Analysis Mathematica, 2 (1976), 77--85
HegoTopb~e TOqllble HepaBeHCTBa B TeOptlll npu6.anaceHua ~yHgnm3
3-I. B. TAIYIKOB
B HeKOTOpI, IX 3a~aqax Teopnn n p a 6 n m r e n n a qbyrrKIm~ TOqrrble I~epaBerrcTBa
no3aonamT ycrarmsr~Tl, Hoaue CBa3H Me~r~ty rOHCTpyKTnBHI, IMU n cTpyKTypub~H
CBO~CTBaMn qbynrlm~. HOaTOMy aasecTnoe nepasencTao ~ reKcoHa , coRep)ratuee
ot~enrn ae~Inqni~u rtan~lyqmero nprI6~rI~eHn~t qbyHrttnr~ IIOJIHIfOMaMH IIocpe)ICTBOM
MO~y~a HenpepbIBnOCTn ee npon3BO~X~HO~ npoa3so) Ino~ (CM., nanpnMep, [1, CTp.
230]), HHTeHCnBHO n3yqazocb s noc~e)IHee BpeM~ C n~e~bm ero OnTHMH3attmi
(H. H. K o p H e ~ q y K [5], [6], [7], H. H. q e p H J , IX [3]). Ho)Io6rma 3a)~a~a nce~e-
~yeTC~ B w 1 nacToatttefi pa6oT~ )IZa HerOTOp~Ix r~accoB aHa~nTnqecrnx qbyrrrtmfi,
r~e ocuoBono~arammnM aBnaeTca p e 3 y ~ T a T K. H. B a 6 e r t r o [2].
B w 2 nposo~aTCa TOqHbIe o~er t rn ae~nqnm,~ IIpoIt3BOJlbIq[O~ qbynrI~nn qepe3
ee MO;ay~, HenpepuanocTn n MO~ya~, nenpep~,mnOCTH ee BTOpO~ nponaaO~HOfi.
3 Z n otteHKrt aHa~ornqnu HaaeCTHOMy HepaBeHCTay 2 / a n ) l a y - - A ~ a M a p a [4], co~ep-
~rattteMy Be~nqnnu qbynrttnn n ee nepBo~ i~ BTOpO~ npon3so~HbIX. B 3ar~roqenne
BUBO)~aTC~ cae)~CTBna, co~ep)ra tuae TO,HUe nepaaencTaa, r o T o p u e a n a ~ o r n q a u
nepaBeHcTBau BepnmTe~Ha a Xap~n.
B ~aJibHefimeM n o r BeanqnHOfi qbyHrttnn rIOnnMaeTca ee UopMa B MeTpnre
IIpocTpaHCTBa Lp, II Bce onpe~e~ei~rla n ~OKa3aTeJlbCTBa npr~cnoco6J~enb~ K 3TOMy
r O~HaKO IIOVlTI, I Bee pe3y~IbTaTI~I cIIpaBe~IHBbI kI B MeTpkIKe IIpocTpaHCTBa
OpJltlqa, lqOCKOJ~bKy ~oKa3aTe~IbCTBa IIO cytueCTBy He MeHJtIOTC~I.
w 1. OueHKH Hax.ayqmux npu6.anxexu~ aHa.aUTnqeCKUX B Kpyre ~yurdm~
I'[yCTb f ( z ) - nporI3ao~buaa aI/adlriTHqecraa BI~JTprl e~nlmqrtoro r p y r a
~yaxttvia,
f(z) = ~ Ck zk, Z = oe it, 0 ~_ O "< 1. k=O
I 'IocT~O 29 oxcra6pa 1974.
78 YI. B. Ta~fxoB
IIpocTpaHCTBO Hp (1 <_--p ~ r COCTOHT It3 acex dpyHKRrIfi f(z) C KOHeqr~ofi HOpMOfi:
f I 2~ "11/,,, ILfll = Ilflt,,,, = f Lf(qe")l" d*J < = .
p - - l - - 0 [gT~ 0
PaCCMOTpHM IcJIacc ~byHKRI4~ {q)(Z)} C lip, yrnoBl~ie rpaH~Hble a~aseH~Ia KOTOpbIX ~o(e ~x) iipe~cTasyieHJ, i cBepTKOfi
1 2~ (1) q~(e~) : -~x / f(e~(~ +t)) K(t ) art,
r~e f(z)EHp, a K(t) - - CyMi~IpyeMa~/ nO Yle6ery qbyHKlarla C pa~OM ~ypl~e
1 K(t) ~ ~-2o+ ~ ~kcoskt.
OTMeTHM rlarl6oJIee npocTym CBg3b Mem~ly HaHYIyqlI/rlMrI IIpH6JIrlXeHrI.qMH ~yv/ztiHH q~ (z) nOm4HOMaMi~
P~(z) = ~ akz k, E,(qO = inf I1~o-- P~}I k=O Pn
H HaH:IyqmrlMa npH6aaxe~HaM~ E~(f).
.rIeMMa. EcJtu cp(z) aenaemca ceepm~ofi u cym~tupye~aa fynmtua Kn(t ) c pabo~t ~ypbe
1 K.(t) ~ ~-2~+ ~ 2~+, coskt
k = l
neompu~ameabna, K. (t) >= O, mo E.(~o) <-- 2,E,(T)
u paeencmeo bocmueaemea na O@tac~uu z".
~oxasaTea~,CTSO. 17ycr~, P*(z) nosmrmM Harlayamero npn6arlxeHns qbyrm-
E,(T) = [I/-P*I],
Tor~a Henocpe~cTBemro~ rtpo~ep~o~ MOXrm y6eaI~r~,ea, nTO 2~
q~(eix)-P.(e ix) = ~ / 2K"(t){f(ei'x+~
r~e P.(efO - - HeKoTop~,IB ilOarmOM, 3aBncamHfi OT f(z). IlprlMei~aa miTerpasli.i~oe HepaBencTsO MnHKOSCKOrO, nozynnM
1 2~ E.(~p) <= {[~0--P.{I < I{f-P*ll~-~n / 2K.(t)dt = 2.E.(f) .
~oxa3aHHaa YleMMa rio cymecTBy co~epxaT TO, •TO 6~,IaO 3aMeqelto aBTOpOM [8], [9] npr~ nsyae~rm o)moro pezyabTaTa K. H. Ba6er txo [2].
HeroTop~,~e nepaae~ tcTsa B T e o p n ~ npe6smTxen~a~ 7 9 ~
I/IMe~ uesIbIO yCTarlOBHTb caa3b uannyqtunx np~6nmreHnfi artaJIrtTrtqecI~rix
a rpyre qbynIglIrl~
f ( z ) = ~ CkZ ~, z = ee ~', 0 < - - 0 < 1 , k = 0
c raa~rocTbiO rpannqHbxx 3naqeunB f ( e ~ ) = F ( t ) , rIoTpe6yeM, qTO6U nporI3Bo~na~t
i f ( z ) no arg z npnnaaaez<a~a npoeTpaueTBy H~. :3~0 yeaoane nO3BOaneT npe~cTa- BnTb 3naqenna qbyrlI~IIrIn B BHae cBepTrrI aByx qbynrttn~, O)IHO~ n3 I~OTOpr~X ~IBJDIeTC~
nponaBo~Haa F' ( t ) rpaur tqaux 3Haqerm~ f (z ) :
! ~ I F ' ( t ) = lira df(z) dz _ lim i ~ ckkz k . Q~I dz dt ~ t k=X
By~eM C~IaTaT~,, ~ITO no oTnomemlm r cTapmnM IIpOH3BO~HbIM F (') (t) Bt,xno~InaeTca
anaaornqaoe ycaoBrie, Re o roaapaaaa 3TO r a ) r ~ i B pa3 B )IaabHeBmeM. Faa)IKOCT~ rpanmInbix 3na~ennfi F( t ) 6y~eM xaparTepn30aaTb nop~)~rOM:
rIpon3BO)~HO~ F(')( t ) n e e MOgyaeM nenpepl, mrtOCTU:
{ 2 . i }~lP o~(F%,~) = sup f J F ( ' ) ( t + h ) - F ( ' ) ( t ) [ " d t
I / I3 JIOMMbI BbXTOKaOT
Cae~aCTane 1. ~ r ~ ~tO6bZX namypastbttbtx r u n cnpaeet)~uebt ~tepaeencmocc
E , ( f ) ~_ n-" E , ( F (~))
u paeencmeo Oocmueaemc~ t)st~ f ( z )=z" .
CsIegCTBrie 2. l lpu yc~onu~x c~et)cmeu,~ 1 uMeeM
E , ( f ) ~_ n -~ IIF(')I{.
~OCTaTOqHO noscnnT~, nepasencTBa cs~e~eTBna 1. Eean noao~rnTb 2 k = k - ' B yeaoBaax aeMM~, TO <p(e ~x) ora~reTea r-o~ n e p s o -
o6pa3nofi rpaHHqna,xx 3HaqeHH~ f ( e ~) no nepeMenrto~ x.
T e o p e M a 1. ,~ ta ,~tO6b~X uamypastett~tx r u n cnpaeeO,~uabt nepanettcmaa
E , ( f ) <= 4rt_ ~ f to(F('), t )d t . 0
Paeeucmeo Oocmueaemea ua 06yutaluu z".
~ o r a s a x e ~ , e x a o . Cor~xacno eze~cxsnm 1
1 E. ( f ) _~ -k-: r ~. (V~'- %
80 JI. B. TafixoB
B CBI;I3It C 3THM AOCTaTOqttO AOKaBaTb HepaBeHCTBO
e , ( f ) <= --~ f ( ~ ( F ' , t )dt .
Hostoz<rrM
l,'IMeeM
n/2n n
A ( f , t ) = A ( F , t ) = ~ f F ( x + t ) c o s n x d x = --~t2n
~/2n R
= T f {F(t + x) + F(t - - x)} cos .x dx. 0
En ( f ) <= E. {F(t) - A (F, t)} + E. {A (F, t)}.
I/IrITerpnpya rio qaCT~lM, noayqHM
nl2n n / {F(t + x) -- 2F(t) + F(t -- x)} cos nx dx = e ( t ) - a ( f , t) = - g
u , n/2n
= 2 / { F ' ( t + x ) - - F ' ( t - - x ) } { l - s i n n x } d x .
llpI,IMeH~leM iiHTerpaJlbHOe tIepaBeHCTBO MFIHKOBCKOrO:
(2) E.{F-A(F)} <= IIF-A(F)ll
1 ~12. <= -5 f IlF'(t W x ) - F ' ( t - x ) l l { 1 - s i n n x } d x .
o
C ~ a r a e M o e E.{A(F, t)} oue~lrtM c n a q a a a corJ iacr lo cJIe~ic+aruo 1, c o r J I a c n o cJIejlCTBHIO 2, IIO.ROX(HB B O60HX cJIyqa~x r = 1:
(3) E. [ f ( t + x) + f ( t -- x)] cos nx dx <- O
<:~ E n i 2 / [ F t ( t - - ~ x ) - l - g t ( t - - x ) ] c o s n x d x =
= E. [F(t + x) -- F(t - x)] sin nx dx <= 0
<- 2 / [ F ' ( t + x ) - F ' ( t - x ) ] s i n n x <=
1 ,~12. <= - i f~ llF" (t + x)-- F' (t-- x)ll sin nx dx.
a 3aTeM
He~oTOpl~le HepaBeHcTBa B Teoprm npn6~mmerm~ 81
O6~,e~mmeM otter~Kn (2) n (3)
l nl2n 1 nln / < f o(F', t)dt. (4) E, ( f ) ~_ [ IF ' ( t+x ) -F ' ( t - x ) l l dx = -4 o
TeopeMa 1 no~nocT~,m )xoKa3arra.
OTMeTnM, ~TO rmpa~ericTBo Trlna (4) 6biao iioay,~eno B 1961 r. H. I I . K o p -
Hefi '~yKOM [5] ~ I a Ksmcca 2zc-nepno)m~ecKnx qbyrmImfi f(x) c m, xny~c~r~iM Mo)xy-
~eM Henpepr~mHocTri og( f ' , t ) B MeTpn~e npocTparmT~a C.
w 2. Crpy~rypmae cnoficTna ~m~@epemmpyeM~,ix ~ynlcm~fi
3~ecb IIpHBO~ITC~ TOq_HbIe oI~eHKtt BeJIHqHHI, I IIpOI43BO~HOfi qbyHralHn qepe3
MO~yYtrI rieIipepbiBrlOCTrI CaMO~ qbyHKlatI1/ H ee BTOpO~I IIpOH3BO~rIO~. BMeCTe
C rpaHriqH/~iMri 3HaqertrlgMri .F(t)=f(e it) aHa~nTnqecKnx qbynK~n~ MOmrtO paCCMOT-
peTr~ qbyHKtmH f(t), 3a~anrmm Ha qnC~OBOfi OCn C HOpMO~
+~
ilfll = { f If(Ol" at} p 1,
nsm 2~-nepHo~HqecKHe ~yHKUtm C I-tOpMO~
I!711 = If(t)lPdt~ , -lz
HOBO Bcex c~yqaax cymecTBOBatnIe npoit3Bojltto~ qbyHKUm~ 03tta,taem, "iTO qbyineumo
MO~HO IIpe}lcTaBHTt, HeonpeTleJIeHHLIM t tHTerpa~oM .rIr OT ee IIpOH3BO}IHOH.
B ~iaCTttOCTt~, npon3BO~tHaa qbyH~UU~ eCT~> Heonpejlenetm~i~ nHTerpas~ J Ie6era er
BTOpO~ IIpOHSBO,/~HOfi. 3aBI4CHMOCTb HOpMbl OT Be.IIHqI, IHbI p MbI He yKasbIBaeM,
rmc~o.ab~y o~onaaTea~,Hme pe3ynbTaTbI n fix ~oga3aTem,cTBa He 3aBI,ICaT OT p.
Ta~, nanprrMep, ) ~ a nepno)maecKnx qbyHKtmfi nMeeT MeCTO
T e o p e M a 2. ,ll~n mo6ofi 2zc-nepuoc)u,.wcnofi ~ymcguu c a6comomno nenpepbta- nofi npoussoOno?t u mo6oeo 2 > 0 cnpaseO~uso nepaeencmao
Hi'li : i <o(i", {'--sint}at+= oi o {i, }sinta', eOe pase.cmeo Oocmueaemr c)~>~ 2=n u f . ( t )=a cos (nt + b).
~oKa3aTe.~IbCTBO. HOYIO~IIM, KaK rl paHee,
:t A (: ' , x) = T f f ' ( x + t ) c o s 2 t d t
- ~ 1 2 ~
~/2;~
= 2 / { f ' ( x + t ) + f ' ( x - t ) } c o s 2 t d t .
6 Analysis Mathematiea
82 .rl. B. Ta~iroa
HMeeM
l l f ' l l ~ I I f ' -a( f ' ) l l + I I h ( f ' ) l l =
~ ~/2~ d t+ = f {--f'(x+t)+2f'(x)--f'(x--t)}cos2t 0
~__ ~/2z cos 2t art + f {f '(x+t)+f'(x--t)} = o
1 nI2Z art + = 2 f { f" (x+t) - f" (x- t ) I{1-s inXt}
+ s ~2~ {f(x + t) - f ( x - t)} sin 2t art 2 { "
Ta~HM o6pa3oM, nocae 3aMeHbI nepeMeHHO~ H nprIMeHeHHn HepaBeHCTBa MHH-
KOBCI(OFO IIO.rlyqI4M
~---~ yz f"[ % } - f " ( - t ] ] {1-sint}dt+ (5) IIf"ll <= x + x 2 ) I
+ 2 ~ ~ f [ x+2} - - f [ x - -2} sintdt,
OTKy~ta c~e,ayeT Tpe6yeMoe HepaBeHCTBO. PaBeHCTBO npoBepaeTca HeIIocpe)ICTBeH-
HblM BbI~II4CYleHHeM.
3aMeTrIM, qTO B c~y,~ae anaznTnqecKHx qbyHKtmh nnI~ qbynKttrih, 3a~anr~bix na
tI~iCJ/OBO~ OOH, HepaBeHCTBO (5) OcTaeTc~ B TOqHOCTH TaKHM ~Ke, a ~OKa3aTeJIBCTBO
aHaYlOYOB TeOpeMbI 2 nOBTOpaeTCg ~OC.rlOBHO. PaBeHCTBa ~ t aHa0/HTHqecKHx
qbyHICI~I4~ AOCTI4ratoTCfl npu 2 = n n f,(z)=z", a B cJIyqae qbyHlCl~rlfi, 3a2IanHbtX Ha
qncJIoBo~ ocrI, rrepaBeHcTBa He yJIyqtuaeMb~ npn ~rO6OM 2 > 0 . Hpr~MepOM MO~eT
c~yzcnTt, noc~e2lOBaTem, HOCTb dpyHKttrI~ f , ( t ) = COS At r lpa -- 2rcn/2 <= t <= 2rm/2 rI rlpOJIOYDI~eHHblX ~OCTaTOHHO FYla,~KO ,/~O TO)t(~eCTBeHHOFO HyYDI BHe yl(a3aHHOFO
oTpe3ica.
HocpeacTBOM nepaBencTaa (5) H HepaBeHcTBa Tuna 13epHmTefiHa MOaCrto nosiy-
qHTb o u e u r n rIpOl, I3BO)IHbIX Ha K~accax ue~,lX qbynKttn~ '~epe3 fix Mo~yJII, I nenpep~,IB-
rlOCTn. HpnBeaeM npnMep
T e o p e M a 3. ,/bin npou~so~bnoeo mpueono ~emputtectcoeo nostunoMa
1 Tm (x) = -~ ao + Z (ak COS kx + bk sin kx) k=l
HeKoTopbte nepaBencTaa B TeOpHrt nprt6nrt~e~Ha 83
cnpaoeO~uao nepaeencmao
IlTm~)ll <=-~m co Tm, dx, r = 1,2 . . . . .
u paaeucmao Oocmueaemca, ec/tu T m ( x ) : a cos (rex + b).
}~oKa3aTe~I, CTBO. B HepaBer~cxae (5) nos~o~r~M 2=m,f (x)=Tm(X) , I~ npr~- Mer~aa nepa~eHcT~o 3nrMyH~aa ([10, cxp. 20----22])
rto3IyqUM:
[T,~' ( x + t } _ T,~' ( x - t ] l <- m ~ [Tm I x + t I - Tm { x - t } [,
IITm'll --< T ~ m / x + -T~ ' x-- { 1 - s i n t } d t +
+ 2 / Tm X+-~ - m X-- s intdt <=
,/:[ora3aTe~bcxBo 3aaepmaeTca, ec~H cua,~aza BOCnOJlb3OaaTbCa nepaaertcTaOM 3nr- MyH~ta
IIT~')II - m'-XllTm'l[,
a 3aTeM HpHMeHHTb ~ o r a 3 a H H o e HepaBeHCTBO.
HpnBe~eM eme O~l~rt nprIMep npnno~rertHn HepaaeHcTaa (5). C 3TOfi tlem,~o orpannnnMCa qbynrttnaMrI f(x), onpe~eaeHnUMn rm sce~ qrlc~oso~ ocri c HopMo~ Ilfll ~ CMUC~e npocTpartcTsa Lz(--o% + ~ ) .
T e o p e M a 4. ]Inn n~o6btx uamypanbnbtx qucen 0 < k < n u am6oeo 2 > 0 cnpa- aeO~uao nepaeencmao
IIf<k)ll = co f , - - e ( t )d t +-~
e0e n - k + l n - k - 1
P(t ) - - - 22 sin tq- - - (1 - s i n t ) ,
n n2 ~
k - I Q(t ) =
n
k + l .. ;t ~ sin t + ~ (1 - sin t).
6*
84 51. B. T a ~ o a
Hepaeencmeo ney~yuutaemoe, mo ecmb ~)na tca~cOoeo 2 > 0 cyutecmeyem nocnec)o-
aamenbnocmb fymc~tu~t fm( t , 2), Ona tcomopbm omnoutenue npaao(t ,~acmu nepaeencmaa
tr .~eeo~t cmpe~fumca tr e3unu~te npu m-~ co.
o r a3 aTe~ii , CTB O. OIIeHr~i I1 f(k)]l cor:IaCHO rrepaBeHCTBy (5)
llf(k)l] ~ x + ~ x - - { 1 - s i n t } d t + = 2 2 o , , [[
2 ~/2 It t
HOpMN qbyHKI~n~ nO)I 3uaroM ~HTerpanoB 6y~IeM oIIeH~BaTb corJIaCHO uepa -
BeHCTBy X a p ~ [4]
lie(re)l[ < !1~011 ("-~>/m II~p(")ll m/", 0 _-< m <-- n,
IIpHMeH~I e ro B 3KB/,IBaJIeI-ITHO~ qbopMe
r l - - m Ilq~r < ~ II~pll +m2"-mll~<")ll.
n
I-loJmraa crla~Iasm m = k + l , 3aTeM m = k - - 1 rI Bl~i6rlpaa B o 6 o n x cJIyqaax
q,(x) = ~ , / ~ (x ) = f x + T
Ho~yqHM
(k+~) n - k - 1 k + l 2 ,_k_ 1 A(~)~ [th2t/~ II < n~,k+l IIA~,/~II-I- 2t/211, n
[[d~7/~ 11 ..--'- 1).. < n - k + l IrA H k - 1 ~n-k+llA(n) I n2k_ 1 1~2t/,~ll-t n "" ,-2t/x
Ec:iri IIO~CTaBHTB 3TH oIIeHKH IlO~ rlHTerpa:Ibi, IlprlBeCTrI Im~O6able tLrlermi
H oIIeHHTt, n p n p a m e r m a MO~yJIeM HelIpepBIBHOCTH, TO IIOJIyqHM HepaBeHCTBO
B T e o p e i e 4.
TOqI~OCTb rlepaBeHCTBa npoBep~IeTC~ rla TOil me noc~e~oBaTe:ibHOCTri ~yn-KIirI~, ttTO H B TeopeMe 2.
.)-lHTepaTypa
[I] H. I/I. A x a e s e p , JIetcttuu no meopuu annpotccumattuu, HayKa (Mocha, 1965). [2] K. I/I. t ; a6e r i ro , O HaHJIyqIIIeM np~6Jmaceuau o~ioro icaacca auaJmT~iecKux qbynmm~.,
lisa. A H CCCP, cepmq MaTeM., 22 (1958), 631----640. [3] H. H. qepHl , tX, O uamlymaleM npH6Jmxceuma I lepao~ecmlx qbymom~ TpHToItOMeTpHtlee-
ICHMa IlommoMavm B L2, Mamem, ~amemtcu, 2 (1967), 513--522.
HeKoTop/,Ie HepaBeHCTBa B TeoprlrI IlpH6JIrt,xem, la 85
[4] F . F . X a p j I a , )Ix.E..J-I/,ITTJIbBy,/~ H F. IIoJlrla, Hepa6eucmaa, HHocTparman anTepaTypa (Mocl~a, 1948).
[5] H. I I . KopHet tqyK, O HaHJiymiieM paBnoMepnoM npn6Jm,xemm /mqbqbepenlmpyeMblX ~yHrtml~, ,lIotr A H CCCP, 141 (1960, 304---307.
[6] H. 17. Koprie~t~lylq To,man rOnCTaaTa B TeopeMe ~. ~meKcona o namlymlieM pamIoMepnoM Ilpn6Jmxemm rienpepl~mnl, IX nep~tojm~iecKax q b ~ , ,/[otcJt. A H CCCP, 154 (1962), 514---515.
[7] H. FI. K o n e ~ y ~ , ~rcTpeMaJIbnBIe 3i-ia,-lem, Ia qbyrlKl~onaJIoB n Ha.BJIymJIee npn6mixenne na iolaccax nepno~m,teclmx qbynmm~, Plsa. A H CCCP, ceprm MaTeM., 35 (1971), 93--124.
[8] 5I. B. Ta~xoB, O Ita~aymlmx Ji~mefmr~Ix MeTo~Iax np~i6Jmxerma i~JlaCCoB B r r~ H ~, 3rcnexu a*amem, naytr 18 (4) (1963), 183--189.
[9] 3-I. B. Ta~xoB, O Hal~lyqnleM llprt6JIrlXerma a cpe~HeM HeKoTopBIx i~aaccoB aaaJmTnqeclmx dpymamrJ, Mame~t. 3amermcu, 1 (1967), 155--162.
[10] A. 3rlrMyr~t, TpueonoMempuuectr pm)bt. II, Map (Mocl~a, 1965).
Some exact inequalities in the theory of approximation of functions L. V. TAIKOV
Exact inequalities are obtained that illuminate the interrelation between best polynomial approximations of functions, analytic in the disk and the modulus of continuity of the derivatives of the boundary values of these functions.
For various classes of functions exact estimates are given for the derivative of a function by means of the modulus of continuity of this function and tbe modulus of continuity of its second derivative.
As application, exact inequalities are deduced, analogous to the well-known Bernstein and Hardy inequalities.
JI. B. TAI~KOB CCCP, MBITHII]I~ 141 001 MOCKOBCKI41~ JIECOTEXHHqECKI41~I HHCTHTYT
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