ~ N p i=Np

C~le~o~aTe~n, ao, npe~es~ B (2.19) cymeci~ye+, H Moxe+ 6J,IT~, ~al< i<o~eam,~M, TaK It ~eCKOHeqHt,IM.

1-[peJIIIOJIO>KrlM IlpOTHBHOe, T. e. qTO :3TOT I lpe~eJ I OTJIHqeH OT HyJ~f Ha HeKO-

TOpOM IIOJIMHO>KeCTBe MHOXZeCTBa E rio.rlO~KrlTe.qbHOft Mep~,I. 1-[O T e O p e M e Ero- p o B a o p a B H o M e p H o ~ CXOJIHMOCT~I MO~KHO Ha~TH qrlC2~O ~ 7>-0, rIOJ1]V~HO)KeCTBO

m 0

F ~ E c m E > 0 rt nozm~oM P ( x ) = ~ th(x) (mo>=N~o) wayne, ,n'o Re(x)<z i = Nko

152 H. 13. Fiorocm~

6 H r p ( x ) < - - 3 npn Bcex x6F, r~Ie e < - - m F -- i~eKoTopoe IlOaOXrlTeal, Hoe uncsto

4 (onpe~e~Ierlrle dpynrlm~ Re(x), re(x) cM. qbopMyJIy (2.10)). YlcHo, ~To Ilpri 3TOM HoMep Nko MOXlm ~Sl~I6paT~, FlaCTO~II~KO 60JII~IUHM, ~ITO B CI4JIy paBHOMeprlO~ CXO- ,,.~I,IMOCTH K rly.rlm {hi(x)}~=~ H (2.16)

1

(2 .20) s u p s u p ih,(x)t + ~ / h,(x)dx < e. i~_ Nko x ~ [0,1] i = Nko

BBe,/leM MIto~(ecTBa:

a = {X,(x) _-> ~}, . = ( - a <= r, (x) <= e ~ (x) < ~},

c = { g , ( x ) < ~} ~ { ,> (x ) < - 6}.

M ~ o x e c T a a A, B , C - - H e r l e p e c e r a l o m ~ I e c n , a(hNk., hNko+~ . . . . , h,,, 0)

Ml~Ie, AUBUC-:[O, 1] r~ F c C . PaccMoTpnM c:Ie~Iyromrl~ MoMenT ocTaHoaI~n n(x) (x([0 , 1])

min { n;

n (x) = ]min {,

J [ e F K O BH,ReTb, t lTO

.(x) hi(x) < 2e

i=Nko

FIO3TOMy

i = Nka

#no, ec.lli~ xE B;

N~o _<- ,, <= , .o, 2 ' h,(~) < - ~ } , e ~ ~ C . i=Nko

n(x) a . ~ Bcex x ~ [ O , l ] H zS h~(x) < - 6

i=Nko

- - H3MepH-

xEF.

/ [ X h~(x)] dx < ---~mF. i=Nko

n(x) C ~tpyro~ cTopouu, no ~eMMe4 qbyrmlm~ ~ hi(x ) npeAcTaBJ~eTc~ B BHAe

i= Nko mo

CyMMI~ ~ 6~hi(x) (3i=0, 1; Nr<o<=i<=mo), rr BcJIe~CT~re oI~cI~rr~ (2.20) rtlvrceM: i= Nko

n(x) m o 1

,,,F- IE z h,(x)l <,x Z ! ,,,(x)dx 2 o i=Nk o o

~ e F o H e Mo~KeT 6]~ITb.

I~oJlyqeHHOe lIpOT~tBopeq~te g 3aBeptuaeT ~OKa3aTe.abCTBO gleMMbI. HaM rloHa,~o6I:fTC~I c~e,z!ytoitta~ JIeMMa O r~epecTaHoBKax IIOJIIoII-IOMOB rio Cl/lC-

r e M a M {h,(x)}~'= 1 .

Tp~roHoMeTpn~ecKne Hynb-pn~b~ 153

5 I e M ~ a 6. Hycmb 96yumluu hi (x), i= 1, 2, 3, ... yOoa~emeopamm yc/loeuro (~z). N

7= N ToeOa 3~ta ~m6oeo no:nmo~v~a ~ dihi(x ) cyulecmeyem nepecmaHoer, a ZN= ~({d~}~= 0 i = l

matcaa, ~tmo n N

(2.21) ~,,~_Nmax [~=~,d, hi(x)l ~[~= dihi(x)[+ max = ~_,<=~'

O~a ecex x~[O, 1].

~o i~a3aTe~i i , CTBO. IlyCTb b~, bz . . . . . b N ( N > 2 ) npoa3~om, n~,m ~r~c~a r~

S , = ~ b j , n = 1 , 2 . . . . , N (So=bN+~=O). By)~eM ro~opr~T~,, ~TO ~ncaa by,

j = 1, 2 . . . . , N y~OBaeTBOpa~OT ycaoBmo (m), ec~r~ cymecxBymT noMepa q, 1 <=q<=N Tagr~e, ~TO S,b,+~<=O npn O<=n<q ~ S.b.+t->_-O np~ q<=n<=N. M~,~ CanTaeM N > 2 , TaK ~aK Aa~t 3roS~,~x ~Byx ~ncest yc~ozrm (co) y~o~3eTBopJ~eTcz, qepe3 G, 6y~eM 0 6 0 3 n a q a T b ~an~em, mrifi r~3 Tarnx HoMepoB q.

C a e ~ y m m n e C~O~CTBa ~ncea bj, 1 <-j<=J~, c (co)-C~O~CTBO~ 3ergo npo~e- p~IOTC~t.

a) q n c ~ a b~, b~ . . . . . 12i, 0, bj+~ . . . . . bu+ ~ npn am6OM 1 <=j<=N TaK)Ke y~OB- aeTBopamT yc~o~mo (co).

6) ) Ia~ ~ a ~ o r o a, ~nc~a bx, b~ . . . . . bq,o, a, b%,+a . . . . , bu+~ o6naaaroT (co)- CBO~CTBOM.

B) max JS~I ~ max Ib,].

Ta~ KaK qHc~a Sqo, bj, j > q o H S N Bce HeOTp~aTeJabHble HJ~H Bce Heaoao- ;K~ITeJ/bHble, TO H3 B) B/~ITeKaeT Ta~>xe

m a x ]S.I ~ max [S,,[+[ >~o bJI ~ l ~ n ~ N l <=n~qo~ J qr

max [b.l+lS~-Sqo~l ~ max [b , , i+ ISG l ~--n~qr I ~_n~N

OTClOAa y~e B/,ITeKaeT cnpaBe~rtBOCTb HepaBeHCTBa (2.21), ec~H MbI rmtca-

>xeM cymecT~oBaR~e nepecTanoBKri ~N=~N({d~}~=I), )~a~ KOTOpO~ qbynK~rm

d,~(oh,~,(o(x), l~i<=N, B xaar To~Ke oTpeaKa [0, 1] y~oBaeTBOpZmT ycao- mIro (09) (npe~noaoraeTca , ~TO N=>3; yTBep~j~emie ~eMM~Z ~:Lq N = 2 o~e- B~!HO).

T T N--1 3aMeTrlM, ~ITO eC.IIIJI llepecTaHoBKa N - l : N-l({d/}i=l ) H3BecTHa, TO BBrI~y nOCTOmmOCTrI dpyaKmrfi hi(x), 1 <=i<N, na ~moz<ecwBe {hN(X ) ~0} , ~mt Ha6opa

~nce~ {d,~,_l(oh,~_l(o(x)}~_~l, y~o~aeTBopam1Jmx (co)-yc~oBnm, cymeCTByeT e~a-

154 H.B. Iloroc~m

vl l~ HOMep qo,(X)=G~ Ilprr x~{h:q(x)#O}. 5ICHO, ~rro Tor~Ia B c t ~ y C~O~CTB a) ~r 6), ~rrc~la

d ~ . _ o ) h . ~ _ . ( # x ) . . . . . d..._.(%~h~_~(,~)(x).

yAot~aeTBopzmr (o)-ycJIom~IO ~a~ zaz x~ {hu(x) #0}, Tar: rI ~ z x~ {hN(x ) =0}. C~e~o~aTea~no, e c z , z~( i )=z~_~( i ) nprI i<=qo, z u ( q ~ , + l ) = N rI z u ( i ) = =rN-~( i - -1) npn q ~ + l <i<=N, TO ~epecTano~r:a ru - - nc~o~ax.

TaK Kar ~ i ~ IlpOrI3BOm~rmlX ,aByx qbynI~ttI, I~ yc.llOBrle (co) y,ao~aeT~aowtewc~ , TO name yTBepxAeH~re cnpage,~.llrlBO ~-iz N = 3 , a C.lle,~oBaTe.libHo, ~i ,~.1i~ Bcex N ~ 3 .

JIeMMa 6 yCTaHOg.lleHa. B ~a.a~,i~efimeM y~o6Ho IIO.Vtb3OgaT~Cg HI, IXelIpI4Be~HHI:,IMH otlpe~Ie.lleI-mmvm.

O n p e A e ~ e H ~ e 3. Pa3 (1.4) Ha3b~eaemcst 7-ynuaepca~btr ea~u O~a ~o6o~ us.~elm~Uoft u no,ram ec~oOy ~ot~e,mo~ ~ytt~guu F(x), cyutecmayem noc~te3oeame~tt,-

nocmb s~tatcoe 7,=_+ 1, n = l , 2, 3 ... , O~a Komopofi paO ~ n f ~ ( x ) cxo3umca

noumu ecmOy na [0, 1] ~ F(x).

Orlpe~eJIeHI~e 4. Yc.~oeumca na3bleamb pa3 (1.4) ao-ymteepca~t,m,t~t, ec:m

npouseo:tbnbdt pa3 2 L ( x ) maKo(t, ,,,no l i m f , ( x ) = 0 no~tmu ecmOy u fk,(x)----

= ~ ( x ) , n => 1, 0d.~ ne~omopblx no~tepoa 1 = k i < k z < k z < . . . o6~ac)aem ceo~tcmeoa: Oia ~im6o~ us~tepumo~ gSyt,K~uu F(x) (He ucKAm~teno, ~tmo F ( x ) = ++_ co na ano-

~lcecmeax no~w~lcumeJtbno(t ~tepbt) .uo~mto max, nepecmaeumb ~t.~em,t paOa 2 f , (x),

t~mo6t, t enoch no~ty~tenm, tft paO cxoc)uxca no*tmu ectoc)y x F(x) u npu 9room nop,qOox c.aeOoeauua 95ynm4u~t f~(x), k , -<k <k,,+ z, n ~ 1 coxpana~ca.

~.~i ~OKa3aTe2IbCTBa ae~fM~I 7 HaM no~a~o6rlTOI TeopeMa A (AOKa3aTeJIhCTBO CM. HanprrMep, B o63opHo~ pa6oTe [6, CTp. 18--23]).

U

(2.22)

T e o p e M a A. Hycmb u3Mepu~uble 96yttm4uu ~o,(x), n = 0 , 1, 2 . . . . matcue, ~tmo

sup I[q,.(x)/h < + o0, q,.(x) -~ o (n ~oo) cza6o e G[0, 1]

liminf f Iq',,(~)l d x > o E

3.az .am6oeo Mno~cecmea E c [0, 1] no.aoatvumeJ~b~to~t .~epbt. ToeOa cyutecmayem noOeucme.~a {q~,,(x)},,~o matcaa, ~tmo c).aa ato6o(t noc.ae-

Tpuro~o~x,~eTp~ecK~e a y n t , - p ~ , ~ 155

i)oaame.,zbnocmu quce.a {0~,}L0 c 2 %--~-- + ~ cnpaaet)~uebt coomtwutenua r~=O

Ig N (2.23) l imsup . ~ e ~ t p , . ( x ) = + ~ u l i m i n f ~ %, q),,, (x) . . . .

N ~ n=0 N ~ n=0

no,onu ecmOy ua [0, II.

~OKa~KeM creM~y.

YleMMa 7. Hycmb ~fiynmtuu q~.(x), n=O, 1, 2 . . . . yOoe:zemeopamm npeOno:to- atceuuam meope~ubt A u A,, ~(x) - - xapa~r OqyHmlua ut~mepeaaa A,, i =

" { j ~ l - ' ' 2 , . ) j = l , 2 , 2". ToeOa cyufecmeyem noOcucme:aa {r (x)}~=o, O ~

Komopo(t npouseost~ttbt~ p.~O ettc)a

2kpn

(2.24) Z X [~,,A% ,~(x)~o% (x)+q~,,+2(x)], 0 ~Po < P~ "< Pz < ..., n=0 j = l

ec)e eo=0, e,,+ ~ = e . + 2~v., n >=O, aezaemca oOnoepemetato 6 u ao-y~ueepca~nb~r nan moa~Ko ~6ynnyuu q,(x), i>=l yOoexemeopmom yeaoeu~o

(2.25) Z Ith(X)l < + i=1,

no,tmu ecmOy na [0, 1], a uucxa ~., n>=O, - - coom~tozueHua~u

(2.26) [e,I <= ( n + l ) -~/~, n = 0, 1, 2 . . . . u

~ o K a a a T e n l , CTBO. B~,i6ep6M ~I~ nm6oro qbyn~arlIO t,,(x) TaKylO, nTO

(2.27) I I%(x)-- t,(x)i]~ < 2-" .

~ . ~ = + ~ . ~=0

n = 0 , 1, 2 . . . . cwyneH~aTym

He orpaHH~rmaa 06IILHOCTI4, MOXHO CqHTaTb, qTO ~rlS~ HeKOTOpOFo )IOCTaTOqHO 6oal, moro HOMepa V n > n dpyHKILHH ln(X ) IIpHHHMaeT IIOCTO~rHBIe 3Ha~IeHH~I Ha Kax)IOM g3 ImTep~a~IoH A~,j , j = l , 2 , 3 . . . . . 2 ~-, a B KOHI~aX HriTepBa.rIOB A,~,,j t , (x)=O.

Tar: KaK sup IItn(X)H2 < + cxz (CM. TeopeMy A I~ (2.27)), TO corJIaCHO JIeMMe 2 O<=n<~

H 3aMe'JaHmo 2 cylIIeCTByeT no~toc~e~oBaTem.HOCTt. {t~ (x)}~= 0 I~ nocJ'Ie~IoBaTesII,- HOCTb ~mcelI s,,~,O (n~oo) TaI~t~e, qTO ~YI$L IlpoH3BOJlbHI~IX HoMepoH O ~ p o (n+l) ~} Pn

156 H.B. lqloroc~

KpoMe T01-O, 6y~eM Cqr~TaTb, qTO HoMepa r,, n >=0, B crI:iy Teopel~mi A m,t- 6pam,i Tar, ~ITO ~SI~ IIO~C~rCTeM~,I {9,, (X)}~= 0 VI ~rO60~ noc:Ie)~oBaTe~r~imCTrI ~mcea

{~,,}n=0 C Z ~ = + ~ BblnO~rI~ttOTC~i COOTHOmeHI,Ig (2.23). 12=1

Onpe~eJirrM pexyppenTHO noMepa m,: m0=r0, m,+l=r~m, s = 0 , 1, 2 . . . . . O~e- Bn~rlO, ~ITO m~+l >ms , msC {r,}H=0 ~ qbyrlim~ix tin(x) nprlrmMae7 rlOCTO~mn~,Ie 3~ia- ~ienrlX Im imTepBaaax Am +,,i, 1 ~j<=2"'+', i>=1 (s>=O).

Ha (2.27) ~SlTeraeT, ~ITO t,,(x)~O caa6o B L2[0, 1], no31oMy Mo)KrlO rla~Tn uo;mocaeaosa~em.HOCT~. HoMepos {S,},7=0 (s0=0) TaK~rx, ~ITO

2msn+l

(2.29) ~=~ f tm,~ (x)dx < 2 - "

= amsn+l, ] ~ i x ~cex i=>l, n@0.

~ o r a x e M , ~TO ~oacr~cTeMa {~o~ (x)}~=0, rae { k , } - {m,,,+~ } - - rmxoMaz. 3aqbrlrcrlpyeM nocaeao~aTenmlOCT~ Bo3pacTaromrlx tIoMepOB {p.}~~ rI nocae-

,~OBaTe~II~IOCT~, ~IrlCe~ {~,},~=0, y~o~sreT~opx~omrlX (2.26).

IlycTt, /0=1, / , ,+a=/ ,+2%' .+x, n = 0 , 1 , 2 . . . . . I l o a o x ~ M npri f,<--i<l.+x (n

lgn Z~mSpn+ l,i--ln+ l (X ) tkpn(X), eCJI//~ XC {t~pn(X ) ~ (n + 1)/~n}; h~(x)

I 0, ec2m x~{t~p,(x) > ( n + l ) e , } .

Y r a x e ~ t~eKoTopl~ie CZOfiCTSa qbyHImri~ h~(x), i=>1, xovop~,ie HaM IIOHa- ~06aTC~ B Jlaal~nefimeM:

a) ~IaI~ 2iio6oro #=>0 cy~ecz~yeT HOMep N u TaKO~, ~TO llp~I 0~tO60M i>=N u MHOXeCTBO {hi (x )#0} LIealVlKOM Ilpgna~0IeXHT O~IFtoMy ~ TOJII, I~O O~I~IOMy ~IHTCp- ~aaly Au, j, 1--<-j<=2u;

6) hf(x)-+O (i-+oo) paBHOMepito Ha [0, 1]; B) CI~CTeMa {h~(x)}~~ O6OlaalaeT C:aO~CTBOM (~);

1

lanAm~p +vj(x) 9~,flx)-ht.+j_ffx)l < + =,

r)

o o

a) 2 2 n=O l ~ _ j ~ l n + l - - I n

llOqT/r Bcro~y Ha [0, 1];

/V

e) N

l i ~ s u p Z h , ( x ) = + ~ ~ l i m i n f Zh~(x) = - ~ i=1 N ~ i=1

IIOqTH" Bcro~y.

r a) H 6) Henocpe~cTBeHHO BblTeKalOT H3 oIIpe~e~CHHH qbyH~Z-

tm~ hi(x). YIyHKT ~) m,ITeKaeT rI3 Toro, qTO qbyi~KmI~ t~p(x) nprI ~rO6OM n_-->_(~

TparoaoMcTpa~ec~ae ay,-a,-pa~l~,~ 157

nptmHMaeT IIOCTO~ItII-It,Ie 3Haqerlri~t Ha rmTepBaJlaX Amsp.§ l<=j~_2msp,+l +1,

aeaIe~IcT~He ~iero MHo~eerBo {t~%(x)<=-(n+l)e,} 3a tmralo~IemteM roae,J~toro �9 ~cola aaaorImm pamtona~tbrmix To~IeK, ~tB.rl~leTca CyMMO~ 14eKOTOpt,IX raKrlX rI14Tep-

B a J I O B .

~ a a e e , y~IriT~,maa cooTaomea~te (2.29), riMeer~

1

In-~-_-i</n + 1 1-<_-j~.ln+l--ln Amspn+l,j

<= l~,,I f Itkp (X)ldx+[~,I Z [ f t% (x)dx < dmspn+l, J {t 2pn(x) > (n + 1) rn} 1 ~ j ~ I n + 1 -- 1. .

< ~ f [te~(X)i d x + 2 - " , ]/n + 1 {t~ (x)>(n+l )~n}

Pa

�9 BMecTe c (2.28) o3rla~IaeT clipaae;~mmOCTb r). ~0I~t ~oKa3aTe~Ii, CTBa ~I) 3aMeTrlM, qTO

1

~o ~ , f Io~nAmspa+l'J(X)~~ t= l<=J<--n+l--.O

~-,,Z[l~,,lil%o(x)-%,(x)h+t~ot f t%o(x)ld~ l < +~. (t~. (x)>(,+l)e.}

Pn

1Io TeopeMe .lIemi OTCrO~Ia B1,ITeaaeT ~I). HaroHeI~, nyl~rT e) caeayeT I43 ~), paBeucTBa

N ln+l--I n N Z Z ~a%o+~,Ax)%~ = Z~%,,(x),

n=0 j = l n=0

~(OTOpOe crlpaBe~armo ;Iaa Bcex ~Borlqao uppatmorm~ibm, Ix To,~er oTpe3Ka [0, 1],

r~ H3 COOTaOmeI~14~ (2.23). ,~alsI )IoKa3aTeYlbCTBa JIeMMbI 3aMeTrlM, vlTO (~ /a a0-yrirmepcam, rmCTa p z a a

(2.24) B~,ITeI(amT COOTBeTCTBermo ~43 5 r~ ao-yrm~epcam, nOCTH p ~ I a (c~. 3a~r

Hrm 4 noc:~e ~Io~a3aTem, cTga ~IeMMbI)

ln+ t--I n ( 2 . 3 0 ) Z Z [~n/Imspn+l,J(X) ~Okpn(X)'~-~l.+J-l(X)]

n = 0 j = l

a a a rtport3Boa~,rmtx qbyu~t~a~ */i (x), i => t, y a o a a e r a o p a m ~ r r x yc:Iounm (2.25). B c143y CBOfiCTBa ,a) aOCTaTOqlm 14oRa3aTb, qTO p~I)I

(2.31) ~ [h, (x) + th (x)] i = 1


58 H.B. llorocn~

~ I e T C a 6 rl a0-yn~mepcaab~f~ea, KaK~4Mn 6~,i tin 61,i~i~ ~3MeprlMI, te ~yHKI~HrI qi(X), i =>1, y~osaexBopammHe (2.25).

) !o~axeM, fi-yHnBepcaabaOCT~ p a a a (2.31). HyCTb F(x) - - liporr3Bo31brraa n z ~ e p i ~ a ~ n0~IIt~ BCm)Iy Konemm~t qbyn~R~a.

~ a n noaynegrr~ cxoaameroc~ nOaTn ~croay K F(x) ~loapn~a pa)Ia (2.31) nocTponM nocaeaoBaTea~nOCT~ nOa~HOMO~ Q~(x), k = l , 2, 3 . . . . , no cr~cTeMe {h,(x)}7=~, y~oBaeTBo!onromux c a e a y m K l ~ yGrlOBH~M:

N~ts N~.s

l) Q ~ , _ l ( x ) = E 6[h,(x), Q ~ , ( x ) = Z 3~'h,(x), i=N~s_ 1 i = N ~ s - i

# t l! N~,~'~, (k~oo) , r ae ~Ii~cJta 6~, ,5, nprmrrMamT 3rra,mnnx 0, 1 IlpttqSM a i �9 a i = 0 a3I.~

~cex i = 1 , 2, 3 . . . . . 2) Q=~_dx)Q~,(x)=O aa~ scex x~[0, 1] u

r;~e

m {]F,_ , (x) -Q~,_l (x) -Q~,(x) 1 >= 2-*} < 2 -*,

N~.s F,(x) = F(x) - Z (6; + 6;) [h,(x) +,l~ (x)],

i=1 s = 1, 2, 3 . . . . (Fo(x) -~ F(x)).

3) m({Re, ,_ l (x) => F ~ + l ( x ) + 2 - s } U {vo~(x) <= F Z _ t ( x ) - 2 - ~ } ) - < 2 -5-1, s => 1,

+ l r ze F [ (x) = ~ - [ G ( x ) +--]Fs(x)]] - - COOTBeTCTBenHO l~eoTpn~aTeabHa~: n a n He~oao-

x n T e n m m a ~aCT~ ~byHKiam~ F~(x). HoamaoMbt Qk(X) rmcTporrM pe~yppenTHJ, IM 0 6 p a 3 o i , npn aTOM Q~s_~(x),

Q2~(x) 6y~eM CTpOHTI, CoBMecTno. HyCTb #1=>1 - - rmMep, 7 ~ KOTOpOro cymecTByeT qbynK~r~z g~(x), paBnaa

glOCTO$[HHOH Ha m~Tep~a~ax A t q , j , j ~ 2 a, TaKO~I, qTO

(2.32) m{[F(x ) -g l (x ) I >= 1/4} < 1/4.

Ha~aeM N0->__l TaKO/~, HTO ,/~JI~ 3HaqeHrll~ i>=No Mno.ecwBo {hf , (x )#0} n p m m ~ e x n T o~rmMy ~ TOa~,~O O)I~OMy I~HTep~a~y Asv j. DTO MOXHO c~les~aT~ B Cl'Uly CBOI~CTBa a ) CI, ICTeMbI {h/(X)}~= 1 . Yloabxyacb crm,m.~a CBOfiCTBOM 6) ClICTeMbI

{hi(x)}F'= 1, a 3aTeM n e), ~a~zeM noMepa N~ >N~ >No, ~yra KOTOp~,IX ~/,moa-rmIOTC~

c:2e~yiomrre yc~o~rra:

r '~e

sup max. lh,.(x)l < 1/4; m{Rel(x) >= gl(x) => tel(x)} > 3/4, i ~ N t xE [0,1]

e , (x)= ~ h~(x). ~=N I

TpriroHoMcTpaRec~le gyJm-ps/i~t 159

Cor~acHo yTBep~eHHtO 3aMeqaH11a 3 MO~KnO Ollpe~eJI11Tb 110~11HOM~I

N~ N 2 a l ( x ) = ~'6[h:,(x), Qz(x) = z~ 6~'h,(x) (6 ;6[ '= 0; 6~6~'= 0, 1; Nl<-i <- N2),

i = N l ~ i = N 1

~rXOTOp~IX: c n p a B e ~ r m ~ c~e~ymmHe COOTHomen~u (CM. pasencTso (2.12) 11 ~epaBeHcTBa (2. ! 3), (2.14)):

(2.33) m {Igl(x)-al(x)-O2(x)] >= 1/4} < 1/4;

(2.34) Rdi(x) < gi ~ (x) + 1/4, rQ, (x) > gi- ( x ) - 1/4;

11 Ql(x) Q2(x)=O z3mscex xc[0, 1]. t tp = l N 1 . Ilono>~nM 6 , = 6 , = 0 np~i 1 < ' <

J le rKo B11~eTB, NTO

{IF(x)- 01 (x) - Qz (x)] ~ 1/2} = {(IF(x) - g~ (x)[ + ]gl (x) - Q1 (x) - Q2 ix) l) => a/2 } c

=> 1/4}U {Igl(x)-Ql(x)-az(x)] >= 1/4},

nO3TOMy o6~ej!1111m{ (2.32) n (2.33), nosIyarr~:

m {IF(x) -a l (x) -a~(x) >= 1/2} < 1/2.

C zpyro~ CTOpOn~I, !13 COOTnOmenn~ (2.34) B~ITeKaeT, qTO

{Re,(x) => F+(x)+ l/2}C {gl~ (X) >= F+(x)+ l/4}= {tF(x)-gl(x)l >-_ 1/4},

{re, (x) <= F - (x) r 1/2} = {gi- (x) =< F - ( x ) - 1/4} c { IF (x ) - g, (x) t =>- 1/4}

11, c~e~oBaVeJn,11o, tBBn)Iy (2132) 11MeeM:

m({RQ~(x) ~ F+(x)+I/2}U {rQ~(x) <- F-(x)-1/2}) < 1/4.

Hoarmo~o~ Ql(x), Q~(x) co c~o~cT~a~11 1)--3) nOCTpOen~L IIpezmoJm~n~ zenep!~, ~TO Ha He~oTopoM mare l => 1 y~e onpe)~eaem,~ no~n-

nOM~ Ql(x), Q~(x), ..., Q~t_l(x), Q~(x), yaoB~eT~opammne ycZO~naM 1)--3) ~ l <=s<=l. Onpezte~rn~ no~nnoM~I Qz,+ l (x), Q~+ ~(x).

,~a~ dpynKm~ Ft(x) Hafi~eM HOMep ,ut+ 1 11 qby/aKK11Io g,+a(x), IIOCTO.ffHI:Iy'Io Ha rmTepBaJIax Au,+,,j, j=<2m+,, TaryIo, ~TO

(2.35) m{lF,(x)~-g__(x)l >-_ 2 - ' -z} < 2- ' -z l

IIOJlb3ygCl~ CSO~CTSaM~ 6) rr e) C~CTeM~ {hi(x)}7= ~, na~aeM n o ~ n n o ~

Nzt+~

i=N~t + l

~glfI KOTOpOFO c r~paBe~ . ~ ~epaBeHcTBa

sup max thi(x)l < 2 -~-"~ 11 m{Re__(x) => g~+~(x) ~ re__(x)} > I - -2 -~-~, x~[O,~] i>=N~t + l

5*

160 H.B. Ilorocnu

IIpH 3TOM HOMep N~t+l BJ, I6npaeM HaCTO.rlbKO 6001bIIIrlM, qTO Ilpl~ i>=N~t+l MHO- X~eCTBO {hi (x) r npnHaax~ex~r~r OanO~y rr TOabI(O O~XImMy riwrep~aay A~,+, d .

CaIe~oBaTe~,Ho, aa:~ no,~naoMa Pt+~(x) rI dpynr~rm gt+l(x) •oxmo ilprr~e- tIHTb yTBepx~ieHne 3aMe~ianH~ 3, cor~Iacao X(OTOpOMy cymecTByeT napa nOJInHO- MOB Q.,t+l(X), Q~,t+2(x), n MeromHx B~)/ 1), yaxoB~ea~op~romrrx rtep~oMy COOTnO- meHnm yc~oBna 2) C S-----l+ 1 ~ Tarrm, ~TO

(2.36) m{[gt+a(x)-Q2t+,(x)-Q2t+2(x)[ >= 2- ' -~} < 2 - ' -= ;

(2.37) Ra.~,+~(x ) < gl++a(x)+2 -*-~, ra_,+~(x ) > gt-+l(X)--2 - t-~.

nycz~ a;=a;'=0 npH N=~<i<N=~+~. HeTpyauo y6eanwr, ca, UTO ~XaUt nOarmo- NOB Qgt+l(X), Q2l+~(x)cnpa~eaalr~,~ yc0mm~ 1)--3).

~e]~cTBHTe.rIBHO, TaK KaK

c_ {[Ft ( x ) - g , + , ( x ) [ -> 2 - ' - ' } U {Ig,+,(x)-azt+~(x)-a=t+~(x)[ >= 2-~-=},

To r~3 (2.35) rr (2.36) B~,n'e~<ae% ~XO

m{IF~(x)-az~+l(X)-O=t+=(x)[ >= 2 -~-*} < 2 -~-*.

flao~ee, ZBrr~Xy U~almuennR (cM, (2.37))

{Re21 +1 (X) ~ F, + (x) + 2 - t - ~ } c

c {g~++~(x) ~ Fff (x)+2-'-~}c{[F~(x)--g,+,(x){ >= 2- ' -~},

{r (x) < FF(x ) - 2 - ' - * } c {gF+~ (x) < F~-(x)-- 2 - ' - = } c {[F~(x)- g,+~(x)[ > 2 - ' -~}

v~ Hepm~enc~a (2.35), rto:ay~IrrM, qTO

m({~o~,+, (~) :~ F? (x) +2-'-~}~ {~,+~(x) ~ ~,- (x)-2-'-~}) < 2 -'-~.

l-[ycwt, IIOJIHHOMbt Q~(x) onpealeo~e~t~,i a : ~ ucex k = l, 2, 3 . . . . . IIooloxrrM 3 i= =5~+5~', i = 1 , 2 , 3 . . . . . / tcuo, ~ ,o a~=0, 1 npH acex i=>l. H0raxeM, ~TO pga

(2.38) 2 6, [hg(x) +~h (x)] i=1

no~Tr~ ZClO)Xy ua orpe3se [0, 1] CXOe~HTC~I K qbyu~nH F(x). 3aMeTrr~ c~a,mJTa, ~TO ~3 ~woporo ycsmmIa a 2) u (2.25) nermcpe)~czaeuuo

BblTeKaeT

[F,(x)[ <= [tF,_l(x)-a=,_~(x)-O~,(x)[+ Z ln,(x)[] --< s = l s = l i=N~s_ 1

s = l i=1

no,~rH ~ClO~y.

TpnronoMeTp~ecK~e nyn~-p~l~ 161

B qaCTHOCTH,

(2.39) lira F~(x) = 0,

1". e. N ~ - a a riO,~yiOC.~ef(OBaTe.nbHOCTb qacTn~m,~x CyMM p~;~a (2.38) CXO~I/ITC,~ K

F(X) nOqTrr ~C~oRy. C~eRo~aTem, no, ec:~H rtora3aT~, qTo ~r H3 pz)xoB

(2.40) ~ 6~ hi(x), ,~ ~'hi(x ) i = l i=1

/{BJDIeTC5[ CXOJI~UIgMCJ1 IIO~ITH BCIOAy, TO p a r (2.38) raK CyMMa Tpex CXO)I~ttlrlXC~

nOaTn BC~O~y pa~o~, Ta I~e 6yJIeT cxo~I~mnMc~ norTh ~cIo~Iy. 5lcr~o, UTO ero

CyM~a p a t n a F(x). 1/13 nepso ro ycJ~o~nz ~ 2) B~Te~aeT, UTO

I a 2 s _ l ( X ) l -k- ] O 2 s ( X ) l =

= IOz~-~(x)+a~(x)] <- IF~-l(X)] + [F~-t(x)-Oz~-~(x)-O~s(X)l, IIOaTOMy

(]a2s-l(X)l "-}-[Oz~(x)l) < +

IlOltT!4 BCtO,~Iy. B c~ay (2.39), ycao~n~ 3) ~ c~oficT~a 6) q b y H ~ hi(x) r~OqT~ BCtOAy cnpa-

Be~armo lim Re2:_ l(x) = 0.

f[eFKO BI/I~eTB, HTO

l imsup sup ~ 6[hi(x)~limsupsup[ ~ IQzl_a(x)]+Ro2~_l(x)]~ m~_N~k_ a i = N 2 k _ 1 ~ s>=k k<=l,l<s

<- lira ~ IQ~i_l(x)l +l i r a sup RQ~ l(x) -- 0. k ~ l = k s ~ s -

r io aeM~e 5 neps~n~ n3 pPOlOB (2.40) CXO~TC~ nOnT~ BC~O~Iy. CXO~rMOCTS

BTOpOrO papa ~OKa3J~IBaeTC~[ aHa~orHqHO.

6-ynrmepca~nocT~ pgjIa (2.31) ycwaHOB~eHa.

)IorameM Tenep~, ero a0-yHnBepca~br~OCT~. Hcao, ~TO BBH~y C~O~CTBa 6) CHC-

TEMPI {hi(x)}~~ n yC~OB~a (2.25) ZOCTaTOUnO ~iora3aTb cymecTBozanne p s a a BrI)ia

~6ihi(x), r~e 6~=0, 1; i_->1, KoTop~r~ noc~e no~Ixo~me~ nepecTa}~oBr~ ,~J~e- i= l nob a0-ynHgepca,qeH.

B~6epeM I~po~3~o~bn~I~ a0-Yn~epcaJmn~fi pz)I (1.4). B ~a~ecT~e wa~0ro p ~ a ~ o ~ n o BaaT~ O ~ ~3 nprnuepoB y a n ~ e p c a ~ n o r o p ~ a , paccMowpenn~x,

�9 mCTHOCTI~, 1~ [I9], [21]. IIOm,3y~tc~, CBOfiCTBaM~I a) rr e) C~ICTeM~ {hi(x)}~'=~ rI

162 H.B. Florocm~

nOCJIe)IoBaTeJlbHO np r rMer~ yTBep~{~IeHHe 3aMeqaHga 3), rmRae~ noc~e)~oBaTem,- HOCTB CylV[M

Nn+I

S,,(x) = Z at hi(x), N1 = O, i = N . + I

N,,+~ > N , , n ~ l , rz~e 31= 0;1, i >= l,

TaR~Ie, HTO j1J'IR m o 6 o r o n = > 1

(2.41) m { { f , ( x ) - S , ( x ) t > 2-"} < 2-" , sup max {hi(x){ < 2-". iNNn+l X E [0,1]

11o ~leMMe 6 ~ Kax~loro n=>l CylIIecTByeT llepecTaHoBKa %, qrlce~ Nn+l , Nn+

+2 .. . . . /V.+I, ~UI~I KOTOpO~

q

(2.42) max [ ~.2~3,h,(x)] <--]S,(x)l+ max Ih,(x)l N,,<q~--N,,+x i=. , Nn<i~--N.+*

IIpI, I ~cex x~[0, 11. T a r ra~ (CM. riep~oe yc~omle ~ (2.41))

I f , , ( x ) - s n ( x ) l < l l= l

nO,~Ta Bcroay rta [0, 1], H p~a (1.4) o0-yrlH~epca~eH, TO

lira S,, (x) = lira f , (x) = 0

IIO~rH BClOjIy ~ pea ~ S,(x) Tar~xe a0-yrmBepcaaei~. Ha 8Toporo ycJIoBrI~ (2.41)

(2.42) Bl~iTeraei, ~TO q

lim max I ~ %6ih~(x)l = 0 ~1~oo Nn<q~__Nn+ 1 i = N n + l

IIO~tTg BClO~ly rm [0, 1], IIOgTOMy, corolacHo 3aNeuannm 4, pmx ~ N"2~%3 i h, (x) n = l i = N . + l

a0-yaI~epcasmm /[eMMa 7 E[OFIHOCTbtO ~oKa3aHa.

3aMe~ar t r Ie 4. l-IycTb p a r (1.4) 6-ym~Bepcaaeti ri q b y r ~ r m f , ( x ) npe~cTaB~a-

IOTClI B BIUIe

k.+l f , , ( x ) = • g,(x) ( k ~ = - O , k . + x > k . , n = 1 , 2 , 3 , . . . ) ,

$=kn+l

r~e g~(x), i=- 1, 2, 3 . . . . , - - npormBOm,~Ie tt3MepitMBIe qbyHKllrlg Tarne, qTO

J lim max Z g,(x)!.__ = 0 (2.43) n ~ k n < j ~ k a + 1 i = k n + l

nOqTrI acm~ly na [0, !]-

Tparo~oMeTpa~ecI~te ~xyz~,-pz~,i 163

I'[yCTb A ~ HeI(OTOpbIX 5 . = 0 , 1 ( n = l ) pa~ ~5 . f . ( x ) CXO~HTC$I IIOqTrI n = l

Bcm~ty Ir F(x) - - naMepnMofi ~OqTn Beroay xoneunofi qbynK~na.

Ta~ Ka~ q k n + l q

Z Z 6. g~(x)= Z S , / i ( x ) n=l i=kn+l n = l

k.+ l

~:~ Bcex q=l, 2, 3 . . . . . To Barrzy (2.43) paz ~ ~ 5,g~(x) CXOaaTC~ nowrr~ n = l i=k .+ l

~cm;ay ~ (1)ymr F(x). CsleaoBaTeSI~.HO, p~ta ~gi(x) 6-ynrmepca~em i = 1

Ananorn~no noKaa~maewc~, qTO np~ BI~InOJ~IteltHtt ycJIOBgg (2.43) n3 ), ~an

ao-yHngepcaabaOCTn pn~a (1.4) Bt,ITeKaeT COOTBCTCTBetIHO ]~ HJIH ao-ynnBepcasm-

nOCTb p ~ a 2g~(x). i = l

B ~IaCTItOCTtt, /43 ~ !4 ao-Ynn~epcaJH, nocTefi p ~ a (2.30) ~nx npoi~3Bon~,I~X

qby~IxlIn~ q~(x), yjio~neT~op~iom~x (2.25), I~ITeI~aeT 5 ~ ao-y~nBepcan~,HOCT~ p ~ a (2.24), eca~ m,moaneno y c n o ~ e (2.25).

3aMe*~a~ne 5. ltc~o, ~TO B yCJIOBI, IIIX JIeMMt,I 7 b I4 ao-yrmBepca~,n~,i~ p~A

(2.24) ~ o x n o npe~CTaB~T~ ~a~ o6~,e~n~enne ~myx ero no~pa~o~, i~ax~i~ n3 KOTOpt, IX TaK~e gBaHeTC~ b I,I a0-yrmBepcam,m, iM. B~,~6paB HeKOTOpym rtepecTa-

HoBby paaa (2.24), aa~ KOTOpO~ nepecTa~eHm,~ p ~ CXO;~nTCZ nO~TrI ~cm~y na [0, 1] K xoneqao~ qbynI(mI~ ~ coxpa~iaeT nopa;xoK c~e;ao~an~z ~neao~ nep~oro

IIOjipa~a, no~y~ma cxo~'~mI~C~ IIO,~Tn ~cm~y P~)l, co~Iepa~attm~ 6-yrmBepcam,- m,N no~p~t)I. Cae~oBaTea~,no, 3TO rt y-ymlBepcam, In,i~ Pa)l. IIOgTOMy Pa)I (2.24)

nocae no)lo6nofi nepecTaHog~H ~nenoB ~IBYI$[eTCR o~rmgpeMenno 5, ~, a0-ynnBep- ca.IIbIt~IM.

w 3. ,~OKa3aTeYlbCTBO TeOpeM 1, 2 . 3

I-[yCTb ~rIcaa 0k, k = 0 , 1, 2 . . . . . y)~OBJIeTgOp~IOT COOTrlOlLtettmo (1.1). ~a,~ ~oKaaaTeal, cTBa TeopeM AOCTaTOqHO noKaaaTb cymecTBoBanue nocJIe)xoBaTem,- ~IOCTrI TprtronoMeTprI~ec~crlX nO~TnHOMOB P,(x), r = l , 2, 3 . . . . , co c~e~ymmrn~ CBO~CT13aM24:

(3.1) e . ( x ) =

(3.2)

(3.3)

mr + 1

Z 0~-'cos 2~(kx+0~), k=mr+l

m i = > l , m r + i > m r , r ~>1,

10~l <= I~l, k = m l + l , m1+2, m1+3, ...;

2 n

Z ItP2"+i-I(x)-A.,j(x)q~.(x)][~ < 2-"-1, j= l

n = 0 , 1 , 2 . . . . ,

164 H.B. 1-Ioroora

r a e {q ' . (x )}Lo l~n.aM ~eMMB~ 7.

,~e~CTBIITe.rlBt~O, IIyCTI, rtOdlgHOMbI P~(x), r ~_ 1, (3.3) m,xreraeT

II . 2 Z lle,(x)[[~ =< 2~iq~,,(x)]]~+A~ <= A~, 2 n ~ r < 2 n + l

r~e A~, A9 - - IIOCTO~HHI~Ie. Cor.~acuo riepa~ertcTBy Kap~iecona--XaHTa [11]

- - HeKOTOpa~ CI~:CTeMa ~yn~I~rr~, y)XOBTteTBOp~Iomax ilpe~rlO3IOXe-

y~e IIOCTpOem~L Tor)Ia rr~

n = 0, ~, 2, . . . ,

TaK rat ~n

max max 0k COS 2re X <__-- k k + 1 mr<m'~mr+ 1 k= 1 2 P n ~ r < 2 Pn

111

--~ ( 2 _~ [m .<max { 2 0 ; , ' c o s 2 z c ( k x + O ; 3 [ ] z ) l/z, gkpn~r .~2kpn , 1 r ~ r + l k = m r + l

m /t

(3.4) l i r a = ( max max I ,,~+ O~c~ , ~ l / n + 1 ~._ k- +1 "~<m~--m.~l 2 Vn~r.<2 I:'n . k = 1

I~OqT~ BcroJly ~a [0, I]. C ~Ipyroti CTOpOI~,I, IIO JIeMMe 7 Moxno BsI6paTs rloMepa po<pa<p2 . . . .

~ i z ~owop~ix pz~ (2.24) ~B~eTCa 6 n a-yHnBepca~sn~xg, ec~n ~Ic~a %, n>=O, y~OSJIeTBOpSlOT COOTHOmemIaM (2.26), a qbyI~rlmH ~h(X), i>=l, yc~oBnm (2.25).

I/I3 (3.3) Z~iTeraeT, aTO

1 2kP"

.= t/ -Tr

rr no TeopeMe JIe~n

~__d 1 2kpn

,,=0 r . Z < + co

TO B

1

/ ( . ~ [ max I . ~ O;,'c~ dx-<=CoA~,

r~e Co - - a6co2imTna~ ilOCTO~nna~. B ctt~y c~e~IcTBe, a 1 cymecTByeT IIO~Iloc~e~oBazea~nOCT~ {k,},~ 0 r a r aa , ~rro

~ i a 2iIo6o~ llOCJIe~oBaTeJiLImCTrr paCTylIIJ~X HoMepoB p,,, n = 0 , 1, 2, . . . , iio~rvx zcro,ay na [0, l] cnpaBe~!znzo

lira ( • [ max t o~'cos2rc(kx+Oi,)t]~) =0 . n ~ n + l k _ k _ + 1 m r < m ~ m r + l k= i

o pn .~r<2 l~n

Ti3groHoMerDiuec~e Hy~l~-p~,~l~l 165

nOaT~ ~cmay r~a [0, 1]. CTaJIo 6bITb, pmI

1 Z P,(x) ,= l / n + 1 ok k +l

P n ~ r - < 2 Pn

Tai~xe m3~eTc~ 5 H a-ym~Bepcaa~,m, Lv. Tor~Ia BB~r~y (3.4) ~ 3aMe~aHn~ 4, Tpt~roHo-

MeTpnnec~m~ P~)I

"~r+l 1 2 Z o,, Z ~ / ~ . k cos2"~(kx+O~.) =-- z~Q~cosaTt(kx+Ok)

n = 0 2 k p n < = r < 2 k p n + l k=inr -k l k = 0

aBa~eTc~ 6 ~ a-yaHBepcaJIbH~U, ftCRO, UTO KOa~p~Hm~e~T~ a r o r o p ~ a uc~o~r~e.

I locTpom~ weHepb noa~~m~r~ P~(x) H c)yH~m~m ~o,(x) co cBO~CTaa~

(3.1)--(3.3). PaCCMOTpnM xp~roHoMerpHnecr~e pH)l~,~ dOypbe qby~r~i~ A,,,x(x), n >=0,

d,,,~(x) = 2 _ , + 2 ~ = ~ sin ~2-"Vv c~ 2~v ( x - 2-"-~)"

Ta~ x a r pa~ CXO~nTC~ ~ Mexpn~e L~[0, 1] (cM. [30, CTp. 423]), TO np~ ~m6OM

n>=0 MOXmO ~a~xu HOMep V~ (V0=0) TaKo~, qTO

iIA,,,~(x)-Q,(x)II~ ~- 4 -" , r~e Qo(x) =- 1;

2 . ~ sin rc2-"v cos 2~v (x - - 2 . . . . ~), Q,(x) = 2 - " + r~ ~=~ v

v . -aa .mcwHmm~t cyxe~a ( n > 0 ) . IlpoAo~.x~a~ qby~m~i~ A,.i(x) c nepno~oM 1 Ha j - - 1

13cto gHCJIOByIO OCb FI 3aMeH1g3t x H a X - - ~ I I O ~ y q ~ " M 2"

(3.5) ]d,,,(x)-Q, Ix-J~-~l]l ~ 4 - " , j = l , 2 . . . . , 2L

Hcnoa~ay~ COOTHOme~ne (I.1), ~ 6 e p e M Bo3pacTammHe HoMepa m,, y a o B a e ~ o p ~ m m u e yC~OBHgM: m~+~ ~4m~ (r>=l); mr >v, 2 " < = r < 2 "+* (n>=0), H Ta~ue, qwo

l r

(3.6) A ~ - [ S'2 z q l / Z > l , r = 1 , 2 , 3 , L ~ r , id ~ - " ' " , i = I

ule

Ir=[mr+l ]--l, [ 2m~ 1

IIyCTB 1 17"

H, (x, y) =

r >= 1; 2r, i = min 2 m r i ~ k < 2 m r ( i § 1)

2,, i R~ (y) cos 2n (2i + 1) m, x, 5=1

r>=l, t~ 1/~ r KpaTItO 2n IipI4

k, = i ~ r"

r = 1 ,2 ,3 , . . . ,

166 H . B . l - l o r o c z ~

rAe {Ri(y)}~=a - - op/oHopMripoBaHHa~t Ha oTpe3~e [0, 1] cncTeMa qby~f~nfI Pa~e. Maxepa. 17o Teopeue q~y6mm r~ ~iepaze~cT~y Xm?mHa ~an CrlCTeMbI Pa~eMaxepa NMeeM

1 2,

i dy / l i l t ( x , y),' dx = 4!" f ~, At" rcos~2~z(2i+l)mrx) dx ~_ 4. o O Xi=ll Ar I

l-IoaTOMy )/JI~I nm6oro r ~ l cymecT~yeT nrlC2aO y,~[0, 1], ~s~ KOTOpOro

(3.7) 1 _ l int(x, Y,)tz <= IH,(x, Yr)i[a <= ]/Z fi-

OTcro~a no r~epaBeHC'rBy FEm,~,epa (CM. em6 [13, CTp. 154])

1 (3.8) lint(x, yDlh => NHr( x, Y,)[]~ In,(x, yDI; ~ >=

~a~Iee, zerto, ~TO ~ Crlay B~,I6opa no~epo~ m,, l~, npH 1 ~_i<=l, I~e~io,trlc~er~- ni le oTpe3I<rI [(2i+l)m,--v,, (2i+l)m~+v,] He nepeceramTc~ r~ co~epxaTca o rpe3re [mr+ 1, mr+a]. Ho~IOXla~ Tenepb

np~

r~e nOarlnoMoB onpeaestamTC~t TaX:

0~, = [(2i + 1) m,- - k] (r2 . . . . 1 + 2 - " - ~ ) ,

H

P~ (x) = / / 1 (x, Y0 ~ Z ~-' cos 2r~ (kx + 0~) k = m l + l

2"_<-r<2 "+1 (n=>l)

r 2" x

i=1 ~ Ri (Y') t /2-"+--z ~=1 ~ v cos 2nv x - 2 "

m r + l

�9 cos2rc(2i+l)m~x -- ,~ O~'cos27c(kx+O~), k = m~.+ 1

�9 tt qr~c~a 0~, ~k coraac~m qbopMaJIr~HOMy npoH3Be)lermm TpHroHOMeTpg~ecIC~X

eca~ [(2i+l)m,-kl <-_ v,, 1 <= i <= 1,

~ = 2" A, R~ (y~),

Ri (y,) sin ~2-" [(2i + 1) m, - k] (2i+l)m,--k

~JIa OCTam~rmrx 3rta~ieHrlfi kE[m,+ l, m.+l]

ec3H k = ( 2 i + l ) m /

,~rtc~a ~k, 0k paBnt,~ t~y.nm.

TparOHOMeTpH'~ecrd~e t~yab-pn,~,[ 167

Cnpa~e~an~ocT~ (3.2) HenocpeRcTBenHo ~,rre~aeT r~3 onpe~e:~ennx ~nce~ O~

rr ycao~rm (3.6). HyCT~

~p.(X) = .~A~,~(x)Hz.+j_~(x,y~.+j_~), n = 0, 1 ,2 . . . . . j ~ l

T o r ~ a rr3 (3.5) ~ (3.7) cae)IyeT:

2n

Z l le~-+j- l (x)- ~,,,, (~) p . (x)h j = l

<= Z IlH~,+j-l(x, yz"+j-~)ll4 Q, x - -A,,,i(x ) < 2 - " + L j = l

Hoxa~eM Tenepb, aTO )lZa qbynrt~r~ ~o,(x) B~mOaHeaOTC~ Tpe6oBanna ~e~vl~i 7.

Taz< r a t qrIcJzo m~ (2"<=r<2 "+1) xpaTr~O 2", To dpyrr~rr~ H~(x,y,) rtMeeT n e p a o ~ 2-" , IIOgTOMy

2 n

IJ~o.(x)h = [ Z IIA.,i(x)Hz"+J-a(x, Y2"+J-1)Ii~] 1/2 = j = l

2,~ 1 = [ Z 2-" llm~ y.-+j-lll~] ~/" = j=l r

~astee, ~ cmay OpTOrOnam,HOCTn qbynrRrr~ Pr(x) H (3.3) 3ar.~ro,~aeM, ,~3ro ~o,~0 c~a6o B L2[0, 1].

~ZL~ 3a~epmenr~ ~ora3aTe~bcTBa TeOpeM OCTa6TC~t yCTaHOBHTb c n p a B e ~ n -

ZOCT~ COOTnomenr~ (2.22))~ax ~ro6oro MnoxecT~a E n o ~ o x ~ T e a s n o ~ Mep~I.

3aMeTnM cnaqa~a, ~TO e c ~ n ~ n 0 > 0 , 1 <=j0--<--2 "o, - - npor~3Bo~,n~,te ~oMepa, TO ~ cnay (3.8)

(3.9) f I~.(x)l dx = Z f [H~"+.i-~(x, Y~"+.i-~)l dx >= l~--J~;9n dn, .i

Ano, Jo An, j c dno ' Jo

1 1 ]~j~_~,, 41/~ mAn'j 4l/~mAno,Jo"

An, j ~ Ano ' JO

~ i 3a~anaoro e > 0 e~,~6epfiM oTrp~,~Toe M~o~ecT~O Ge, COCTOa~ee n3 KOHeqltOFO ~lI;lCJIa nenepecera~omrrxc~ ITHTepBaJIOB H Taroe, HTO ITIS-<g, r~e

S=(E'NG~)~(Q\E) ~ CrrM~ea-pn~ecraa pa3nocT~, MnomecTz E n G~. 5IcHo, ~ITO IIpH 3TOM MO~KHO CqHTaTB, HTO MHo~eCTBO G~ npe~cTaB.riaeTc~ B BH~e

i o

G , = ~ A,o,~- ~, I < ~ j l < j ~ < . . - < j ~ o ~ 2~~ (l ~ io <= 2"*) I=1

168 H.B. FIoroca~a

A~Ig ]~oeTaTO~HO 6 o ~ I n o r o Ho~epa n o. 143 (3.9) cJIe~yeT, qTO

1;m inf f te.( )l dx _-> inf Z f dx- lim f :> ~ c o ~

E n ~ i = 1 Ano, Jl S

I ~o 1 >= ~ mA., j - l im sup ( m ~ ) :/"~ II cp,, (x)]t ~ >= " ~ ( m E - s) - (s/2) :/~. ~ '

BBrr~y I I p O I ~ 3 B O J I b H O C T H ~ : > 0

1

T e o p e ~ i 1, 2 H 3 DOJIHOCTBIO ~oxaaaHN.

w 4. ~rnimepea:mm, m p~2!b~ no 6a3acaM npocTpancTna L~[0, 1]

B yCTaHOB~'IeHHblX JmMMax 1- -7 (w 2) He qbrirypHpytoT xaxHe-~II46O CBO~CTBa

TprIrOHOMeTprr~ecKo~ crrcTeM7,i qbyHK~H~. 17poaHa~rr3gpoBaB ~oKa3aTe~SCTBO Teo-

peMBI 3~ MOXCHO yBH~eTb, qTO B Heft[ H3 CBO~CTB TpHroHOMeTprlqecKo~ CHCTeMBI

qbynKIIn~ ~cnoJ!l,3yeTcs nepaBeHCTBO Kap~ecoHa--XaHwa i~ HeKoTopoe armpo~cH-

MaTHBHOe CBOHCTBO TpHFOHOMeTpHHCCXHX (~yHKIIHH, BBiTexaronlee H3 IIOdIHOTI~I B

L2[0, 1] TprrronoMeTpr~ecxofi CnCTe~aJ, I ~ i~3 T o ~ e c T B qbopMam, H~Ix r~porr3Be~e-

Hrr~ TpHroHoMeTprI~ecKgx nO~HHOMOB. IlpHBe~eHHoe no~po6Roe ~oxa3aTe3r_bCTBO

MO~KHO HecKoJTBI<O ylTpOCTHTt,. Tax, nanprrMep, HcnoJiB3OBaHne HepaBeHCTBa Kap-

necona- -XanTa c noKa3aTe~eM p > 2 (BMeCTO p =2 ) llO3BO3I~eT yCTaHOBrrT~, COOT-

~omeH~e (3.4) 6e3 npH~eneH~z ~eM~S~ 2 (c:~e~cTm~e 1), npn 3TOM ~ ~a:mHe~me~

~oKa3aTe~TBCTBe TeopeM~,I 3 MOZ~O IIpHMeH~ITB cJIaglb~ BapHaHT :IeMMI~I 7, JIOKa-

3aTea~,CTBO KOTOpO~ raE~e ~3'm~IaeTcz B :~eMMe 2. TaKo~ r~yTs AOKa3aTe~cTBa TeopeM 1, 2 ~ 3 noz~oaaeT noay~aw~, 6o~ee o6m~re yTBep~AeHrrx. Moz<no

XaK y C ~ T ~ yT~ep~)le~ne TeopeMs~ 2, Ta~ r~ yc~a~oBnr~ ee aHaaor ~ a ne-

KOToporo ) l o c r a T o ~ o m~poKoro x~acca OpTOHOpMr~po~aHH~IX CgCTeM. MsI He

6y~eM 3)lecb OCTaHaB~THBaTbC~ ~a 3~OM Bonpoce, a aHaJ~orHqH/,~M MeTOJIOM ~Ioxa-

men cymecTBoBanne e-yHgBepca~sHOrO pz)Ia no IIpOtI3BOJibttOMy 6e3ycJmBHOMy

6a3~cy npocTpancTBa L~[0, 1]. YIycT~ T = {0~(x)}~=~ ~ npo~r c~cre~a ~byH~mr~, ~p~na~ae~amaz

L2[0, 1]. O6oaHa~rr~ nepe3 L~(TN), N = I , 2, 3 . . . . , aaM~maH~e ~ MeTpr~xe L~[0, 1]

;mHet~r~or~ o6o~Io~r~ crrcTeM~I N={0~(X)}k= ~ ( T ~ = T ) . Hano~r~M, ~TO ,~B:~aeTC~ 6a~ucHo~ C~CZeMO~, ec:m ~ro6az d?ynxttna g(x)~Lz(T) pa3aaraeTc~

B e~rr~CTBeHH~ p ~

(4.1) g(x) = ~.~ a~(g) ~k(x),

TpnrolmMerpn~ecxne HyAb-p.-'L2bl 169

cxoR~mmRc~ rio r~opMe L~[0, 1] K qbyIJKI~H~4 g(x). l~a3r~crlax CrlCTeMa T na3~,macTcx 6e3yc:rozrm,2, ec.nr~ OHa OcTaeTcYI 6a3/JCI:IOITI IIplcI .rIIO60~ nyMepa~Hr~ qbyn~t~r~ ~k~(x), k=>l.

HyCTb q

- - Maz<opaHTa 'mCTRUH~IX CyMM pa3.~o~eu~X (4.1) ~yH~Rr~rt g(x)EL2(T). H ~ e pacCMaTp~IBaR3TCX 6a3ttCttbte cucmeJvtb~ T, O,~n RomOpblX cyu4eemeytom nocnegoea-

meabnocmb ~yttKgu~ {ul}t=lc l l 0 (onpe~e~eHHe ~nacca 1A 0 CM. ~a cTp. 31) U nocae- OoaameAbnocmb ~vmoatcecme {~}~=1 c mEl-~l ( l - ~ ) ma~cux, ~mo

(4.2) fu,(a (g, x))dx _~ []g(x)![~, 1 =1, 2, 3,... 0:1.,~ ecex g(x)CL2(T). El

BBe~eM c.,Te,~ymmee CBO~CTBO (fl) anx C~CTeM ~ y u K u ~ ~,

([3) Cyu4ecmeyem r, one~maa ~,aepa p, 9Keueaaemnnaa ~vtepe !le6eea, (T. e. Mepa # onpe~e:leaa na H3MepHMblX B cM~Ic~e I le6era nO~M~lo~ecTBax oTpe3/~a [0, 1] H #E=O Tor~a ri T03~,~O Tor~a, r o r ~ a m E = 0 ) , maKa& ,4too O~a ~m6~tx unmep-

ea:toe A=A, ,~ (n~_O, l ~ j ~ 2 " ) u ~tuc:ta e > 0 :aoa~c,o ~m~mu ffiynmtuw t ( x ) = =t(A, z, x)~L~OP), c)~a xomopo(t

(4.3) i[t(x)ii, ~ (]aA)'t~ i[t(x)li~,

(4 .4 ) l[t(x)ll~,~ao~ < a ~,lt(x)h (A ~ = [0, 1 ] \ A ) .

3 a ~ e , a a n ~ e 6. 1-IycTs T - - 6a3nc~m~t cnc'reMa co C~O,~CT~O~ (fl). Y 6 e a r ~ c z , '~XO rr am6a~ crmTe~vm Tu , N ~ 2 , o6sm~aeT CBO~CZ~OM (fl) C 3aMe~o~ p na p/2, )IOCTaTO'~HO no~a3aT~,, "~TO ~ t ~ca,m,~x nH~epzaxoB A =A,,,. i rr ,~r~c~a gr 1) cynmcxByeT ~]~ynx~; t(x)~L~(T~), y~tozneT~op~mnla~ ~epa~encT~aM (4A) u

(4.5) /It(x)h ~ ( 1 - e ) ( p A ) '/2 [[t(x)l},~.

O6o3Ha~m,'v~ ~mpe30(x) rlporla~,oa:~nyro qbynK~lJm TaKyto, q-ro

1 1

f O( ) q(x)dx = . f 0 a . . oex 2. 0 0

B ~a,~ecT~e O(x) _~o:~no m,~6pax~, n e p w m qbynxurrm n3 CHCTeM~,~ 6nopToro~ansrmfi c T (o cymecT~osa~rm 6rmpToroHazsn~,~x CnCTeM cM., ~anpmaep, [9, r,Ta~a 6]), T o n m ~oaqbqbmmer~T a~(g) pa3~mKemm (4.1) qbynxmm g(x) paBen

1

a~(g) = f O(x)g(x)dx . 0

170 H.B. Horocmt

BOgbMeM )~OCTaTOqHO 60~3_bIs noMep m > n TaKo~, tITO B C~Ty a6co3ItOT-

ao~ nelIpepl, mHOCTH Mep]~I /~

(4.6) max (l~Am, i) ~!~ < ~-min {1, #A,, j} l]O(x)II: ~ ll(/~ (x)!]~ -~ ix < i ~ = i 2 ~

(ia = ( j - 1) 2' . . . . , ie ----- j 2 " - " ) .

HyCTb T(x ) = Z Y~,,~tm,~(x), t (x) = T ( x ) - - a l ( T ) ~ q ( x ) ,

r~e {~y~KHm4 tm, i(X ) ( i~<i~i~) coraacno ycao~mo (/~) Ta~ne, ~To

(4.7) #Am, i = Iltm,,(x)lh ~ (]2z~t.,i) 112 iit.,.(x)ll~,

(4.8) II t,,,,, (x)ll ~.(~ ~, ,) < ~ min 0 , ~A,,, 3,

a ,mcna y. , , i= • 1 (i~<i-~iz) ~ cnay paBencT~a

a t ( T ) = Z 7,.,iat(t.,.i) i~ < i ~ i ~

B~16paHbI TaK, qTO ia~(T)t <= max lai(tm,3l.

i~ < i ~ i z

O,~eBn~IHo, .tTO a~ ( t )=0 (T. e. t (x )s .

i '1 . . ~ laa(T)l < ( max i t , , , , , ( x ) h ) { l o ( x ) i l ~ ~ " = - ~ m m {~, . a , . 3 i!e.(x)lIP - i l ] I '

] l t ( x ) I h = ilT(x)[h-la~(T)l Ii'P,(x)lh > Z ht,,,,~(x)lh-Ilt,.,~(x)l]L~(a7.,,)-

- Z iit,,,,,(;)!l~,(.~ i I < : l ~ i z

I# - i

Z [~,'~,,,,~- Z I!t,o,,(x)ilL0~,~7. , ) ] - ~ a , , , ~ > i l # A , , , j - 4 P A , , , j - - ~ t t A , , j = 1 - pA, , j .

Ana.~orH,Jno, neTpy~rIo 3aMeTRTb:

-<: ~ X l ilt(x)ll~ = !lT(x)ll2+ {ax(T)t t~$1( ),12 <

i '] 2 1/2 g < ( Z [Lit,,.~(x) L=<~,,. O+ Z ilt,,,.,(x)i,L=<~.~ ,)] ) + T (ua"';)~/s < i ~ < i < i 2 ~ , i l < i ~ i 2 "

<(Z i l < i < = i 2

Tp,arOHoMeTpaqec~eae Hysl~-pa~,l,i 171

~WO COBMeca-Ho c oRenKo~ )IJ~ []t(x)l]~ Bae:~t6W (4.5). HepaBencTso (4.4) noa~y'~nM cJ~ei~ymmnM 06pa3oM :

Ht(x) i t~(<) <= Z iit,o,~(x)liz~(~ a+la~(T)ll[Oa(x)ih < ' i ~ < i ~ i ~

< < , lit(x)[l <-- [It(x)ll .

~oKaxeM c~ae~iylomylo weopeMy.

TeopeMa 4. a) ~:ta npouseo:tbnoft 6asuc.o~t cucme~bt T, yi)oeaemaopmouleO

nepaaencmay auaa (4.2) u o&tac)am~qe# ceoO, cmeo~t (fl), cyuteemayem 3 u a-yttuaep-

ca./lbHbtft pro)

(4.9) ~ C~ ~Pk (X)- k = l

6) Ho :uo6o(t 6ezyczoetto(t 6asucno~t cucme~e T co ceo(tcmeo.~,t (fl), cyutecmeyem

p~,t6) (4.9), tcomopb~ noc:te coomeemcmeytou4e~t nepecmartoetcu ~meHoe ae~aemca

oOttoepe~euHo b, 7, a-ynueepcct./tbttbt.,~t. Kpo.~r moeo, ec:tu T - - Hop.~tupoeauuaa cuc-

me~wa (m. e. !l~(x)llz=l, k>=l), mo

Z {ckl ~+~ < + co a.~ft ~ m 6 o e o ~ >0 . - k = l

KaI< cae,n;CTm, re no.JIyuaerc~i

TeopeMa 5. Ho mo6oMy 6esycJzoeno~y 6a3ucy npocmpaucmea L2[0, 1] cyufe- cmayem a-ynuaepcaAbub~ paO.

~oKa3awea~cxBo T e o p e ~ 4 dpaKTnaeeK~ ~BJIgeTC~ noBTopenneM aora3aTe~- CTBa TeopeMt,I 3, 3a rlCKJiIOqeHgeM ~racTn yCTaHOB.IIeHg~ oI~enr~ MaJIOCTI~I KOa~(I)H- ~rtelaTOB p,q~a (1.2). Hpri aTOM B AoKaaaTeal~czBe yTzepx~eHn~r a) ImpaBencTBo Kap~ecoHa--XaHTa 3aMeHzexca HepaBeJlCTBOM (4.2), a B ~oKa3aTeJI~CTBe yTBepx~e- Hrl.'t 6) TaKmv~ xe HepaBerlcTBoM AJI~ nepecTaBJieHH~tX pa3Jio~KeHrlfi (4.1), m~iTeKaro- mmvt H3 peayJIt,TaTOB FapcrI~t (CM. [7] ~Z [8, r.aaza 3]).

H3 TeOprm 6aaHcm,lX CHCTeM ~IaBeCTIm (CM. Hanpi~Mep [9, rnaBa 6, CTp. 381, 375])

TeopeMa B. I lycmb T - - c)e~ycetoet~aa OasucHaa cucme~ua. ToeOa cyutecm-

eyem no~toatcume~tbHaa nocmoamtaa Bo= Bo(T) ma~caa, ~tmo 3~a npou~eo~uoft o6ynK-

u uu g(x)~ L2W)

a~(g) ][r <= B0 I] g(x)!l

172 H.B. Ilorocm~

~oi~azaTe~I t , CT~O TeopeMl~I 4. Hcrlom,3ya CXO~mV~OCT~ ~ L~[0, 1] pa3- ~ ioxem~ (4.1) rI yT~epx~erme 3aMe~larm~t 6, Ha~6M IlOC2iexomaTe~ir~HOCTl~ rto~IrI- nOMO~ {Q, (x )} ,~ no CnCTeMe ~, rtMeIomnx rler~epecexalomrrecz pa3~lox.erma:

H T a K I t e , N T O

Q~(x)= Z b~,~b k(x), k = m v + l

~'11 = 0~ I n v + 1 > D'lv, V ~ 1

1 1/-,. , 1 [IQe-+~-~(x)[[~ => w(~A.,~) " ][Q~,+j-l(x)li2 = wI*A.j ,

t ~ ~ 2 - ' p A , , , ~ , tO~"+3-x(x)[!L~(z.~,) < aa.~ scex n:>0,= 1 =< j =< 2".

IIo.llOX~tM 9n

%(x) = 2?. ,~A, ,d(x)Qz,+j_~(x) , n = O, 1, 2, ... j = l

r~e *mcna 7,,,~.= _+ 1 (1 ~j=<2") olIpejIe~3eHsi TaK, qTO

(4.10) max ~ ~,,. f O.~,,+~_~(~)d~ I ~_ max I fQ~,.+j_~(x)a~], l~--q~2 n j = An, j l<=J~--2n An, j

~I noraxeM, ~TO ~ISm q'by~IIHfi q~,,(x) BI.IIIO~IHamTC~I TpefoBann~ aeMam,t 7. T a r v.aI< it([0, 1 ] )~ I (3To s~.ITeraeT ~3 aepasencTza (4.3)), ~Ierxo Bnae:r~., ~ o

(n=0 , 1, 2 . . . . ) 2n

(4.11) Z [lQ~-+j-~(x)]l~ <= 1, j = l

(4.12) ~! ,o.(x)~L!~ = [ ~ ' IQ~"+j-~(x)il~,(~. )]1/~ ~_ 1, j = l

(4.13) I! ~, I]A..~i(x)(p.(x)-?.,jQ2.+j_l(x)i[2 = ~ I!Q~%j_I(X)tiL~(~ ) < 2-" .

j = l j = l '

]IJm IIpO~3Bo:Ibm.Ix n > q > 0 , 1~i<=2 q, nMeeM ( .A=(i - -1)2"-q , A = i 2 " - q )

f k~ dx = Z frQ~"+J-~(x)l dx->- Z [llQ~,,+:-l(x)h- Aq, i Ja <J<--J~ An, j Ja <J~J~

1 1

' ,11 .," ~32

(cM. ~t,mo~ (2.22) • )ioKa3aTe.fft, CTBe TeOpeM~,I 3), OTKy;aa HeTpy)IuO m,mecT~,~ cnpa- BeJIJIIIBOCTb COOTHOIlIetIttI~ (2.22).

Tpr~roaoMerpnaecvdae Hyas-pa~b~ 173

C apyrof i cTopoa~,I (cM. (4.10)),

~j<= 7.,i f Q2.+j-l(X)dx[ <= [~~ = Ij. =,= ~..,

<=2, ~ax a f <=2 l~_j-~max . . . . (itA,. ~) 1/2 .

Ima ro~y ~ cnay (4.12), a6coaImTUOfi uenpepJ, maOCTri ~tepJa /, rI aa~t~ttyrocrn B Lz[O, 1] anue~Imfi o 6 o a o ~ r i CI,ICTeM't,I {{zla, i(X)}~=l}~=o, 3arnm~aeM, a~ro ~o,(x)-~0 ( n ~ ) cna6o B L2[O, 1].

CorJmcrto JIeMMe 7 cyIIIeCTWeT no~cncTeMa {q)k,(x)}~=0, /IJI~t rOTOpO~ npori3- Bo~,ri~,~ PX)I ~ ) I a (2.24) ~ a ~ e T c x o~Ho~peMeano 5-, a-yHrmepcaa~r~mv~, a nocae I~eXoTopofi nepecranoB~n ~Iae~o~ 8-, 7-, a-yng~epca~ibH~,IM (CM. 3aMeqasne 5).

H3 (4.13) no TeopeMe JIe~n ~,IveraeT, ~TO

2 n

.~ ~ IA.,j(x)%(x)-y.,.iQ~,,+~_~(x)l < + oo n=0 j = l

rlO~Wn ~acm)ly na [0, 1], rto3woMy pa~

gkP n

(4.14) .=~o % "~ v~' i 0 ~ . (x) j=l n' -~m ~.tj--i

~JIl t JIIO~blX IIO~IIOCJIe~OBaTeJIbI-IOCTkI HOMepOB {P.}~=0 i~ IIOC2lellOBaTe.rlbHOCrIff ~IFICeJI

{%}~=0 co C~O~CTBaMn (2.26) TaK~e ~tBnaeTca 6-, a-ymmepcanbnr~I~, a rtocne r~epe- cTano~r~I ~ste~io~, rpoMe ro ro , rI ;~-yr~rlBepcam, rn ,~ .

~a~ee, TaK Kar (c~. (4.2) r~ (4.11))

u, ( max f2=(Q~,x))!lq(~. , <= H ~--s ut(fa'(Q~,x))l[~.(E,) < = i ; 2knNv<2kn -t-1 2kn~ kn+l

n ~ 0 , l = > l ,

TO rio ~eMMe 3 cymecTayroT ~o3pacTammne noMepa p . n 'mcaa ft. c

~ ROTOpblx

1 (n>O) . ~ / L = + ~ , O < fl,, <= n + 1

n~O

lim B,, [ max fa(Q,,, x)] 2 = 0 n ~ okn .kn +1

IIOqTI/I Bcm,ay Ha [0,1]. CTa~o 6~,ITb, B CgJ~y 3 a g e q a H ~ 4 p~t~ ~o CHCTeMe

r~,o,,--,, o+~ Z b~g,~(x) -- Zc~O~(x) n=0 k k ~1 k=mv+l k = l Pn~v<2 Pn'

~n~ieTcn (3, a-yH~rBepcanbH~,tM. IlepBag qaeTb TeopeMbI )Iora3a•a,


174 H. 13. Horocn~

,/~oga:~eM Tenepb BTOpytO ~IaCTb. IlyCTb BHOBb I/OYIHHOMI:,I

mv+l

Q~(x)= Z b k~p~:(x) (m~+~ > m~) k = m v + l

no CHcTeMe T rt qrrc~la 7 n , j ~- ~___ 1 m,I6pam,i Tar, ~TO ~IO60fi p~t~ Bn~a (4.14)

RI~5[eTCg O~HoBpeMCHHO 0, ]~, a-yHHBepca~II~ItBIM IlOC~Ie HeI(OTOpOfi IlepecTaHOBK~

�9 I~Ie~Io~ (o6osnaqrrM ee ~Iepe3 %), 3TO - - IlepecTarlom(a ~ii~ce~ {{v}2%,~_,<z~, +x},~=0,

3a~14c~tma~ OT IlOC~IeAo~aTem, I~OCWri {a~}~= 0. KpoMe TOFO, MOX<HO Cq14TaTB, qTO

~ no~irmoMoB Q~(x), v -> 1 cnpa~e~r t~a eme oI~eHt~a (4.1 I).

IJ3 weopeM~i B c~Ie~IyeT, ~iTo

m.s+l

(4.15) Z Z b~ll~'k(X)l!N<=Bo Z ][Q,(x)li~<=Bo; n = 0 , 1 , 2 . . . . . 2n~v-<$ n+ l k = m v + l 2 n N v < 2 n+ l

Cor~IacHo ~epaBeHc~y Yapcmi (CM. [7] I4 [8, czp. 93, ~epa~encT~o 3.8.1]) rip14

KaX~OM V=>I CylII~eCTByeT iIepecTattoBKa % qiiceJi my+ l, m y + 2 , ..., m~+l, ~JI~

KOTOpO~ 1 m v + l

--< 16 f Z b~,O~(x)Jdx, 0 k = m . + l

rae q

- - MaxopanTa ~tacTrI~nl~IX CyMM IIO.IIHHOMa Q~(x) IIp~t IlepecTa~IoBxe %. I-IOaTOMy

It max Y2,~(Q~, x)]]~ <= Ill Z Y2L" (Q, , x)i[~ =< 1 6 ( B o + 1 ) , n =>0. 2kn<v_ < 2kn+ 1 2kn<=v<2kn+ 1

B c~ay JleMM],I 2 (cJleAcT~re 1) MO:H(HO B],I6paTI, Bo3pacTammHe HoMepa p , ,

raxee, nTO ~O~T14 BCm;ly na [0, 1]

lira 1 . ( max Q~(Q~ x ) )2=O. n ~ r / + l k k +1 ' 2 Pn~_v<:2 Pn

Cn.eaoBaTe~sno, ~epecTaBaennrifi pz~ no CnCTeMe T (cM. 3aMe~a~Ke 4), T0- nepecTanoBxa naTypaJlLnoro px~a, nH~y~npoBanI~ax cynepnoaHrmefi % ({%, v ~ 1 })

my + 1 bk

2 Pn~V<2 Pn

aBJI~eTc~ O,Z~IoBpeMeHHo ~, 7, a-yHr~Bepcam, m, rM.

TpnronoMerpnqecr.~e nysm-p~am 175

~J I a 3aBepmeH14R )~ora3aTe~r~cTBa TeopeMJ, i 3aMeTffM, qTo ecJ!14 ~ - - HOpM14-

pozau14an cI, ICTeMa, TO BB~I~y (4.15) ) ~ t IlpO143BOJlbHOrO e :>0

= ~ ' ( n + l ) : t + ~ / 2 2 ,~' b~ ~ < + ~ , . k. k- +1 k=mv+l n=O 2 Vn<~V<:2 Pn

3aMeqa1414e 7. 5Icao, qT 0 yTBep~,~eI-I14~I TeopeMt,I 40CTaIOTC~ B c/,I3Ie IIp14

6oJTee caa6J, ix npe~noJmxe1414ax OTHOC14TeJIb/-IO C14CTeMbI ~.t. B qaCTHOCTI,I, KaK

B14)!HO I43 )IoKa3aTeJIbCTBa TeopeMJ:,i, ,~OCTaTOtI140, qTO6/:,I ~ o6Jm~aaa CBOHCTBOM"

Cylt~ecTByIOT roHeqHa~ Mepa p, alCB14BaJIeHTHa~ Mepe 5Ie6era, l'tOC31e~oBaTe3IbHOCT14

dpyurwafr ulE~[o, l>--1, H MrloxecTa El, I>=1 c mEta l (I-*~) Tar14e, qTO ~JI~t

m o 6 o r o rmTepaaJm A = A , , j ( n ~ 0 , 1 <~j<_--2"), ~ c ~ a ~ > 0 14 aoMepa N : > I MOZ<UO

y~a3aTl, CyMMy

t(x) = t(A, ~, N, x) :- ~ bg ~ (x) (m >: N), k = N

yj~o~Jleaa~opammym nepaaencTBaM (4.3), (4.4) n uepa~encTsy (4.2) c g(x)=t(x) ~ cnpa~e)!JmaOCTU yTBep~en14J~ a) 14~14 Hepa~eHcTny

u~ b~p~(x) ~ ]lt(x)[]2 np14 ~cex t ~ 1 Xk =N L~(Et)

~ s cnpa~e~rmOCTU yTBepX~Ie1414Z 6).

B 3aKy[roqermc OTMeT14M, qTO pe3ys~bTaTbl HaCTO~tt[e~ CTaTb14 C HeoSxo,~14-

M~IMH 3JIeMeHTaMH I4X )IoKa3aTe21bCTBa 6bi~14 H3JIOXeH/~I B 1976 r o s y ~ IdlHCT/~TyTe

MaTeMaT14KIr A H ApM~14C~O~ CCP, a ~ora3aTea~cT~a TeopeM 4 ~I 5 (14 ~IeMM

1, 2, 3, 6, 7) - - B TOM x<e r o s y a a ceMrlHape ]I. E. MeH~IUOBa 14 17. JI. ~ g ~ r u o B a

17o Teoprm qbyHru~fi B MOC~OBCKOM y~mBepc14TeTe. B CBa314 C ~aeMMO~ 60TMeT14M,

qTO cy14~eCTBoBarI17e IlO:IrlHOMOB C onT14MaJii~itl, iMrl nepecTaHOBICaMl~ 17o crlcTeMe

X a a p a (a TaKxe nO C14CTeMaM qbyHrI~14fi, paccMoxpeaH/~lX B pa6oTe [1]), 17pmi14-

MatoIIIHX 3Haqerme 1 Ha pacm~p~mml~xca IIO)IMHO~KeCTBaX 3aj!a14rloro i~TepBaJla,

ycTarloBJIeHo My17ter~tt~OM [17].

JInTepaTypa

[1] d~r0 F. ApyTrOH~tH, IlpeJ~craBnerme H3Mep~tMblX qbyHIc/_q'r,~ NOqTI/[ BCJO~y CXO~ffLI~NHJIC~[ p n ~ a ~ , MameM. c5., 90 (1973), 483--500.

[2] H. K. ]3apr,, Tpueono~tempuuecKue p.~Jbt, qbrt3MaTrr~3 (Mocrma, 1961). [3] Y. S. CHow, Convergence theorems of martingales, Z. IYahrsch. verw. Gebiete , 1 (1963),

340~346.

6*

176 H.B. Horocsn

[4] J. L. DOOB, Notes on martingale theory, Proc. Fourth Berkeley Sympos., Math. Statist. Prob., Univ. Press, Los Angeles, Calif., 2 (1961), 95--106.

[5] N. DUNFORD and J. SCHWARTZ, Linear operators, v. 1, Wiley-Interscience (New York, 1958) - - H. ~artdpop~ ~ ~ar IIInapt~, flune~nbte onepamop~t, T. 1, Hnocxpamm~ smxepaTypa (Mocra3a, 1962).

[6] B. ~. Fanonticr~n, YlaKy~mprtbm psgbi n ~ie3aBHcrlMbLe ~yrzka~n'n, Ycnexu ~4ame3r naytr

21 (6) (1966), 3--82. [7] A. M. G~RSIA, Existence of almost everywhere convergent rearrangements for Fourier series

of L~ functions, Ann. o f Math., 79 (1964), 623--629. [8] A. M. GARSIA, Topics in almost everywhere convergence, Markham (Chicago, 1970). [9] I/I. IA. F o x 6 e p r r~ M. F. I(pe~H, BeeOenue e meoputo r neca~foconpaatcettubtx onepa-

mopoo, ~ri3MaTrtt3 (MocI<Ba, 1965). [10] R. F. GUNDY, Martingale theory and pointwise convergence of certain orthogonal series,

Trans. Amer. Math. Soc., 124 (1966), 228--248. [11] R. A. HUNT, On the convergence of Fourier series, Proc. o fS . L U. Conf. on OrthogonaI Expan-

sions, Southern Illinois University Press, Carbondale, 1968, 235--255. [12] O. C. HBameB--MycaToB, O roaqbdpnt~e~Tax TprcroHoMeTpH*leclcaX ~ym,-pa~oB, H~a.

A H CCCP, cepa~ MaTeM., 21 (1957), 559--578. [13] S. KACZMARCZ und H. STEINHAUS, Theorie der Orthogonah'eihen (Warszawa--Lwow, 1935) - -

C. K a ~ M a ~ ~ F. I I ITe .~ ray3 , Teopaa opmoeona/tbnbtx paOoa, r (MoCKBa, 1958).

[14] J. E. LIT'rL~WOOD, On Fourier coefficients of functions of bounded variation, Quart. y. Mar

7 (1936), 219--226. [15] D. MENCHOF, Sur l'unicit6 du d6veloppement trigonom6trique, C. R. Acad. Sci. Paris, 163

(1916), 433~436. [16] D. MENCHOr, Sur la repr6sentation des fonctions mesurables par des s6ries trigonom6triques,

Mame~lr c6., 9 (1941), 667--692. [17] F. UI. Mymer~H, O ~o~HurIenxax ncro~y cxo ;~r~xcz p ~ o ~ no HeKoTop~,~M nepec~a~e~-

r~st~ cneweMaM, PI~e. A H CCCP, cepx~a MaTe~., 42 (1978), 807--832. [18] J. Newu, Bases mathOmatiq~es du ealcul des probabititds (Paris, 1964) ~ :)K. He~6, Mame-

Mamu~ectcue ocuoa~t meopuu eeponmnocmeh, Mr~p (Moc~Ba, 1969). [19] A. M. One~c~H~, O ner~ozop~,LX oco6e:mocTnx p ~ o ~ ~yp~,e ~ npocrpancr~ax L p (p<2),

Mame~,t. c6., 77 (1968), 251--258. [20] H. B. H o r o c z n , O60Z(HOM eBO:~CTBe nOe'lH~,lX opTormpM]~po~anH~,rx CHCTeM, M a m e ~ ,

za~emmt, 17 (1975), 681---690. [21] H. 13. I I o r o c z n , Hpe~cTaenew,~e n3MepHMs~X dpyH~un~ opzoro~mns~arcr~ pn~aMrr, Mame~a,

c6,, 98 (1975), t02-:112.

[221 R. S~L~M, On singular monotonic functions of the Cantor type, J. Math. Phys., 21 (1942), 69--82.

[23] A. C. SCHAZrr~g, The Fourier--Stieltjes coefficients of a function of bounded variation, Amer.

J. )Y[ath., 61 (1939), 934--940.

[9_./4] A. A. T a n a ~ H , Ilpe~cTaB~emte ~3MepnM~,~X ~bya~c~n~ pnaa,-vna, Ycnexu ~atne~vt. ~ay~r 15 (5) (1960), 77--141.

[25] A. A. Taaa.,~ar~, Tpnro~o~eTpn~ec~ne p z ~ , ym~nepcaa~,~ae OTnOCHTenbHO ~o~p~r~O~, II~e. A H CCCP, cepn~ MaTeM., 27 (1963), 621--660,

[26] H. 5 I . Y ~ , n ~ o ~ , Pe~eHm,~e r~ rmpemer~n~e r~po6~aeM~ Teop~[~ Tpr~ro~oMeTp~4~JecKa~ :~ opToroua~'u, nr, ix p~o~ , Ycnexu Mame,~. nayh, 19 (1) (1964), 3--69,

Tp~roi~oMeTpi4,~ec~e Hyal~-p~a 177

[27] t7.5I. Y ~ , a n o B , 17pe~cTamlem~e d~ynKlm~ pa)~aMH n K2Iacct,i 9(L), u ,uame3t. nayK, 27 (2) (1972), 3--52.

[28] N. WIENER and A. WINTNER, Fourier--Stieltjes transforms and singular infinite convolutions, Amer. J. Math., 60 (1938), 513--522.

[29] N. WIENER and A. WINTN~R, On singular distributions, J. Math. Phys., 17 (1938), 233--246. [30] A. ZYGMUNO, Trigonometric Series, v. 1 (Cambridge, 1959) - - A. 3~irMyrr~, TptteonoMem-

puuecKue pa3bt, T. 1, Map (Moc~ZBa, 1965).

On the coefficients of trigonometric null-series

N. B. POGOSYAN

Let {ek}~~ be a numerical sequence satisfying the conditions

Ok40 ( k o ~ ) and ~ a ~ - : + ~ - k=O

It is proved tha ~, there exists a trigonometric series

a~ cos 2~ (kx + 0~) k=0

where lek] <--Ok, k = 0 , 1, 2, ..., possessing the following property. For each measurable and a.e. finite function F(x), xC[0, 1], the numbers dk=0 or 1, k=0 , 1 . . . . . may be chosen in such a way that the series

dke;, cos 27r(kx+Ok) k=0

converges to F(x) a.e. on [0, 1]. In addition, if F(x)~O, then 8goe'~o~O for at least one k0=>0. Certain generalizations are discussed, too.

CCCP, EPEBAH 375 010 YJI. XA~)K,qHA 5 APM$:IHCKHITI FOCY.~,PCTBEHH/alIYl IIE,I~AFOFI~qECKI417I HHCTHTYT FIM. ABOB,qHA

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