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Збіжність рядів за деякими ортонормованими системами та коефіцієнтні оцінки: Автореф. дис… канд. фіз.-мат. наук / С.О. Кирилов, Одес. держ. ун-т ім.

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Oaeanueeee aea?aeaaiee oi?aa?neoao ?i. ?.?.Ia/ieeiaa

Ee?eeia Na?a?e Ieaenaiae?iae/

OAeE 517.518

Ca?aei?noue ?yae?a ca aeayeeie i?oiii?iiaaieie nenoaiaie

oa eiao?oe??ioi? ioe?iee

01.01.01- iaoaiaoe/iee aiae?c

Aaoi?aoa?ao

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa

o?ceei-iaoaiaoe/ieo iaoe

Iaeana-1999 ?.

Aeena?oaoe??th ? ?oeiien

?iaioa aeeiiaia a Iaeanueeiio aea?aeaaiiio oi?aa?neoao? M?i?noa?noaa
ina?oe Oe?a?ie

Iaoeiaee ea??aiee :

aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani? Eieyaea A?eoi? ?aaiiae/,
Iaeanueeee aea?aeaaiee oi?aa?neoao, i.Iaeana.

Io?oe?ei? iiiiaioe:

aeieoi? o?ceei-iaoaiaoe/ieo iaoe , aeioeaio Aiae???iei A?oae?e
Iiaianiae/, I?aaeaiiioe?a?inueeee aea?aeaaiee iaaeaaia?/iee oi?aa?neoao,
i.Iaeana;

eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe, aeioeaio Ioaeaeueiai Aaeieueo
Aa?aiiae/ , Iaeanueea aea?aeaaia aeaaeai?y aoae?aieoeoaa oa a?o?oaeoo?e,
i.Iaeana.

I?ia?aeia onoaiiaa:

?inoeooo iaoaiaoeee IAI Oe?a?ie, a?aeae?e oai??? ooieoe?e, i. Ee?a.

Caoeno a?aeaoaeaoueny “8” aeiaoiy 1999 ?. i 15 aiaeei? ia can?aeaii?
niaoe?ae?ciaaii? a/aii? ?aaee K 41.051.05 i?e Iaeanueeiio
aea?aeaaiiio oi?aa?neoao? ca aae?anith: 270026,i.Iaeana , aoe.
Aeai?yinueea, 2.

C aeena?oaoe??th iiaeia iciaeiieoenue o a?ae?ioaoe? IAeO ca aae?anith:
270026,i.Iaeana, aoe. I?aia?aaeainueea, 24.

Aaoi?aoa?ao ?ic?neaiee “6” aa?aniy 1999 ?.

A/aiee nae?aoa? niaoe?ae?ciaaii? ?aaee
A?othe I.I.

Caaaeueia oa?aeoa?enoeea ?iaioe

Aeooaeuei?noue oaie.

O iao?e aeena?oaoe?? aea/athoueny aeaa eiea ieoaiue. Ia?oa iia’ycaia c?
ciaoiaeaeaiiyi iiiaeiee?a Aaeey aeey ca?aeiino? iaeaea anthaee ?
aacoiiaii? ca?aeiino? iaeaea anthaee ?yae?a ca i?oiii?iiaaieie nenoaiaie
?c iaaiiai eeano. Ae?oaa iia’ycaia c iaea?aeaiiyi ioe?iie ii?i ooieoe?e
o aeayeeo neiao?e/ieo i?inoi?ao /a?ac eiao?oe??ioe Oo?’? oeeo ooieoe?e
ca caaaeueieie i?oiii?iiaaieie nenoaiaie.

Iniiae caaaeueii? oai??? i?oiaiiaeueieo ?yae?a aoee caeeaaeai? ia
ii/aoeo iaoiai noie?ooy, eiee iii?oeee, ui oe?eee ?yae aeanoeainoae
?yae?a ca o?eaiiiiao?e/iith nenoaiith iiaea aooe ia?aianaiee ia ?io?
nenoaie ooieoe?e, nie?ath/enue o?eueee ia oiiao i?oiaiiaeueiino?.

Ao?i, ?ioaineaii ?icaeaaoeny ye naiino?eiee iai?yiie oai??y
i?oiaiiaeueieo ?yae?a ii/aea eeoa a 30-?, 40-? ?iee iaoiai noie?ooy. ??
?icaeoie o iao?e e?a?i? a ia?oo /a?ao iia’ycaiee c ?iaiaie
I.I.Eieiiai?iaa, Ae.?.Iaioiaa.

O aea/aii? ieoaiiy i?i ciaoiaeaeaiiy iiiaeiee?a Aaeey aeey ?yae?a ca
i?oiii?iiaaieie nenoaiaie naia Ae.?.Iaioiaei ?, iacaeaaeii a?ae iueiai,
?aaeaiaoa?ii aoa iaea?aeaiee ia?oee ooiaeaiaioaeueiee ?acoeueoao:
ciaeaeaiee oi/iee iiiaeiee Aaeey aeey ca?aeiino? iaeaea anthaee ?yae?a
ca i?oiii?iiaaieie nenoaiaie, ui iineoaoaaei a?aei?aaiith oi/eith aeey
aaaaoueio ?ioeo aeine?aeaeaiue. I?ioa aei oeueiai /ano ?nio? aeoaea iaei
?acoeueoao?a, ye? a aecia/aee oae?, aeineoue caaaeuei? oiiae ia
i?oiii?iiaaio nenoaio, ui aeicaieyee a i?aeneeeoe oeae ?acoeueoao.

Ia?o? ioe?iee ii?i ooieoe?e o i?inoi?ao Eaaaaa /a?ac ?o eiao?oe??ioe
Oo?’? ca o?eaiiiiao?e/iith nenoaiith aoee iaea?aeai? Oaonaei?oii ?
THiaii, a oaeiae, aeaui i?ci?oa, Oa?ae? ? E?ooeaoaeii o 20-? ?iee. Aeae?
oeae ?acoeueoao ?icaeioee a aaaaoueio iai?yieao.

O ?iaioao Iae?, Ia?oeeieaae/a ? C?aioiaea a?i aoa ia?aianaiee ia
caaaeuei? i?oiii?iiaai? nenoaie. Aiaeiae oeeo ?acoeueoao?a aoee i?ci?oa
iaea?aeai? a ?ioeo i?inoi?ao ooieoe?e o ?iaioao ?yaeo iaoaiaoee?a. Ao?i,
oe?ea iecea ieoaiue, iia’ycaieo c oaeeie ioe?ieaie, caeeoeeenue
a?aee?eoeie.

Oaeei /eiii, iaoa ?iaioa i?enay/aia ?ica’ycaiith caaea/, ye? iathoue
oai?aoe/iee ?ioa?an, ui iaoiiaeth? ?? aeooaeuei?noue.

Iaoith ?iaioe ?:

iaea?aeaiiy iiiaeiee?a Aaeey aeey ca?aeiino? oa aacoiiaii? ca?aeiino?
iaeaea anthaee ?yae?a ca i?oiii?iiaaieie nenoaiaie eeano S(p,();

iioe?aiiy a?aeiieo ?acoeueoao?a Iae? oa Noaeia ia i?oiii?iiaai? nenoaie
c AII;

anoaiiaeaiiy iiaeo ioe?iie ii?i ooieoe?e o i?inoi?ao Ei?aioea /a?ac ?o
eiao?oe??ioe Oo?’? ca caaaeueieie i?oiii?iiaaieie nenoaiaie, a oaeiae
aeai?noeo ioe?iie eiao?oe??io?a Oo?’?;

ioe?iea inoaoi/iino? io?eiaieo ?acoeueoao?a.

Iaoiaee aeine?aeaeaiiy.

A ?iaio? aeei?enoiaothoueny iaoiaee oai??? ooieoe?e, a naia, oai???
iac?inoath/eo ia?anoaaeaiue, oai??? i?oiaiiaeueieo ?yae?a, ?ioa?iieyoe??
iia?aoi??a, oai??? aeeaaeaiiy ooieoe?iiaeueieo i?inoi??a.

Iaoeiaa iiaecia.

On? iaea?aeai? iaoeia? ?acoeueoaoe ? iiaeie. A aeena?oaoe??:

aaaaeaii iiaee eean i?oiii?iiaaieo nenoai, yeee ? ?icoe?aiiyi a?aeiiiai
eeano Sp – nenoai, oa ciaeaeai? iiiaeieee Aaeey aeey ca?aeiino? oa
aacoiiaii? ca?aeiino? iaeaea anthaee ?yae?a ca nenoaiaie aeaiiai eeano;

aia?oa iaea?aeai? ioe?iee eiao?oe??io?a Oo?’? oa aeai?no? ioe?iee ii?i
ooieoe?e uiaei i?oiii?iiaaieo nenoai ?c AII;

aeaii ?aaoey?ia canoinoaaiiy iac?inoath/eo ia?anoaaeaiue aeey
iaea?aeaiiy ioe?iie ii?i ooieoe?e o aeayeeo ooieoe?iiaeueieo i?inoi?ao
/a?ac ?o eiao?oe??ioe Oo?’? ca caaaeueieie i?oiii?iiaaieie nenoaiaie.

Oai?aoe/ia oa i?aeoe/ia cia/aiiy.

?acoeueoaoe aeena?oaoe?? iathoue oai?aoe/ia cia/aiiy. Aiie iiaeooue aooe
canoiniaai? o oai??? i?oiaiiaeueieo ?yae?a, oai??? ooieoe?iiaeueieo
i?inoi??a.

Ca’ycie ?iaioe c ieaiiaeie iaoeiaeie aeine?aeaeaiiyie.

Oaia aeena?oaoe?? ? neeaaeiaith /anoeiith iaoeiaeo aeine?aeaeaiue, ye?
i?iaiaeyoueny ia eaoaae?? iaoaiaoe/iiai aiae?co ?inoeoooo iaoaiaoeee,
aeiiii?ee oa iaoai?ee Iaeanueeiai aea?aeaaiiai oi?aa?neoaoo ca oaiith
‘Iao?e/i? oa oiiieia?/i? aeanoeaino? ooieoe?iiaeueieo i?inoi??a’

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??.

Iniiai? ?acoeueoaoe aeena?oaoe?? aeiiia?aeaeenue ia I?aeia?iaei?e
eiioa?aioe??, i?enay/ai?e 100-??//th c aeiy ia?iaeaeaiiy ?.ss.?aiaca o
i.??aia (1997), ia ui??/ieo iaoeiaeo eiioa?aioe?yo aeeeaaeaoeueeiai
neeaaeo a Iaeanuee?e aea?aeaai?e aeaaeai?? oa?/iaeo oaoiieia?e
(1997,1999), ia nai?ia?ao i?ioani?a A.I.Noi?iaeaiei a Iaeanueeiio
aea?aeaaiiio oi?aa?neoao?.

Ioae?eaoe?? c iniiaieo ?acoeueoao?a aeena?oaoe??.

C iniiaieo ?acoeueoao?a aeena?oaoe?? aaoi?ii iioae?eiaaii i’youe ?ia?o a
iaoeiaeo aeo?iaeao.

No?oeoo?a oa ianya aeena?oaoe??.

Aeena?oaoe?y neeaaea?oueny c? anooio, aeaio ?icae?e?a, aeniiae?a oa
nieneo aeei?enoaieo aeaea?ae. Ia?oee ?icae?e aeena?oaoe?? iiae?eaii ia
i’youe i?ae?icae?e?a, ae?oaee – ia aeanyoue i?ae?icae?e?a. O nieneo
aeei?enoaieo aeaea?ae – 48 iaeiaioaaiue. Caaaeueiee ianya aeena?oaoe?? –
108 noi??iie.

INIIAIEE CI?NO

Ia?aeaeaii aei a?eueo aeaoaeueiiai iaeyaeo ?acoeueoao?a aeena?oaoe??.

Iaoae {fn(x)}-i?oiii?iiaaia nenoaia ooieoe?e ia a?ae??ceo [0,1], i?e/iio
fn(Lp[0,1] (n=1,2,…) aeey aeayeiai 2(p((. Aeaii ianooiia icia/aiiy.

Nenoaia (= {fn(x)} iaceaa?oueny Sp-nenoaiith , yeui ?nio? noaea n,
oaea , ui aeey aoaeue-yeiai iie?iiio ca nenoaiith (

ni?aaaaeeeaa ia??ai?noue

((Pn((p(((Pn((2 .

Iiiyooy Sp-nenoaie aoei aia?oa aaaaeaii N.A.Noa/e?iei ? aeieeei ye
ocaaaeueiaiiy oiai oaeoo, ui oe??th aeanoea?noth aieiae?thoue eaeoia?i?
i?aenenoaie o?eaiiiiao?e/ii? nenoaie.

Aeae? i?oiii?iiaaia nenoaia (= {fn(x)}, x((0,1) iaceaa?oueny nenoaiith
ca?aeiino?, yeui aoaeue-yeee ?yae aeaeo

((k=1 akfk(x)
(1)

c ((k=1 ak2 2)
aa?aoueny i?ino?? , a?eueo “aeecueeee” aei i?inoi?o L2. A naia, iaoae

((t)=t2[ln(e+t)/(ln(e+t-1)]2+( , (>0.

ssnii, ui ((t) iiiioiiii c?inoa? ia (0, +() , limt(( ((t)=( , limt(0
((t)=0 .

-nenoaiith, yeui ?nio? oaea noaea N, ui aeey aoaeue-yeiai iie?iiio

Pn(x)=(nk=1akfk(x)
,n=1,2,…

ni?aaaaeeeaa ia??ai?noue

(01((Pn(x))dx(C((((Pn((2).

O.I.Aaeeeaa?a iieacaa, ui eiaeia S( -nenoaia oaeiae ? nenoaiith
ca?aeiino?.

Caa?iaiiny aei iaoeo ?acoeueoao?a.

Iaoae (={(k}-ianiaaeath/a iine?aeiai?noue ae?enieo /enae. sse i?e?iaeia
ocaaaeueiaiiy iiiyooy S? -nenoaie aaiaeeii eean S(p,()-nenoai ianooiiei
/eiii.

i?oiii?iiaaia nenoaia (= {fn(x)} (fk(Lp[0,1], 2

0 . Oiae? iine?aeiai?noue

(n=ln2+( (e+(n)

? iiiaeieeii Aaeey aeey ca?aeiino? iaeaea anthaee ?yae?a ca nenoaiith (
.

A ?acoeueoao? aea/aiiy ieoaiiy i?i inoaoi/i?noue oai?aie 1.3.1 iai
aaeaeiny aeiaanoe ianooiio oai?aio.

Iaoae (={(k}- ianiaaeath/a, iioeea aeiai?e iine?aeiai?noue aeiaeaoi?o
/enae.

oai?aia 1.4.1. ?nio? oaea i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1]
{(n(x)} , yea ? S(p, () -nenoaiith (2

0 aeeiio?oueny oiiaa

(=((k=1 ak2 ln2+( (e+(k) 2 ?

(q(c)=(((n=1(cn(qnq-2)1/q2, r(1) ? f noia
?yaeo((k=1 ck(k(x) ca ii?iith L2[0,1], oi f(Lq,r ?

((f((q,r(cq,r{((f((2+M1-2/q(({cn}((q’,r

Aeiaaaeaiiy oe??? oai?aie aaco?oueny ia ianooii?e ioe?ioe? ia?anoaaeaiiy
noie ?yaeo.

Eaia 2.2.2. Iaoae {(n(x)} – i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1],
(n(AII ? (((n((*(M (n=1,2,…).

?) sseui f**(L1, 02) .

O i?ae?icae?e? 2.3 ie ie?aii ?icaeyiaii aeiaaeie, eiee i?oiii?iiaaia
nenoaia {(n(x)} ? iaiaaeaiith a noeoiiino? o i?inoi?? Ls(20) ? {cn} – iine?aeiai?noue
eiao?oe??io?a Oo?’? ooieoe?? f. Oiae?

(q,r,s(c)=(((n=1 cn*rn()1/r(cq,r,s Ms/(s-2)(2/q-1) ((f((q,r’.

??)sseui (q,r,s2, r>0) ? f noia ?yaeo((k=1
ck(k(x) ca ii?iith L2[0,1], oi f(Lq,r ?

((f((q,r’(’ (cq,r,s Ms/(s-2)(1-2/q) (q,r,s(c).

Ioe?ieo ia?anoaaeaiiy, ia ye?e aaco?oueny aeiaaaeaiiy oe??? oai?aie,
iiaeaii o ianooii?e eai?.

Eaia 2.3.1. Iaoae {(n(x)} – i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1],
(n(Ls ? (((n((s(M (n=1,2,…).

?) sseui f*(Ls( ,02 ciaeaeaoueny
iine?aeiai?noue {cn} , aeey yei? aeeiio?oueny (3) , ?yae (2)
ca?aa?oueny a L2[0,1] , aea eiai noia ia iaeaaeeoue Lq[0,1].

O iiaeaeueoiio ci?no ae?oaiai ?icae?eo iaoi? ?iaioe oaeiae o?nii
iia’ycaiee c oai?aiith A.

aea/aiith ??ciiiai?oieo ieoaiue, ye? iia’ycai? c iath, i?enay/aiee
oe?eee ?yae ?ia?o noaeia ? Aaena, E?ooeaoaea, Aoee?ia, Iiioaiia??,
Eieyaee .

Oae A.?. Eieyaea iaea?aeaa ioe?iee ii?i a Lq ooieoe?e, ye? ? noiaie
?yae?a ca aeayeith i?oiii?iiaaiith nenoaiith a oa?i?iao ii?i iia?aoi??a
/anoeiaeo noi. O i?ae?icae?e? 2.5 iaoi? ?iaioe i?noyoueny ioe?iee, ye?
ocaaaeueiththoue ?acoeueoaoe A.?.Eieyaee ia aeiaaeie i?inoi??a Ei?aioea
Lq,r[0,1] (q>2,r>0).

aaaaeaii aaee/eio

(n(s) =sup{(((nk=1 ck(k((s:(nk=1 ck2=1} .

Aiia yaey? niaith i? ui ?ioa, ye ii?io iia?aoi??a /anoeiaeo noi ?yaeo
(2) Sn:l2(Ls.

Iaie aeiaaaeai? ianooii? ?acoeueoaoe.

Oai?aia 2.5.1. Iaoae {(n(x)} – i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1],
i?e/iio aeey aeayeiai s((2,+(] : (n(Ls[0,1] i?e an?o n=1,2,… .
sseui 20 , (=r(q-2)s/(q(s-2)) ? iine?aeiai?noue a={an}(l2 oae?,
ui

(q,r(a)=(((n=1((rn-(rn+1) ((n)1/r 2 , r>0 i iine?aeiai?noue
a={an}(l2 oaea,ui

(q,r(a)=(((n=1((rn-(rn+1) (nr(q-2)/q)1/r 0). aeey i?eeeaaeo iaaaaeaii oaeee
?acoeueoao.

Oai?aia 2.5.3. Iaoae {(n(x)} – i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1]
, i?e/iio, aeey aeayeiai s((2,+(] : (((n((s(Mn , (nk=1
M2k=Bn(n=1,2,…) . sseui 2q( ? iine?aeiai?noue a={an}(l2
oaea, ui

Dq,r(a)=(((n=1 (an(rBn( Mn2-r)1/r2, f(L2,r[0,1] ? {an} -eiao?oe??ioe
Oo?’? ooieoe?? f ca nenoaiith {(n(x)}, oi

(r(a) (cr,s((f((2,r ,

a iicia/aiiyo oai?aie 2.5.6.

Aeey i?inoi?o AII iaie iaea?aeaii.

Oai?aia 2.6.3. Iaoae {(n(x)} – i?oiii?iiaaia nenoaia ia a?ae??ceo
[0,1] ? (((n((*(Mn , (nk=1 M2k=Bn (n=1,2,…) . sseui q((1,2] ,
r([1,2] ? f(Lq,r[0,1] ? {an} – eiao?oe??ioe Oo?’? ooieoe?? f ca
nenoaiith {(n(x)} , oi

Dq,r(a) (cq,r((f((q,r .

O i?ae?icae?e? 2.7 i?noeoueny oaa?aeaeaiiy , yea iieaco? , ui oai?aie c
i?ae?icae?eo 2.5 a aeayeiio ?icoi?ii? inoaoi/i?. A naia, iaie
aeiaaaeaii, ui oai?aia 2.5.3 oi/ia o caaaeueiiio aeiaaeeo (aeey nenoai
?c L( ia iaiaaeaieo a noeoiiino?) a oiio ?icoi?ii?, ui noai?iue
r-r/q-1 o aaee/eie An , yea aoiaeeoue a Dq,r(a) ia iiaeia cai?ieoe
i?yeei iaioei noaiaiai. A?eueo oiai, ni?aaaaeeeaa

Oai?aia 2.7.1. Iaoae {Mn} – aeayea iine?aeiai?noue ae?enieo /enae, Mn(1,
(nk=1 M2k=Bn (n=1,2,…), q>2,r(2 ?

M2n+1(cBn

Ciaeaeaoueny i?oiii?iiaaia ia [0,1] nenoaia {(n(x)} ?
iine?aeiai?noue {cn}(l2 , oae?, ui (((n((((Mn ? i?e aoaeue-yeiio
(1 oa ( oaea, ui (0.

I?eionoeii, ui i?e p>1 ((t)(( caaeiaieueiy? ianooiii oiiae

B’=sup01 ,

iine?aeiai?noue {cn} ? iine?aeiai?noth eiao?oe??io?a Oo?’? ooieoe?? f ?
((t)(( caaeiaieueiy? (2 -oiiao. sseui p=2 ? ((t)(const, aai p>1, a ((t)
caaeiaieueiy? oiiae (4), (5), oi

((n=1 cn*p np-2(p(1/n) (c(01 f*(t)p(p(t)dt .

Na?aae ?acoeueoao?a, ye? i?enay/ai? ia?aiino oai?aie A ia i?ino??
I?ee/a, a?aecia/eii ?iaioe Iaeoieeaiaea, Ianeiaa, Eieyaee. Aeey nenoai,
o yeeo ii?ie a L( ??aiii??ii iaiaaeai?, ? ooieoe?e ((t) niaoe?aeueiiai
aeaeyaeo, oaeiai oeio ioe?iea i?noeoueny a ?iaio? iaeoieaiaea. A ?iaio?
Ianeiaa iiae?aia ioe?iea aeaia aeey nenoai iaiaaeaieo ooieoe?e. I?e
oeueiio oiiae, ye? iaeeaaeathoueny ia ((t), iinyoue ii??aiyii c iaoeie
?ioee oa?aeoa?.

A iniia? aeiaaaeaiiy aieiaiiai oaa?aeaeaiiy aeaiiai i?ae?icae?eo
eaaeeoue ianooiia eaia.

Eaia 2.9.1. Iaoae {(n(x)} – i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1] ,
i?e aeayeiio s((2,+(] : (n(Ls[0,1], (((n((s(Mn , (nk=1 M2k=Bn
(n=1,2,…) . Iaoae oaeiae

f(s)(t)={1/t(t0f*s(u)du}1/s i 1/s+1/s’=1 .

Oiae?

(nk=1 ck2(A(1Tf(s’)2(u)du. (T=Bn-s/(s-2)).

Aeei?enoiaoth/e oeth eaio ie caeiaoee ianooiio ioe?ieo eiao?oe??io?a
Oo?’?.

iaoae {(n(x)} – i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1] ? i?e
aeayeiio s((2,+(] : (((n((s(Mn , (nk=1 M2k=Bn (n=1,2,…). Iaoae
oaeiae ( – iiiioiiii c?inoath/a iaia?a?aia ooieoe?y ? ((0)=0. Aoaeaii
oaeiae aaaaeaoe, ui

Bn+1(cBn .

Oai?aia 2.9.1. sseui ((s)=((s1/2) oaiooa ia i?ii?aeeo (0,+() ? {ck}–
eiao?oe??ioe Oo?’? ooieoe?? f , oi

((k=1 ( ((ck(Bk(r-1)/(r-2) /Mk)Mk2/Bk2(r-1)/(r-2) (C1(01((C2f(r’)
(t))dt.

Inoaii?e i?ae?icae?e iaoi? ?iaioe i?enay/aiee ia?aiino ?yaeo
?acoeueoao?a A.I.?iae?ia ia aeiaaeie nenoai, ye? ? iaiaaeaieie a
noeoiiino?, ca aeiiiiiaith iaoiae?a, ye? aaeeaaeeny a iao?e
aeena?oaoe??.

I?ino?? E aei??ieo ooieoe?e ia [0,1] iaceaa?oueny neiao?e/iei, yeui ?c
ia??aiino? (f(t)(((g(t)( ? oiiae g(t)(E aeieeaa?, ui ((f((E(((g((E ?
?c ??aiiaei??iino? ooieoe?e f ? g aeoiaeeoue , ui ((f((E=((g((E.

I?eeeaaeaie neiao?e/ieo i?inoi??a ? i?inoi?e Eaaaaa ? Ei?aioea, ye?
?icaeyaeaeeny ?ai?oa. iaaaaeaii aeaye? ?io? i?eeeaaee neiao?e/ieo
i?inoi??a.

Iaoae ((t) – c?inoath/a, oaiooa ia [0,1] ooieoe?y ? ((0)=0 . Caaaeueiei
i?inoi?ii Ei?aioea iacaaii iiiaeeio aei??ieo ooieoe?e, aeey yeeo

((f((((()=(01f*(t)d((t)0 iine?aeiai?noue
{ln2+( (e+(n)} ? iiiaeieeii Aaeey aeey ca?aeiino? iaeaea anthaee ?yae?a
ca aoaeue-yeith S(p,()-nenoaiith. A?eueo oiai, oey iine?aeiai?noue ?
oaeiae iiiaeieeii Aaeey aeey aacoiiaii? ca?aeiino? iaeaea anthaee. Oaeei
/eiii anoaiiaeaii oaea, aeineoue caaaeueia, iaiaaeaiiy ia i?oiii?iiaaio
nenoaio, yea aeicaiey? a aeayeeo /anoeiieo aeiaaeeao i?aeneeeoe
oaa?aeaeaiiy a?aeiii? oai?aie A?aeaoa-Noa/e?ia.

Iai ia aaeaeiny c’ynoaaoe, /e caeeoeoueny caaaeaia oaa?aeaeaiiy a??iei
i?e (=0. Oea ieoaiiy iiaea aooe iaoith iiaeaeueoeo aeine?aeaeaiue.

O ae?oaiio ?icae?e? iao? ?acoeueoaoe noinothoueny ioe?iie ii?i ooieoe?e
/a?ac ?o eiao?oe??ioe Oo?’?. Iai aaeaeiny iioe?eoe a?aeii? oai?aie Iae?
oa Noaeia, uiaei ioe?iie ii?i ooieoe?e o i?inoi?ao Eaaaaa oa Ei?aioea,
ia i?oiii?iiaai? nenoaie c AII.

Ie aeiaaee, ui aeey i?oiii?iiaaieo nenoai, o yeeo ii?ie a L( ia ?
iaiaaeaieie o noeoiiino?, ia ni?aaaeaeo?oueny a?iioaca, uiaei
i?aeneeaiiy a?aeiii? oai?aie Ia?oeeieaae/a-C?aioiaea, yeo aenoioa Aoee?i
ia ii/aoeo 50-o ?ie?a.

Canoinoaaiiyi ioe?iie iac?inoath/eo ia?anoaaeaiue iaea?aeai? iia?
ioe?iee ii?i ooieoe?e o i?inoi?ao Ei?aioea /a?ac ?o eiao?oe??ioe Oo?’?
ca caaaeueieie i?oiii?iiaaieie nenoaiaie, oa aeai?no? ?i ioe?iee
eiao?oe??io?a Oo?’?. Aeiaaaeaii inoaoi/i?noue, a aeayeiio nain?,
icia/aieo ?acoeueoao?a.

A inoaii?o ?icae?eao ?iaioe ie io?eiaee iia? ioe?iee eiao?oe??io?a Oo?’?
aeey ooieoe?e ?c eean?a I?ee/a, aaaiaeo i?inoi??a Lp , caaaeueieo
neiao?e/ieo i?inoi??a. Cacia/eii, ui iaoiaee aeine?aeaeaiiy a oe?e
/anoei? ?iaioe oaeiae aacothoueny ia ioe?ieao iac?inoath/eo
ia?anoaaeaiue. Oaeee i?aeo?ae aeyaeany aoaeoeaiei aeey iaea?aeaiiy iiaeo
eiao?oe??ioieo ioe?iie ii?i ooieoe?e o ??cieo ooieoe?iiaeueieo
i?inoi?ao.

NIENIE IIOAE?EIAAIEO I?AOeUe ca oaiith aeena?oaoe??

Ee?eeeia N.A. I oai?aia Ia?oeeieaae/a-Ceaioiaea//Iaoai.
caiaoee.-1998.-O.63.-?3.-N.386-390.

Ee?eeeia N.A. I iiiaeeoaeyo Aaeey aeey iaeioi?uo eeannia
i?oiii?ie?iaaiiuo nenoai //Eca. aocia. Iaoai.-1994.- ?7.-C.28-34.

Kirillov S.A. Some estimates for orthonormal systems from BMO//Acta Sci.
Math. (Szeged).-1998.-V.64.-P.223-230.

Ee?eeeia N.A. I oai?aia Ia?oeeieaae/a-Ceaioiaea// Aieeinueeee
iaoaiaoe/iee a?niee.-1997.- ? 3.-C.43-45.

Kirillov S.A. Norm estimates of functions in Lorentz spaces//Acta Sci.
Math. (Szeged).-1999.-V.65.-P.189-201.

AIIOAOe??

Ee?eeia N.I. Ca?aei?noue ?yae?a ca aeayeeie i?oiii?iiaaieie nenoaiaie oa
eiao?oe??ioi? ioe?iee.-?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.01 – iaoaiaoe/iee
aiae?c.-Iaeanueeee aea?aeaaiee oi?aa?neoao, Iaeana, 1999.

A aeena?oaoe?? aaaaeaii iiaee eean S(p,()-nenoai, yeee ? iioe?aiiyi
a?aeiiiai eeano Sp-nenoai, oa aeey nenoai c oeueiai eeano iaea?aeai?
iiiaeieee Aaeey aeey ca?aeiino? oa aacoiiaii? ca?aeiino? iaeaea anthaee.

a?aeii? oai?aie Iae? oa Noaeia, uiaei ioe?iie ii?i ooieoe?e /a?ac ?o
eiao?oe??ioe Oo?’?, iioe?aii ia i?oiii?iiaai? nenoaie c AII.

aeiaaaeaii, ui aeey caaaeueieo i?oiii?iiaaieo nenoai ia ni?aaaeaeo?oueny
a?iioaca, uiaei i?aeneeaiiy a?aeiii? oai?aie Ia?oeeieaae/a-C?aioiaea,
yeo aenoioa Aoee?i ia ii/aoeo 50-o ?ie?a.

Canoinoaaiiyi ioe?iie iac?inoath/eo ia?anoaaeaiue io?eiai? iia? ioe?iee
ii?i ooieoe?e o i?inoi?ao Ei?aioea /a?ac ?o eiao?oe??ioe Oo?’? ca
caaaeueieie i?oiii?iiaaieie nenoaiaie, oa aeai?no? ?i ioe?iee
eiao?oe??io?a Oo?’?. Aeiaaaeaii inoaoi/i?noue, a aeayeiio nain?,
icia/aieo ?acoeueoao?a.

io?eiai? oaeiae iia? ioe?iee eiao?oe??io?a Oo?’? aeey ooieoe?e ?c eean?a
I?ee/a, aaaiaeo i?inoi??a Lp , caaaeueieo neiao?e/ieo i?inoi??a.

Eeth/ia? neiaa: i?oiii?iiaaia nenoaia, iiiaeiee Aaeey, iac?inoath/a
ia?anoaaeaiiy, eiao?oe??ioe Oo?’?, i?ino?? Ei?aioea, AII.

Ee?eeeia N.A. Noiaeeiinoue ?yaeia ii iaeioi?ui i?oiii?ie?iaaiiui
nenoaiai e eiyooeoeeaioiua ioeaiee.- ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.01-iaoaiaoe/aneee
aiaeec.- Iaeanneee ainoaea?noaaiiue oieaa?neoao, Iaeanna, 1999.

Ionoue (={(k}- iaoauaathuay iineaaeiaaoaeueiinoue aauanoaaiiuo /enae.
I?oiii?ie?iaaiiay nenoaia (= {fn(x)} (fk(Lp[0,1], 2

0 iineaaeiaaoaeueiinoue {ln2+(
(e+(n)} yaeyaony iiiaeeoaeai Aaeey aeey noiaeeiinoe ii/oe anthaeo
?yaeia ii i?iecaieueiie S(p,()-nenoaia. Aieaa oiai, yoa
iineaaeiaaoaeueiinoue yaeyaony oaeaea iiiaeeoaeai Aaeey aeey aaconeiaiie
noiaeeiinoe ii/oe anthaeo.

Ii iiaiaeo ieii/aoaeueiinoe yoiai ?acoeueoaoa ooaa?aeaeaaony, /oi
oiiiyiooay iineaaeiaaoaeueiinoue ia iiaeao auoue caiaiaia ie eaeie
iineaaeiaaoaeueiinoueth {(n} n (n=o(ln2 (e+(n)). Aii?in i oii, iiaeao
ee auoue (=0, inoaeny ioe?uoui.

Aoi?ie ?acaeae ?aaiou iinayuai ioeaieai ii?i ooieoeee /a?ac eo
eiyooeoeeaiou Oo?uea ii i?oiii?ie?iaaiiui nenoaiai. Aeey iieo/aiey oaeeo
ioeaiie a aeenna?oaoeee eniieueciaai iiaeoiae, iniiaaiiue ia ioeaieao
iaaic?anoathueo ia?anoaiiaie.

Iieacaii, /oi a oneiaeyo oai?ai Iyee e Noaeia i?oiii?ie?iaaiioth
nenoaio, ia?aie/aiioth a niaieoiiinoe, iiaeii, aiiaua aiai?y, caiaieoue
nenoaiie, o eioi?ie ia?aie/aiu a niaieoiiinoe iieoii?iu a AII. A eiaiii,
ionoue {(n(x)} – i?oiii?ie?iaaiiay nenoaia ia io?acea [0,1], (n(AII e
(((n((*(M (n=1,2,…). Anee f (Lp,r (1

2, r(1) e f noiia ?yaea ((k=1
ck(k(x) ii ii?ia L2[0,1], oi f(Lq,r e

((f((q,r(cq,r{((f((2+M1-2/q(({cn}((q’,r

Ae?oaei iai?aaeaieai ?aaiou yaeyaony iaiauaiea yoeo ?acoeueoaoia ia
iauea i?oiii?ie?iaaiiua nenoaiu. Iaie aeieacaii, /oi aeey
i?oiii?ie?iaaiiuo nenoai, o eioi?uo ii?iu L( ia yaeythony ia?aie/aiiuie
a niaieoiiinoe ia aa?ia aeiioaca i aiciiaeiinoe oneeaiey ecaanoiie
oai?aiu Ia?oeeieaae/a-Ceaioiaea, eioi?oth auaeaeioe Aoeeei a ia/aea 50-o
aiaeia. A eiaiii, nouanoaoao i?oiii?ie?iaaiiay ia [0,1] nenoaia ooieoeee
{(n(x)}, (n(L([0,1] (n=1,2,…) , oaeay, /oi aeey ethaiai q>2
iaeaeaony iineaaeiaaoaeueiinoue {cn} , aeey eioi?ie aa?ii

((n=1(cn(qn(q-2)(((n((((q-2))

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