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 <( ca?aa?oueny iaeaea anthaee ia [0,1]. N.A.Noa/e?i iieacaa, ui anyea Sp -nenoaia ? nenoaiith ca?aeiino?. A?eueo oiai, ine?eueee i?e aoaeue-yeiio ia?anoaaeaii? ooieoe?e i?oiii?iiaaii? nenoaie , Sp - nenoaia caeeoa?oueny Sp -nenoaiith, oi iiaeia noaa?aeaeoaaoe, ui c oiiae ((k=1 ak2 <( aeieeaa? aacoiiaia ca?aei?noue ?yaeo (1) . O iiaeaeueoiio ?acoeueoao N.A.Noa/e?ia aoa iioe?aiee O.I.Aaeeeaa?aei ia aeiaaeie, eiee o aecia/aii? Sp -nenoaie cai?noue i?inoi?o Lp (p>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], 20 . 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 (20 aeeiio?oueny oiiaa

(=((k=1 ak2 ln2+( (e+(k) <(, oi aeey aoaeue-yeiai ia?anoaaeaiiy (={((k)} iaoo?aeueiiai ?yaeo, ia?anoaaeaiee ?yae ((k=1 a((k)f((k) (x) ca?aa?oueny iaeaea anthaee, ? aeey iaaei?aioe /anoeiaeo noi S*( oeueiai ?yaeo ni?aaaaeeeaa ioe?iea (( S*( ((2(c(1/2, aea noaea n ia caeaaeeoue a?ae ( . iai ia aaeaeiny ae??oeoe ieoaiiy, /e ni?aaaaeeea? oai?aie 13.1 ? 1.3.2 i?e (=0. O i?ae?icae?e? 1.5 iaie iieacaii, ye, nie?ath/enue ia a?aeii? oai?aie i?i nenoaie iacaeaaeieo ooieoe?e, iiaeia iiaoaeoaaoe i?eeeaae S(p,()- nenoaie , yea a oie aea /an ia ? S(p,(()- nenoaiith aeey aeiaeii? iine?aeiaiino? (( c (‘n=o((n) i?e n((. Ia?aeaeaii aei iaeyaeo ?acoeueoao?a ae?oaiai ?icae?eo. Ia?o? ioe?iee ii?ie ooieoe??, yea ? noiith aeayeiai ?yaeo ca o?eaiiiiao?e/iith nenoaiith, aoee iaea?aeai? Oaonaei?oii ? THiaii, a oaeiae Oa?ae? ? E?ooeaoaeii. I?ci?oa Iae? iioe?ea oeae ?acoeueoao ia an? i?oiii?iiaai? nenoaie , iaiaaeai? a noeoiiino?. Oai?aia A. Iaoae {(n(x)} - i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1] ? (((n((((M (n=1,2,...). Oiae? , yeui q>2 ?

(q(c)=(((n=1(cn(qnq-2)1/q<( , oi ?yae ((k=1 ak(k(x) (2) ca?aa?oueny aei f(Lq[0,1] ? ((f((q(c(q(c) . I?ci?oa oeae ?acoeueoao ?icaeaaany a ??cieo iai?yiao. Iai?eeeaae, Noaei iaea?aeaa ioe?iee ii?i ooieoe?e o i?inoi?? Ei?aioea. Iniiaiei caniaii iaea?aeaiiy ioe?iie oaeiai oeio ? aeei?enoaiiy ?ioa?iieyoe?eieo iaoiae?a. O iao?e ?iaio? ie canoiniao?ii i?aeo?ae, yeee aaco?oueny ia ioe?ieao iac?inoath/eo ia?anoaaeaiue ? yeee, iaaooue, aia?oa cai?iiiioaaa Iiioaiia??. O i?ae?icae?e? 2.2 ie aeiaaaeaii, ui a oiiaao oai?ai Iae? oa Noaeia i?oiii?iiaaio nenoaio, iaiaaeaio a noeoiiino?, acaaae? eaaeo/e, iiaeia cai?ieoe ia nenoaio, a ye?e iaiaaeai? a noeoiiino? iai?aii?ie o AII. Iaie iaea?aeaia Oai?aia 2.2.2. Iaoae {(n(x)} - i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1], (n(AII ? (((n((*(M (n=1,2,...). ?) Iaoae f (Lp,r (12, 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,s<( (q>2, 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 <( , ((n =(n(s) , (n=(((k=n a2k)1/2) , oi ?yae (2) ca?aa?oueny a L2[0,1] aei aeayei? ooieoe?? f , i?e/iio ((f((q,r(cq,r,s (q,r,s(a). Iniaeeai oe?eaaee a?aie/iee aeiaaeie oe??? oai?aie, eiee q=2. Ooo ni?aaaaeeeaa Oai?aia 2.5.6. 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 02 , 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 <( , ((n =(n* , (n=(((k=n a2k)1/2) , oi ?yae (2) ca?aa?oueny a L2[0,1] aei aeayei? ooieoe?? f, i?e/iio ((f((q,r(cq,r{((f((2+(q,r(a)}. Aeiaaaeaiiy oeeo ?acoeueoao?a a?oioothoueny ia aeanoeainoyo iac?inoath/eo ia?anoaaeaiue ooieoe?e. Canoinoaaiiy oaeiai iaoiaeo aeicaiey? c?iaeoe aeiaaaeaiiy aeinoaoiuei i?inoeie ? aeineoue oi?aa?naeueieie. e??i ia?a?aoiaaieo aeua ?acoeueoao?a, o i?ae?icae?e? 2.5 i?noeoueny oaeiae ?yae ioe?iie, ye? ? i?yiei ocaaaeueiaiiyi oai?aie B ia aeiaaeie i?inoi??a Ei?aioea Lq,r(q(2,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/r<(, , (=r-1 - r(s-q)/(q(s-2)), oi ?yae (2) ca?aa?oueny a L2[0,1] aei aeayei? ooieoe?? f, i?e/iio ((f((q,r(cq,r,s Dq,r(a). O i?ae?icae?e? 2.6 oeueiai ?icae?eo iaaaaeai? ioe?iee eiao?oe??io?a Oo?’? ooieoe?e ?c i?inoi??a Ei?aioea a oa?i?iao aaee/ei, ye? aeei?enoiaoaaeenue o i?ae?icae?e? 2.5. Oe? ioe?iee aeai?no? oai?aiai ?c 2.5. Oai?aia 2.6.1.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,...) . sseui q(( s/(s-1), 2] , r([1,q(), f(Lq,r[0,1] ? {an}– eiao?oe??ioe Oo?’? ooieoe?? f ca nenoaiith {(n(x)}, oi Dq,r(a) (cq,r,s ((f((q,r . O a?aie/iiio aeiaaeeo q=2 ia?ii ianooiia oaa?aeaeaiiy. Oai?aia 2.6.2. Iaoae {(n(x)} - i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1] ? i?e aeayeiio s((2,+(] . sseui r>2, 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)<( . sseui ((t)(0 c?inoa? ia a?ae??ceo [0,1] ? ((t)/t - niaaea? , oi iiiaeeio ooieoe?e, aeey yeeo ((f((M(()=sup0<((1 1/((t)((0f*(t)dt<( iaceaathoue i?inoi?ii Ia?oeeieaae/a. Caaaeueiee i?ino?? Ei?aioea, ye ? i?ino?? Ia?oeeieaae/a, ? e?i?eieie ii?iiaaieie neiao?e/ieie i?inoi?aie. Iaoae oaia? {(n(x)} - i?oiii?iiaaia nenoaia ia a?ae??ceo [0,1] , i?e an?o n=1,2,... (n(L( , (((n((((Mn , (nk=1 M2k=Bn (n=1,2,...). O aeaiiio i?ae?icae?e? ie iaea?aeaee oae? oaa?aeaeaiiy, ye? ocaaaeueiththoue ?yae ?acoeueoao?a A.O.?iae?ia ia aeiaaeie nenoai, ye? ia ? iaiaaeaieie a noeoiiino?. Iaoae (f(s/(), 0(s(min(1,(), (( f(s)=( (0, (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], 20 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 (12, 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))<(. ?yae ((k=1 ck(k(x) noiaeeony a L2[0,1] , ii aai noiia ia i?eiaaeeaaeeo Lq[0,1]. Aeaeaa, i?eiaiaieai ioeaiie iaaic?anoathueo ia?anoaiiaie iieo/aiu iiaua ioeaiee ii?i ooieoeee a i?ino?ainoaao Ei?aioea /a?ac eo eiyooeoeeaiou Oo?uea ii iauei i?oiii?ie?iaaiiui nenoaiai e aeaienoaaiiua ei ioeaiee eiyooeoeeaioia Oo?uea. Enneaaeiaaia ieii/aoaeueiinoue iieo/aiiuo ?acoeueoaoia. Iaie onoaiiaeaiu oaeaea iiaua ioeaiee eiyooeoeeaioia Oo?uea ooieoeee ec eeannia I?ee/a, aaniauo i?ino?ainoa Lp , iaueo neiiao?e/iuo i?ino?ainoa. Eeth/aaua neiaa: i?oiii?ie?iaaiiay nenoaia, iiiaeeoaeue Aaeey, iaaic?anoathuay ia?anoaiiaea, eiyooeoeeaiou Oo?uea, i?ino?ainoai Ei?aioea, AII. Kirillov S.A. Convergence of series over some orthonormal systems and estimates of coefficients.-Manuscript. Thesis on the degree of Candidate of Sciences (Physics and Mathematics) by speciality 01.01.01- mathematical analysis.- Odessa State University, Odessa, 1999. In the thesis we introduce a new class of S(p,()-systems which is an expansion of well-known class of Sp -systems. We obtain Weil’s multipliers for convergence and unconditional convergence almost everywhere for systems from this class. Well-known theorems by Paley and Stein which concern with estimates of norms of functions over their Fourier coefficients are generalized to systems from BMO. We prove that a hypothesis which was proposed by P.S.Bullen on the beginning of 50-th and had deal with improvement of a Marcinkiewicz and Zygmund theorem is not true for general orthonormal systems. Applying estimates of rearrangements we get new estimates of norms of functions in Lorentz spaces over their Fourier coefficients with respect to general orthonormal systems. We also get conjugate estimates of Fourier coefficients. We prove the ultimateness of our results in some respect. A new estimates of Fourier coefficients for functions from Orlich classes, weighted Lp-spaces and general symmetrical spaces are obtained. Key words: orthonormal system, Weil’s multiplier, nonincreasing rearrangement, Fourier coefficients, Lorentz space, BMO. PAGE PAGE 14

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