On the order of growth of rectangular partial sums of double orthogonal series
We obtain estimates of the order of growth of rectangular partial sums of double orthogonal series and establish their unimprovability on the set of all double orthogonal systems.
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4728 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510888117665792 |
|---|---|
| author | Andrienko, V. A. Андриенко, В. А. Андриенко, В. А. |
| author_facet | Andrienko, V. A. Андриенко, В. А. Андриенко, В. А. |
| author_sort | Andrienko, V. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We obtain estimates of the order of growth of rectangular partial sums of double orthogonal series and establish their unimprovability on the set of all double orthogonal systems. |
| first_indexed | 2026-03-24T03:04:09Z |
| format | Article |
| fulltext |
Y/IK 517.5
B. A. AHRpI4eHKO (IOa<Ho-yxp. heir. yu-T. OJtecca)
O I I O P H ~ K E P O C T A
IIPJ:IMOYFOJ-IbHbIX HACTHbIX C F M M
~BOI~I-IBIX OPTOFOHAJIbHbIX P H ~ O B *
Estimates of a growth order of rectangular partial sums of double orthogonal series are obtained. The
fact that these estimates cannot be improved on the set of all double orthogonal systems is established.
O'rpHMallo oItillgH nopaltgy 3poc'ralnl~i rlp~lMogy'rllliX NaGTKOBHX cyM nolu,iflmtx OpTOFOIIaJn~IIHX
paltin. BcrauoJute.o ix ocraroqHicrh ua MIIO~mli ncix noludtlmtx oprol'olla~ll,llnX CHC'rCM.
1. Bne~eime. B HaCTOatae~t CTaThC pr B. H. Ko~a~u [1, c. 38 - 44] ncpe-
HOCaTCa Ha ASOflH~r OpTOrOHa~bHt~r panbL
I'IyCTb (X , ~, I~) - - n p o c r p a a c T n o c no~ox14TC~bHo~t Mepofl, Z+ 2 = { n = (n 1,
n 2 )} -- MIIO>KCCTBO TOtleK IIZIOCKOCTII C ~CZIMMI, I HCOTpI.IIIaTe.rII, HI.,I~II, I KOOp/lllHaTa-
M14, tp = { Cp,(X), n e Z~ }--]18OflHaa opToHopMrtponaaxaa CHCTeMa (OHC) aa X,
a �9 MHOmeCTSO 8cex Tamxx CHCTCM, { a,, n e Z+ } ~nsotlHaJ~ noc~qenosaTe~qb-
HOCTIb nC~CTBHTCZIbHblX HtlcezL FIoJIO~'KHM ~ZI$1 TOqeK m, n, N E Z~, 0~ -- ( 0~ I,
~ 2 ) , x = ( x l , x 2) e P~, no onpene.ne14mo, m + n = (m I + n I , m 2 + n 2 ) , Ilxll =
Ct =x,~2, �9 = ( ~ ' , x~;), ~c~m xi2+ a ~ > 0 ( i= ~,2), N = (N,N) ( N = 0 , l ,
2 . . . . ) , a y c z o m x ~ c z , aXO m < n r o r z l a 14 TO:Z~KO Torzla, K o r e a m i -< n i ( i = 1, 2 ) ,
14, c:~e/IosaTeStbHO, n > 0 <=> n e Z+. KpoMe TOrO, 06o31ta~rlM p (n ) = ~ 1 2 + n 2 .
PaCCMOTp14M ~moflnoti opTorona~butaf l p a n
~.~ a,,q~n(x), x e X, (1)
n~0
KOTOpbI~t r, iO~KeT pacxonrr r~ca =m MHO:~.eCTBe 14OJlO~'~rlTe:lbHO~ Mepbxo H3yqrlM 3a~a-
qy 0 n o p a n g e pocTa e ro npJ~MoyroomH~X qaCTHUX CyMr, l
s,,(x) = ~ a,,,~p,,,(x), m, n E Z2+,
I I |~ / I
B 3aBllCHMOCTI.I OT n o p a n K a y 6 u n a m ~ a Ha~ Bo3pacTarma KO3~rIm~eHTOB p a n a (I).
B HaCTO~tRee SpeM~ H3BeCTH~ J]nUab qac'rHMe c.r[yqar~ pellleHH..q 3T0~,~[ 3a/laqH (CM. [2,
3]). 3 / lecb M~a I'IpHBeneM, Ha HallI B3I'JI$ln, rlcqcprlblBalottlHe i40KOHqaTC.rlBHSIe pe-
3yJlbTaTM.
Yc.rlOBHMC~I B naJlbtle~tl/IeM n.q~! 3a~aHHO~ ~BO~HO~[ IIoc.rlCnOBaTe.rlbHOCTH C~yHK-
L 2 Rm't { fn(X), n e Z.~ } a s a ~t~;armoil n~x~flrm[4 nocncnosaTe~r,HOCTH { 7n, n e
e Z+ ~ } noaoa~aTe.rll, r lux ~ l c e n nrlcaTl, f n (x ) = Ov(Yn I ), ec.arl y , J , , f x ) --> 0 n . s . ,
�9 qa~mqHo nolutep~ana rpmrroM N ~ APU 061002 Me~ltyllapolmOll Copoco~cgofl nporpa~Mra no/l-
ltepaggH c~pa3omulxa B o~la~t~4 TO~Ulb/X lmyg ~5 Ygpamle.
�9 B. A. AHJ1PHEHKO, 1999
I$SN 0041-6053. Yl;p. ,num. ~.'vpn.. 1999, m. 51, IV'-' I0 1299
1300 B.A. AH/:IPHEHKO
KOr/Ia max (n t, n2) "-> oo (~taH Kor/la rain (n t, n~_) ---> o, ) It c y m e c ' r s y e r d~yHKttrm
F(x) ~ L2(X) TaKa.,q, qTO
supT, I f , , (X)I < F (x ) n.~. na X.
I !
~IJIJI ~BOI~ltOI~ IIOCJle/3,0BaTeJ'IbHOCTH {~.(tl), I1 E Z+ } 6yAeM llHCaTb
A l 0 X ( n ) = Z , ( n l , n 2 ) - X ( n l + l , n 2 ) , A 0 1 X ( n ) = X ( n l , n 2 ) - X ( n l , n 2 + l ) ,
A tlMn) = Z , ( n t , n 2 ) - X ( n j + 1 , n 2 ) - X ( n l , n ~ _ + l ) + Z , ( n l + l , n 2 + 1)
H 6yneM roaoprrrb, qTO {gOt)} He Bo3pacTaeT, ecJIH AlO~,(n) >_ 0 I4 AOl ~,(n) >_ 0
a./ia 8cex n > 0; {X(n)} c-rporo y6bmaeT, ec.rl~! Alog(n ) > 0 14 AOl ~.(n) > 0 /IJlZ
Bcex n_>0; {X(n)} lle y61,maeT, ecJIH Alog(n)__. 0 u A01k(n)< 0 / I~z Bcex n>_0;
{X(n)} CTpOrO aoapacraer , e c J~ Am~,(n) < 0 r~ a0t3,(n) , : 0 a.rt~ ncex n > 0-' :Flo-
c~lejaoBaTeabltOCTi, { X (n)} Ha3SIBalOT BblnyK.rlo~l, ec.rlrl All ~.(n ) _> 0 /IJ-tz Bcex n >
> 0 .
lIyc 'rb { v,~ }, i = I, 2; ni = 0, I . . . . ; v 0 = 0 , ~ cagoro Bo3pacTamtmle no-
cJle/~OBaTeJIbHOC'rH lleJlblx qrlceYl. TOF21a ~BOi4HylO nOC.tle,~oBaTe.qbHOCTb { V, =
= ( V, , , V,,., ), n ~ Z~ } 6yae~ naabmaTb pemeTKo~i n Z~.. PaccMoTpm, x c ' rporo ao-
3pacTalomym rt Henpepumlym no K a ~ o ~ i ~oopammTe ne~Top-qbyHKUr~m ~ V (t) =
= (V ~(t t) , V z(t~_i) "raKy~o, qXO V (n) = V,, H o/~Ho3na'mo onpetm-~em~ym o6paTaym
t~e~Top-dpytmmno v-~ (t) = ( v7 ~ (t I ),, v- ~1 (t 2))" O6o3na~H~ ~epe3 [ x ] ttezy~o ~acaa,
qHCJ-la X, IIOj'IO)KIIM
q,, = ([vll inl) 1, [v~'(n2)] ) - (q,,,, q,,,) (2)
I.t(n) = v , , + l - V , =-- ( I . t l (n l ) . I '=(- 7.)) = (v,, +l - V,, , v,,z+l - v , , , ) . (3)
2. BcnoMOraTe.~bHb~e yrnep~Remt: :
dIeM~a I . Hycmo cxoOumc~ Oooimoii ,~uc~ooo~ p~O c neompu~ame~,m~tu
~t./leltahttl
E ~ a i ' " (4)
i=0 k=0
TozOa cyu4ecmaytom mamw noc.~eOooame~bnocmu
0 < l~(i) "~ ~ , 0 < 12(k) 1" ~ , (5)
~mo p.~O
~ ail~l~(i)12(k) (6)
i=0 k=0
cxoOumca.
~oga~me~t~cmeo. Ecmt papa (4) He CoRep~KHT OTJIHqIIbIX OT Hy.rl~t *-IJ-IeFIOB, YtH-
60 TaKOBIalX KOHeMHOe qHC.Y[O, TO B Kaqec'rae l i(i) rI /2(k) MO)KHO B3$1Tb TllO6bll~ l10-
cJaeztoaaTeJn,HocrH (5).
I'IycT~ Tenepb papa (4) coaep~r r r 6ec~:oHe~Hoe ~ a c a o ~J~enos a ik> 0 . Pacc~oT-
prim TpH B3aHMOnCKJ'llOqaI0mnx cJayqa.a: 1.) a ik = 0 IX~lJ~ acex i >. i o > 1, k >_ O, a
aae~en 'na a ik > 0 t t ~x 6eCKOHeqHoro MHOT, CeCTBa HI-I/~eKCOB (i, k), norta/lalott~rix B
,.no~ocy'" i < i o, k__.0; 2) aik=O /UDIBcex i>_O, k>ko>_ 1, a aJ~e~errr~ aik>O
~aJ~ 6 e c K o n e ~ o r o ~Ho~eeTaa nn;aeKcoa (i , k), nona~ammax a n o a o c y i > 0 , k <
< k 0 ; 3 ) .~Jm J~o6oro i_>l na~i~eTca k=k(i)>.O TaKoe,~'ro a i~> 0, n a a a am-
6oro k>_ 1 nalttteTCa i=i(k)>-O Taxoe. qTo aik>O.
ISSN 0041-6053. YKp. ~am. ~'vpu.. 1999. m~ 51. i~- I0
O I]OPfl/1KE POCTA HPflMOYFOJIbHblX HACTHIalX CYMM ... 1301
PaccMOTpnM cHaqaJm noc~ezlHnit cnyqaIt. HOCKOnbKy cxoIIrrrc~ p~tt (4), CXO-
~aTCa nosTopnee pa/Ir~
}~a~ec, B czyqae 3 HMec~I
m=i+l n=0
Tor~a. CoF~acHo TeopeMe
FloJmraz
~ ( ~ a i k ), ~ ( ~ a , k ) .
i=0 k=0 k=0 i=0
> 0 , ~ ~amn>O ~i>O, k>O.
m=0 n=k+l
s A~a Jzm6oro ~ e (0, 1 ) cxoA~rrca paara
i=0 k=O m=i+l n=0
{ ~ a i , ( ~ ~ a , , , , , ) - r �9
k=O i=o m=O n=k+l
ml(i)= ( ~ ~a,,=) -ct, m2(k)= ('~, ~am,,) -a,
m=i+l t~=0 m=0 /l=k+l
aaan~L q ro 0 < m t (i) $ **. 0 < m 2(k) 1" ~ rl CXO/I~ITC~I nosxopnue p z a u
(~ml(i)aik). ~ (~m2(k'ai,).
i=O k=O , k=O i=O
a c.rte~OBaTedlbHO, CXO/1ZtTC.q/~BO~tllOl4 pJl/I
~ [m,(i) + m2(k )]aik.
i=0 k=0
HoczoasKy
TO. no,nara~
ml(i)+m2(k ) >_ 2~/ml(i)m2(k),
l l ( i ) = ~ , 12(k) = ~ ,
noJtyqaeM, wro mano.nHaloxc~t yc.nomla (5) H cxo.awrca pz.a (6), q'ro a xpe6osa:tocb
/~OKa3aTl~.
B c~yqac 10qeBrl/~HO. qTO B KaqCCT~e 11(i) MO~HO B3$1Tb nIO6ylo nocne~oBa-
TenbnOC~ ll(i)H3 (5). a n Kaqecrse /2 (k) - -nocneaosaTen~aocaTs m2(k ). Aaa-
JIOFHttHO. B cnyqae 2 B Kaqec'rBe 12(k) ~IO~KHO B3nTb nIO6ylo noc~eRoBaTenbHOCTb
12(k) n3 (5). a ~ Kaqeca~e l l ( i ) - - nocneaosaTen~noe~ m I (i).
d-Ie~Ma 2. llycmb nono.,~ume/lbnaa noc.ne~oeame.nbnocmb { 3.(n). n e Z+ 2 } ma-
Ko6a, ,~mo
L ( , 0 = II In (n + 2)II / 1"
u 8,a.~ 8 a , n o a n o c n e O o ~ m e m , n o c m u { a n, n e Z~ }
Z a~ ;k2(n) < =o.
n__.0
TozSa O'utecmsyem no.no.'rumenbna.~ nocneOoeameabnocmb { A(n ) , n e Z~ } ma-
Ka.~, ~mo
ISSN 004!-6053..Yrp. ~tam. ~.'vpu,. 1999, m. 51. N e I0
1302 B. A. AHnPHEHKO
A(n)/~.(n) I" ,=*. (7)
n>0
II ~n(,, + 2 / i l / h i , r ) "t ** . at, t o y o t a . (9)
]][oKa3ameAbcmeo. Cor.rlaCHO JIeMMC 1, Haii~yTC.~ ABe I1oc.rlC/~OBaTCJIbHOCTH
ISSN 0041-6053, Y~:p. ~tam. ~ . p n , 1999. m. 51. N e I0
0 < li(ni) "~ ~ , i = 1 ,2 ; n i ---0, 1 . . . . . TaKtle, qTO
~., a,', ~.-" (n)l~Oh)l~(n2) < **. (10)
n>o
Pacc~toTpHM K•acc ~yHKUrltt ~ , onpeaezerm~x Ha nozyocH [0, + **):
~F = { V ( x ) : W(x)> 0, V(X)'I" +~ , V(x)x-i , l , 0 npn x--->**}, (11)
14 ero IIORKJIaCC
W0 = {W r qJ : v ( L ( n ) ) = L ( n ) s 0 <eiOli) ,[, 0
npH ni'-->~, e l (h i )< a / " ~ ' ~ l i ( n i ) / I n ( n i + 2), i = 1 , 2 } . (12)
Tor~a (cM. (10) - (12)) a.nz noc.nettoBaTe.abrlOCT14
A(n) = ~ . (n )v(L(n) )
manoaaalOTCa ycaomm (7)-(9) . ~e|tc'mnTeaSno,
A(n) /~ , (n) = v ( L ( n ) ) $ **.
IIa~ee,
E = E <_ E <
n~O n20 n>O
HaKOHe1.~, l'IOC JIg/lOB aTeJlbHOffl'b
II tn (,, + 2) II/A(n) = II tn (,~ + 2) II/~,('~) L( . ) ~ ,(n ~) e~_(n 2) =
'- 1 ]~:l(nl)E2(n2)"[ "~,
14 etanyz.rta zaK npottaBe~emle BoapacTammHx nocae/~onaTeabaocTet~ 1 /El(hi), i =
= 1 , 2 .
JIeHMa 3. Hycmb nono~umenbnaa nocneSooamenbnocmb { ~.(n), n r Z~. }
cmpoeo oozpacmaem tr ~, . Tozc3a ycnooue
3 . ( n + l ) = O { X ( n ) } . n e Z~., (13)
neoaxoau.~to u Oocmamo,~no c3n~ cyu4ecmoo~anu~ peutemKu { v n = ( vnl , vn~. ). n
r Z~}, yaoonemoop~lot.eflyc~oou~t
vt~+l -Vn> 4, n_>O,
~'(ml' v"-'+l) <_ r (14) I < p < ),.,(V,, +l, m2) _< r, 1 < p < ~.(ml, Vn.,)
~. (V,, I , m2)
~n~ n,oa,,.~ (m t, m2) ~ Z~.
] Ioxa~amenbcm~o . Heo6xoOu~tocmb ycJma14a (13) oqeBrl~Ha, ~OKa3tc.eM ero
8ocmamo~nocmb. rlyCT~ ~,(t t, t2) ~ CTporo Bozpaeratomaa no Ka3K/I0tt nepeMeu-
HO~, Henpeptannaa t~3yHKHH$1, x~la~omaxca rrpo~o~eHneM noc~e~tonaTe~bHOCTa
7~(n ~, n2) Ha ~aa~parrr t] >_. 0, t 2 >- 0. CoraacHo ycaosato, cymecreyer rtoeroan-
Ha~ C > 1 TaKag, trlo ]~Jl#l l~ex n r Z~
0 I'IOP,q]IKE POCTA nP$1MOYFOJ'IBHHX qACTHHX CYMM ... 1303
Ho~o~nM
Tor~a
~ , ( n I + 1, n 2 + 1 ) < C~,(n 1, n2).
[3( t l , t 2) = l l o g c k ( t l , t 2 ) .
~,(n I + 1 , ~ ) < 1
[ 3 ( n l + l , n 2 ) - f J ( n l , n 2 ) = log c g(nt, n2) - ~ ,
~,(n l , n 2+1) < 1
f j ( n l , n 2 + l ) - f J ( n t , n 2 ) = log c ~(nl, n2) - ~.
PaccMoTprlM QbyHKRHIO ~ (t t , t2), 3aCI~HKCHpOBaB O]~Hy H3 nepeMeHmax, nanprIMep
t2 = 4 - (l)yHKtl]4$1
1 IOgc k ( t l , n~) B(t~) = I~(',. ,~) = ~
CTpOFO Bo3pacTaeT H, cJIeAOBaTeJII,HO, ameer O]IHOaHaqHO onpeRe.rIeHHylo o6paTnyao
qbyHztm10 V (t I)- IIo.no~nM Vn~ = [v (n I )], rAe [x] ozHaqaeT LIeJ'lylo qac-n, aHc.na
x. TorAa
B(Vn,) = B(v(n t ) -Sn I) -< B(v(nl)) = h i , 0 < 5n, < 1,
1 > B(v(nt)) - 1 1 B(Vnt ) > B(V(nl)-8, , , + 1 ) - ~ _ ~ = n 1 - ~ .
H3 3THX HepaBeHCTB BHTeKaCT
6 _4 _< ~(v,,,+0 - B(v,,.) _<
5
c 4 < ~.(v,,,+b ,~)l X(v,,,. ~) ~ c 6.
KpoMc TOrO, IIOCKOOlbKy
Vnl+l--|
1 B(Vnl+I)-- B(Vn,) --" Z [ [~(k+|' n0)-l~(k' n20)] < 5(Vnl+l'Vn,)'
k=vn I
TO ]~JD! Ka3K~OFO n I = 0 , l . . . . BblHOJIH~IffFC~ HcpaBeHCTBO
v,,,+,- Vn, >- 5[B(Vn,+,)- a(Vn,)] > 4.
AHaJIOrHHHO onpoAe.rlRe'rc$1 HOCJIe]XOBaTeJIbHOCTb {Vn2 } H yC'I'aHaBJ'IHBalOTr He-
paseHcTna
C + < },,(n O ,v,,.,+t )/~.(nl 0,vn2 ) < C 6 . vn,+ l-vn2 >_. 4.
B wrore peme~a {Vnt = (vnl, Vn2) } $IBJI$[OTC~I HCKOMOfL
3. Ocuoenue peayolbTa~. Cnpane~mmo c~eA}qomee yrsepxcseHne.
TeopeMa 1. Hycmb { m n, n r Z 2 } - - p e r u e m K a o Z 2, a nocaeOoeameab-
nocmb { ~. (n) > O, n e 22+ } maroaa, ,#no
X(m.)/II In (n + 2) II J, o. (ts)
1. Ecau
15$N 0041-6053; Yrp: .uam. ~3'pn., 1999, m. 51, bff 10
1304 B.A. AH/~PHEHKO
~. a~.2(n) < ~, (16)
n20
mo On.~ mo6ofl OHC (p H (~
sin(x) = { I1 in (. + 2)II / }. (17)
tcozOa max ( n l, n2) ''> o. (u, sna,um, ~ozcga rain (n~, n2) ~ oo).
2. Ecnu ( X , "~', p.) , - npor c ~one~noa. nono~rumenbno~ neamo~ume-
cKoa ztepo~ u % (n) >_. q > 0 cgnn 8cex n >_ O, mo cOn~ mo6oa nono~umenbno~ no-
c/zecgoeamenbnocmu { v ( n ) , n .H Z~. }, v (n) = o (ll In (n + 2)[[), tcozcga max (n t ,
n 2 ) .--> .0, cyu4ecmo3;em opmozonam, m, tff pnc9 (1), ~:ogqbdpu~uenm~ t~omopoeo yOoo-
nemoopmom yc.~oeuto (16) u c3n~ ocex x H X
lim sup X(m.) [v(n)]-I s,,,,(x) = ~*.
max(n I ,n2 )--~**
fl[oKa3ame.abcmeo. 1. Pacc~OTpHM c H a q a n a cny.qa;i m n = n . COFJIaCHO neMMe
2, 9ytt~ecxByeT nOaO~KaxenbHaa nocneao~aresmHOCT~ {A (n), n H Z 2 }, y/IoBneT-
aopx~maa ycn0nnZM (7)=(9).. Ha ocnoBaHm~ O606USenHor~ TeopeM~, Men~tuo-
~a" Paae~axepa (CM.. "iianpn~el 5, [4], cneacr~ne2), n cn~ny (8),~:na mo6oWOHC q) H
H �9 n. ~. perynapHo cxoz;nTCa pan;
Z a.A(n)II In (n + 2)II -! %,(x).
. >0
OTcmaa n H3 (9), cornac.o neMMe KpoHeKepa-Mopmta (c~t. [3], TeopeMa 1), cneay-
eT (CM. a:aK~e (7)) oueH~a (1.7):
s,,(x) = o {llln(n+ 2) l l /A(n)} = m a x ( n ~ , n 2 ) ~ * * .
B c.ny,~ae FIpOH3BO,,rlbHOIYI pemeTKn-{ m ,,} /~oKazaTenbcrBo aHanor'HqHO .ao~:aaa-
Tem, crBy q . l Teoper, tu 3 a3 [5, c. 1313-1314].
2. B cny~ae, Korea nocne~oaaTem, aocT~ { t ( n ) } 0Tt~eneaa OT nyna, ~oKa;~e~
o~on,~aren~noea'~ OaeHKn (17), T. e. yamepaUlenne 2 Teope~a. 3aMeTnM, qTO COOT-
aeTCTaylOl/IeC yraep.,-~enne inz O/XHOKpaTHblX p,.q/IOa Bepao (cM. [1], veopeMa 8) He
TO.m,KO na oxpe3Ke [a, b], no n a e.ay,~ae npoaa~o.m, Horo npoc'rpb.n6"raa ( X , ~,:p.)
c KoHeqnoa noaoxrlTem, not~L HeaTOMnqecKo~ ~epoll (CM. [6], aaMe~arme 1). IIo-
~TOMy nonara~
co(nl) = v(nl,2 ), ~,l(nl) = )~(n~,2),
m,m~M, .TO X~ (n 1) >- q > 0 rt (o(nl) = o ([n(n 4 2)) r,, cne~o~aTen~Ho, n anz;eTca
OpTOrOaanbnbI~ pa~
' ~ c,,, 4,,i,(x) (18)
nl=0
TaKOit, qTO
hi=0
rl ~Jt~. BCCX X E X
lim,,,..~sup s.,(x)l = lim.,~.sup I~oCk,= ~k,(X) = **.
PaccMcrrpHM ~aoJl~oll olrrorona~sn~II p~a
ISSN 0041-6053. Yh'o. ~tum. ~'ypn., 1999, m. 51. N e 10
O FIOPR~KE POCTA I'IPJ:IMO~YFOflbHblX HACTHblX CYMM ... 1305
rlo.rlaraJt
~.~ ancpn(x), (19)
n>0
I Cnl, C C . ~ n 2 = 2;
a n = anln2 = n 1=0,1 . . . . .
[ 0, ec~n n 2 ;~ 2,
H orlpeae.rlaM/~BOt~HylO OHC {%z(x) = cp.l,,2(x) } Ha X Tax :~e, KaK B TeopeMe 3 H3
[6], pacno~o~Krm ee B mlae MaTpHll~ (~0nln2 (X)), n 1, n 2 = 0, 1 , 2 . . . . . rtporI3so.rm-
H~aM 06pa3oM TaK, q'ro6ta ~m~ 3JICMCHTOB CTpOKH n 2 -- 2 BhlI'IO21H~JIOCb yc~omle
(p,,j,2(x) = ~,,j(x), nl = 0, 1 ,2 . . . . .
Tor~aa ~aBOtlHOil pan (19) no 3TOit CHCTeMe cosnaaaeT Ha X c OaHoKpaTakrM paaOM
(18) H HO3TO~J ero nacTnaa cyMHa Sn(X) = SnI(X), CCZH n _> 2, a ZO~dpqbHIl/4CaTbl
y]~OB#-I~l'BOp.qlOT yC./IOBHIO (16). ~a~ee, zVm YlIO6OFO X �9 g HMeeH
iim sup X(n) ~ l max(nt,n2)-~** V(n) $n(x)L >- iimnt_~**sup t~ snf(x)[ = o0.
B cJ1yqae npOn3BOJ-IbHOt4 pemeTgn { mn}, acnoJ]~3ya paccHoTpeHHloll~ csiyqalt
m n = n, npHMeHaeM cxeMy aoza3aTe~bcTBa q.3 Teopesu 3 a3 [5].
3a.~e,~auu~ 1. Ecz• r a n = n , TO BCzyaae { a n } e 12 H X ( n ) - - 1 no~JaHM
qacTmalt pe3yabraT Mopm~a [2], a B cJlyqae, Korea a,, = 1 ~tza scex n > 0, a BMe-
CTO X(n) s3Jrro X-I (n) I I In (n + 2)II , rz~e X(n) $ o. H y~tosJ1eTBOpZeT aeZoToptaM
aOnO~HHTe~HUM ycnosrtaM peryaapHoc 'm, ~ ~acTau~q pezy~mTaT Mopmta [3].
2. B ycJlOBa~X TeOpeMl~ 1 1~OCJle/~OBaTeJ]bHOCTh in n HOUGH0 3aMCtlHT~ 2no6olt
noc~e~tosaTe.abHOCTbIO a (n) $ **, HMexomet~ TOT ~ge nopa~aox pOCTa.
PaCCMOTpHM Tcneph noapo6Hee c~yqai~, zor~ta ~ (n) ~ 0, max (n ~, n 2) -~ **-
T e o p e e a 2. l'lycm~, cmpozo oo3pacmatouca.~ K ** nocneOo~amenbnocmb { 0 <
< X ( n ) , n e Z 2 } upezuemKa {ran, n e Z~.} ma~coot~, ~mo A ( n ) = X ( m n ) yc~o-
oaemaop.~em yc.weu~o (13) ,pememga { v n, n ~ Z~. } onpeOenena no A (n ) ne -
paoencmoa~tu (14), a noc~ec~ooameabnocmu { qn, n e Z~. } u { ~ ( n ) , n e 7.2+ }
onpe~enen~ coomoemcmoenno s (2) u (3).
1. Ecnu
9 --9
a~ X "(n) < .o, (20)
n~0
mo cg.a.~ np~t~toyzonbnbtx qacmttbtx cy~t~t s n(x) p~tga (1) no n~oSoa O H C r a �9
n. a. na X oepna ottenKa
Smn(X) ---- Ox{~'(mn)II (q.)II }, max(,,~, n2) .--> -0. (21)
2. Ecau ( X , ~ , I t ) ~ npocmpancmoo c Kone~mo~, nonoJKume~bnoti, neamo~tu-
,~ecKoa ~tepoa, mo On.~ n~oaoa nocneOo~amenbnocmu (o ( n~, n 2 ) - - > 0 . , KoeOa
max ( n ~ , n 2) ''> **, naaOemc.~ opmoeonanbn~a p.qz3 (1) , yOoenemoop.atou4utl yc no-
su~ (20). maKoa, ,~mo ~c~o0.3' na X
lim sup _ . ~ . . . . ]sm. (x)I = **. (22)
max(nl,n2~'>" Z(mn) U In lX(qn) II
,l~oga3amentcmeo. 1. PaccMoTpHM cHaqaJta c,ayqatt m n =" n . l ' Io~KoJlbXy H3
ISSN 0041-6053. YKp. ~lara. ~.'vpn.. 1999, m. $1. N~ ]O
1306
orzpezle.nenr~a (2) rtoc.ne,~o~aTe.abnocTH { q,,} Bbrre~amT nepaBeHc~a
Vq, <--n < Vq.+l, l! ~ Z+,
TO
r/~e
B.A. AH~PHEHKO
Ql(n) = {m ~ Z 2, Vq, m<-m I_<n 1, O_<m2_< vq,,},
Q2(n) = { m ~ Z~., O-<ml-< v%,. Vq,,. +l_<m2<n2} ,
Q 3 ( n ) =-{m~ Z~,, Vq,,i + l < ' m l < n 1, vq,,. +l_<mz_<n2}.
B ca~y (14) n (20) HMeeH (3/Iecb rl z~a~ee anTerpa:l~ 6epyzc~l no X)
Z X-2(Vq,, ) f S2q,, (x)d~t(x) = Z ~'-2(Vq,, ) Y' a~n =
II~0 n>O In~Vqn
= Z a~,. Z ~-2(Vq,, ) = 0 (1 ) Z a~, )C2(m) < ""
m>O Vqn~m m>O
OTC~O/~a cors~acHo TeopeMe JIeBH BI, mO~M, qTO npH max(n I, n2) --'> oo
s%,(x)= Ox{X(Vq,,) } = o.,.{X(n)}. (24)
B cn~y O666ttlenHol~ ~eMMLa Metmmo~a-P:~teMaxepa [7], ~ ~ 6 o r 0 n e Z~ cy-
mec~yer qbyaKuilJl 5,,(X) > 0 TaKaJl, qTt: (CM. (3))
t.~meQ3(n) �9
5~(x)d~t(x) = 0 {lllntx(q.)ll} ~ a~..
Vqn <ra<Vqn+l
Tor~a B crt:ly (14) nMeeM
~, ~-'-(Vq,,) Illn ~t(q,,)ll--" f ~2(x)dl.t(x) = O(I) Z X-2 (Vq,) Za,n2 =
tZ_>0 It~O Vqn ~m< Vqn+|
- - o(1)Z Z ' - " _- v,. , o, 1" a~ X "(m) O ( 1 ) X-'0n) < .o.
11~0 Vqn ~nl<Vqn§ 11l>--0
O T c ~ a cor~acHo TeopeMe Yleea npn max'(n 1, n 2) -'~ 0o c~egyeT otteHKa
Z amCPm(X) = ~ )llln~t(qn)[[} = ~
meQ3(n)
~aaee npm~eHaa neMMy MeHbtUOBa--Pa~eMaxepa K CyMMe
~_~ am q~m(X),
meP.i (n)
B CriSP/(14).no.ml'~aeM
(25)
ISSN 0041-6053. Y~p. ~lam. ~.'vpu,, 1999, m. 51, bl ~ I0
3
Sn(X) = Sv.(x) + ~ ~ amCPm(X), (23)
i=l m~Qi(n)
0 I'IOP,fl,/2KE POCTA FIPJ:IMOYI"O./IbHblX qACTHblX CYMM ... 1307
{ II E ~'-2(Vqn)ln-2itl(qn,)f max ~_ a,.cp,.(x) dit(x) =
n>_O vq., ~m<vq,+ I ,nEQl(n) IJ
= oo) E ~-2(v~,,) E ~?~ = o(1) E a~-2(m) < ~ ,
n>O Vqn "gm<Vqn+ I #n~O
OTKy~ta C rioMoml, lo TeopeMu fleBH BUBOaHM ottermy (max (n l, n 2) -'> 00) :
]~amtPm(X) = ox{~, (v , i ) i n i t l ( q , , i ) } = o . r { ~ . ( n ) l n i t l ( q n t ) } . (26)
meQl(n)
AHa.nornqno no~yqaera ottenKy
E a m t P m ( X ) = ox{~.(n) lnkt2(q ,L, ) }, m a x ( n l , n 2 ) - - > *o. (27)
meQ2(n)
Tenepb (21) cneztyeT rl3 (23) - ( 2 7 ) .
2. BTopoe yTBep:,K/leHHe TeOpeMbl /ioKaautaaeTca Tax :~e, xax B TeOpeMe 1, C
yqeTOM TOFO, qTO B c.ny,~ae O/~HOKpaTH~X p.~/tOa r162 yTBep,~JIeltl.le
nepHo (CM. [1 ], ~oKaaaTe.rmc'rao TeopeMu 9).
C.nyqait rlpOH3BO.rlbHOfl pemeTKrl {m, ,} ltcqeprlblaaeTca npriMenemleM cxeMbl
noKaaaTe.nbc-raaTeopema 3 xla [5].
C nO~,tOtUblO Teope~,lU 2 no.ny,qaeM "raKo~ peay.rlbTaW.
TeopeMa 3. Ilycmb cmpoeo aospacmatouta.q K o. noc.aeOo6ame.abnocmb { 0 <
< X ( n ) , n ~ Z~ } u pememtza { m . } yOos.aemeopmom oOno,vy u3 c,aeaytou4ux yc-
Ao6u~:
a ) bnx ne~omopoeo a = ( a I , a 2 ) , 0 < a i < I , i = 1 , 2 , noc/lecgoaameat,-
nocmb )~(m.) II e - ' : ' II noqmu yfbtoaem:
~) O,ucecmoyem ot = ( a I , a 2 ) , 0 < ~ i < 1. i = 1 ,2 , matzoe, ~mo
In ~.(m,,) $ , a In X(m,) .1..
p (n ct) p(n)
1. Ecnu ~,tnonneno ycnoeue (20). mo cgn.~ /uo6ot~ OHC tp ~ r
,,,,.(:,) = ox { z(,,,,,)II in (1 + n/In ~,(mn))II } , (28)
tcoe0a m a x ( n 1 , n 2 ) --> ** (u me,v 6onee. roeDa min(n I, n2) -'> *")-
2. Ecnu ( X , '~, I t ) ~ npocmpancmoo c rone~no~, no,ao~umem, no~, neamo~tu-
qectzm'i ~tepoti u noc.aeDo~'tmem, nocm~, to ( n ~, n 2) "-> "*, tzoeba m a x (n t, n 2) ~ oo
mo naaOemcn opmozonanbntaa p.~O (1), &a.~ romopoeo aunonneno ycnoaue (20) u
~ctoOy na X
lira sup to (n)X-~(m, , ) l l ln (1 +n / In~ , (m, , ) ) l l -~ l Sm,(X) i = **. (29)
max (nl,n2)---~
,D[ora.~me.abcm~o. Hyc'rb BUnO.nHeao yC~OBHe a ) . rloKa.~e~, wro peuieTKa
v,, = (t,,I/~'l, t,,~/~:~),
c/Ie [ x ] o3HaqaeT Ue.ny~o qaCT~ qncJ~a x , y/~on.rleTnop.~eT yC.nOBn.~IM (14). ~[eflc'rBn -~
TO.,rlbHO, .rlel"KO BH,/Ie'rb, q ro noc,neao~are.rmHOem
v,,, = [n) /a'] , i = 1,2,
y/Ioa~eTBOpaIOT yc.nomllo
cxp(v,,~i+l- v.~ ~) = 0 ( I ) , i = 1,2.
ISSN 0041-6053. Ytzp~ ,uam. ~.'Vlm., 1999. m. 51. tW i0
1308 B.A. AH/~PHEHKO
Orcmz~a B cnny yC~0Bna a) nonyqaeM
f l ff'i IXi X(mv,,+~)/~,(mv,,)< C exp(v ,~+~- v,,~ ) = O(1 ) ,
i=l
a Tor~ta, cor~acno neMMe 3, name yrBepa~tenne Bepao. Ho n Ta~oM cny~ae, ecnn
BtanonHeno ycnosne (20), Ha TeopcM~ 2 cne/~yeT oReltKa (21).
FIocKo.qbKy
= , , , , , + , - , , , , , = -
a {qni }, i = 1 ,2 ; n i = 0, 1 . . . . . onpc/~eneH~i I/epaBeHCTBaMH
[ , /e~ [(q,, + 1)l/e,] q"i J <" n i <
TO
C ~tpyrot~ CTOpOHbl,
lngi(qni ) - lnqn ~ - Inn/ , ni-->o~. (30)
In n i - In (1 + ni/ln~,i(m,, i)) , ni.--->o. (31)
rae ~'l (n 1) = L I (n l, 0) , L 2(n2) = L (0 , n2) - - caea~a nocae~toBaTeabaOCTn ~ ( n )
Ha KOOp]~HHaTHH x OCSX.
~/~CTBnT031bHO, ~ OHeBH~HO, RTO ,/]JIH HeKOTOpOFO C > 0
ln( l + n i / i n Xi(m,,,)) < Clnn i,
a a c n a y ycnoBns a ) r~MeeT MeCTO ueno,~Ka aKBrmanetmmx HepaBenCTB
~.i(mni). <_ C 1" exp n~ q r , In ~i(mni) <. C2n~i
r "i / in Z,i(m. ,) >-- C3n) -e' r In (1 + n i / In Z.i(m.,) ) >_. C 4 In rtb
r/ze C k, k = 1, 2, 3, 4, - - }xeKOTOp~ae nonoa<rITenhmae nocToaHmae.
TaKHM 06pazoM, a3 (30) H (31) no~yqaeM
In I.ti(q,,,) , In (1 + n i / In Li(m,,,) ) . (32)
Ha (32) cneayeT, qTO ouenKa (21) ~oaceT 6blTb 3anacaaa B Bnae (28), a (29) maTeKa-
eT n3 (22).
B czyqae [3) nOClle/~OBaTeJlbHOCTb { A ( n ) . = ~,(mn) } y~aoBne~opaeT yC21OBHIO
(13). CneZoBaTeilb14o, cornacHoneMMe 3, HattzxercspemeTga {Vn} S Z+ 2 TaKaa,
wr0 { A (n) } yaoBzea~opJ~eT HepaBencTaaM (14) n, cneaoBaTenbao, nep~inenc~aM
Ai (vn/+l)
Vni+i ~ Vni > 4, 1 < p <- Ai(V,,~) < q ' i = 1 , 2 ; n i = 0 , . 1 . . . . . (33)
OTCIOIIa, Ha OCHOBaHHH ./10MMbl 5 H3"[6], flbaKTnqecKn ]~oza3aHHOi~l B [ 1 ] (Teope-
Ma.4), m, neeM
BIvq,§ l / lnAi(Vq~+l) <_ I.ti(qni) <- BeVq~/ lnAi(Vq~) , i - 1 , 2 .
OTclo/Ia I4 I43 T0rO, qTO { q,~ } orlpe/Ie2DleTc~t HCpaBeHCTBaMH Vq,~ ~ n i < Vq,+ 1, B
c14ay (33) nonyqaeM
I.ti(qni) - ni / InAi(ni) = ni / ln~.i(mn~ ) , ni-.->~.
ISSN 0041-6053: Ytcp, ~tam. :~. pn.. 1999. m. 51. I~ I0
O I'IOPJ:I/1KE POCTA I'IP~IMOYFO.rlbHbiX qACTHblX CYMM ... 1309
B nTore OTClOIla c~ettyeT (32) H/xa~brIeflmne paccymaenna TaKne me, zaK 8 c~y-
qae a).
S a , , , ~ a , , , , , a. B yc~oB~,,, ~) Tc0pcM~ 3 BM0CTO p ( ~ ) - - ~ ? + ~,.'0 MO~O
B3$ITb ~m6ym noslo~nTe~bHylO qbyHZUHIO v ( n ) ~IJls ,I ~ O, nMexomym c~e~u
vi (n i ) , i = 1 , 2 . y a o ~ e T a o p z m m a e nepaBencTaaM n i - a i <_vi(ni) <_ n i + b i , r/Ie
a i, b i >_ 0. i = 1 , 2 . ~ aeKoTopue nOCTOZnn~e.
Kaz rIoKa3EaBaeT c~e/lymulaa TeopeMa./I~a ,.6h~CTpO y6bmaloulHx" nocole/IoBa-
TeJIbHOCTefl ~.-1 (n) ~IorapnqbMnqecKn~ MtlO~HTeIIb a ottenKe HcqeaaeT.
TeopeMa 4. Hycm~, { m,, = ( mnl , ran2 )} - - p e u l e m K a o Z~. npu~te~t m o = 0
u noc~eaooamem,nocmt, { ~,(n) > O, n e Z2+ } manoebt, vmo 3 q > 1 V n = ( n 1,
,,:) ~ z?; .
~. ( m,~,+ l , m~,,_+ l ) > q max {~.(mn,. m,_,+l). ~(mn,+l , mnz ) } . (34)
Ec.au tcoa~qbu~uenmbt p a a a (1) yaoe.~emeopatom yc.~oeuto (20), mo O.aa moSo~
O H C tO e �9 oepna o~enra
s,,,,(x) = O x { ~ ( m , , ) } , m a x ( h i , " 2 ) - ~ ~ . (35)
,RoKa3ament, cm~o. HycTs m,,( k ) = ( m,,, ( k ). m,,,. ( k ) ) - - 61mma~tuax K TOqZe
k = (k 1, k2) Toqza pemeTKn {m, ,} . y~oaae r~ops~ tuaz Hepa~eHcray
m, , (k ) >_ k. (36)
To raa za (34), (36) n (20) noay~aeM
Z Z -" " " = ~.- (mn) Z a~ =
n>O n>_O k<m,
= E a~" Z 7"-2(m,,) = E a~ Z Z~'-2(mn,'mn~_ ) =
k20 m n 2k k>-O ran, >k I ran: 2k 2
= 0(1) Z a/~ Z k-2(m,,, ' m,,2(k)) =
k~O m,h }k I
-- o(~) ~ a~ X-~(m,,,(~). m~:(~))= O0) ~ a~ Z-'-(~) < --.
k~0 k>0
OTcmzta Ha OCHOBaHHH TcopeMbl J'IeBH czle/ayeT (35).
YCTaHOBHM OKOHqaTe~IbllOCTb TeOpeMbI 4.
Teope ~a 5. l I y c m b ( X , ~, g ) - npocmpancmao c rone,~noa, no.~o~ume.~bno#,
neamo~tu,~ectcot~tepoa. { m , , } - - p e u t e m K a o Z+. a {~.(n)~'**, n e Z~} u
{v(n) , n e Z2}. v ( n 1, 1,2) ~ oo , t~ozt)a max(n 1, n2) ---) o* . - -noao . , rume .abn~e
noc.~eOoeamem,nocmu. Tozcga cyu~ecmayem opmozona .~n~a paO (1), roaqbdpu,uen-
mot Kornopozo yOo8aemoopatom yc.~oou~ (20). ma~oa. ,~mo oc~oOy na X
lira sup v ( n ) 7 ~ - t ( m , , ) i sm,,(x) I = 0..
max (n~ .n 2 )-~**
3a~e'mM, ,4+o ~ a a o a n o z p a ' m u x pgaoa noao6nas TeopeMa acpna (c~. [1], xeo-
pcMa 11). M~z ~ocno~,3ycMca ~'rolt rcope~ol l H Ha3oBer~ ee ~e~rnola Ko.ns/~t.
,iloga3ame.~t, cmeo. I-[poBo,~HTC.~ nO TO}[ ~Ke CXCMC, qTO H ~OKa3aTC./ISCTBO BT0-
poll qacTri TcOp~MH 1. IIycT~ cnaqa~a m, = n. He orparm~nna~ O6ttmOCT~, MOCK-
nO cqrrraT~, wro v ( n ) ~ * * . Ho~o~xnM t o ( n ~ ) = v ( n ~ , 0). ~. ~(n~) = ~.(n i, 0) . Co-
r~acno ~eMMe Koza~tu. na}taeTca pa a (18), z o a q b q b m t u e ~ ~0TOpOrO y~toa~eTaO-
pa~aT yc~oanio
ISSN 0041-6053. YKp. .uam. ~.3,plh, ! 999. m. 51, N e I0
1310 B.A. AHZIPHEHKO
,Y_., <
nl=O
~t~a K o T o p o r o BC~]Iy n a X
l i m s u p t O ( h i ) XI l (n i ) I s,,,(x) [ = **.
f l | --.~ oo
PaccMoTpHM ~BOI~HOR o p T 0 r o n a ~ b n H | . | p ~ ~ a n r n o ~ a r a ~
n20
~ c,l" /I 2 = 0;
= n I = 0 , 1 . . . .
an anln2 = ~ 0, It 2 > O,
n o n p e ~ e . a a a ~BO~.Inylo O H C {r = %,~,., ( x ) } t ta X TaK, KaK B T e o p e M e 3 rta
[6] . 3 w o w pz]1 c o a n a , a a e T Ha X c o u n o K p a r n b t M pazlo~t (18 ) rt, c ~ e / x o a a r e . a b n o ,
s , , ( x ) = s,~ ( x ) , a e r o Koaqbqbnmlenwta y ~ o a . a e w B o p ~ i o r y c . a o m n o ( 2 0 ) . T o r ~ a a . a z
. n m 6 o r o x ~ X Hr, teeM
l im s u p u(n)~.-I(m,,)ls,(x)[ > l im s u p c o ( n l ) X'~ t ( h i ) [ s ,h(x) [ = *~.
max (n I ,n 2 )'-~** 111 ..--) oo
B c . a y q a e rlpoH3aOJlbrlO~.~t p e u l e r K r l "[ r a n } , n c n o n b a y z paccMoTpe l t r lb l f l c . n y q a ~
m, , = n , nprtMeHJ~era cxeMy aoKaaaTeJiI , CTBa TeOpeMLa 3 rla [5] .
!. Ko,a~Ba B. H. CKOpOCTh CXO/IHM0Z-,2.'I'I'I II cy~l.~nlpyeslocrn op'roronaJn,max pa/toa n nJio~em4e ue-
Ko'ropldx K.IlaccoB qbyllKl[n|l MIIOIHX nepe.~emtux: s .... KalUt. dpna.-Ma'r, nayK. - O/tecta,
1973. - 129 c.
2. Mdricz F. On the growth of the rectangular partial sums of multiple non-orthogonal series // An.
Math. - 1980. - 6, N a 4. - P. 327 - 341.
3. Mdricz F. The Kronecker lemmas for multiple series and some applications // Acta math. Acad.
set. hung. - 198 I. - 37, N ~ 1 - 3. - P. 39 - 50.
4. Mdricz F. On the convergence in a restricted sense of multiple series II An. Math. - 1979. - 5.
N e 2 . - P . 1 3 5 - 147.
5. AntgpuenKo B. A. 0 cxopoc', 'n CXO/[HMOCTH Kparmax oprovo.a~u,m,~x p~l/to]~ / / Yxp. r4aT.
acypn. - 1990. - 42, N~ 10. - C. 1307 - 1314.
6. Andrienko V.A. Rate of approximation by rectangular partial sums of double orthogonal series //
An. Math. - 1996. - 22, N ~ 4. - P. 243 - 266.
7. ]Tr H. O reopeMe Mem,tuona - Palte~axepa ]t21~l ]tBO~lllhlX p.tl/toB ff COO611L AH
FCCP. - 1965. - 39. N'-' 2. - C. 277 - 282.
l'IoJ~yqeuo 17.04.97
155N 004/-6053. YKp. ,~tam. ~'y. p, . . 1999, m. 51. I~P /0
|
| id | umjimathkievua-article-4728 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:09Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f0/fc63e64ab65729eecb5a10397b7429f0.pdf |
| spelling | umjimathkievua-article-47282020-03-18T21:12:54Z On the order of growth of rectangular partial sums of double orthogonal series О порядке роста прямоугольных частных сумм двойных ортогональных рядов Andrienko, V. A. Андриенко, В. А. Андриенко, В. А. We obtain estimates of the order of growth of rectangular partial sums of double orthogonal series and establish their unimprovability on the set of all double orthogonal systems. Отримано оцінки порядку зростання прямокутних часткових сум подвійних ортогональних рядів. Встановлено їx остаточність на множині всіх подвійних ортогнональних систем. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4728 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1299–1310 Український математичний журнал; Том 51 № 10 (1999); 1299–1310 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4728/6157 https://umj.imath.kiev.ua/index.php/umj/article/view/4728/6158 Copyright (c) 1999 Andrienko V. A. |
| spellingShingle | Andrienko, V. A. Андриенко, В. А. Андриенко, В. А. On the order of growth of rectangular partial sums of double orthogonal series |
| title | On the order of growth of rectangular partial sums of double orthogonal series |
| title_alt | О порядке роста прямоугольных частных сумм двойных ортогональных рядов |
| title_full | On the order of growth of rectangular partial sums of double orthogonal series |
| title_fullStr | On the order of growth of rectangular partial sums of double orthogonal series |
| title_full_unstemmed | On the order of growth of rectangular partial sums of double orthogonal series |
| title_short | On the order of growth of rectangular partial sums of double orthogonal series |
| title_sort | on the order of growth of rectangular partial sums of double orthogonal series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4728 |
| work_keys_str_mv | AT andrienkova ontheorderofgrowthofrectangularpartialsumsofdoubleorthogonalseries AT andrienkova ontheorderofgrowthofrectangularpartialsumsofdoubleorthogonalseries AT andrienkova ontheorderofgrowthofrectangularpartialsumsofdoubleorthogonalseries AT andrienkova oporâdkerostaprâmougolʹnyhčastnyhsummdvojnyhortogonalʹnyhrâdov AT andrienkova oporâdkerostaprâmougolʹnyhčastnyhsummdvojnyhortogonalʹnyhrâdov AT andrienkova oporâdkerostaprâmougolʹnyhčastnyhsummdvojnyhortogonalʹnyhrâdov |