Некоторые точные нер авенства в теории при ближения функций

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<ul><li><p>Analysis Mathematica, 2 (1976), 77--85 </p><p>HegoTopb~e TOqllble HepaBeHCTBa B TeOptlll npu6.anaceHua ~yHgnm3 </p><p>3-I. B. TAIYIKOB </p><p>B HeKOTOpI, IX 3a~aqax Teopnn npa6nmrenna qbyrrKIm~ TOqrrble I~epaBerrcTBa </p><p>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 </p><p>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. KopHe~qyK [5], [6], [7], H. H. qepHJ , IX [3]). Ho)Io6rma 3a)~a~a nce~e- </p><p>~yeTC~ B w 1 nacToatttefi pa6oT~ )IZa HerOTOp~Ix r~accoB aHa~nTnqecrnx qbyrrrtmfi, r~e ocuoBono~arammnM aBnaeTca pe3y~TaT K. H. Ba6er t ro [2]. </p><p>B w 2 nposo~aTCa TOqHbIe o~ertrn ae~nqnm,~ IIpoIt3BOJlbIq[O~ qbynrI~nn qepe3 ee MO;ay~, HenpepuanocTn n MO~ya~, nenpep~,mnOCTH ee BTOpO~ nponaaO~HOfi. 3Zn otteHKrt aHa~ornqnu HaaeCTHOMy HepaBeHCTay 2/an) lay- -A~aMapa [4], co~ep- </p><p>~rattteMy Be~nqnnu qbynrttnn n ee nepBo~ i~ BTOpO~ npon3so~HbIX. B 3ar~roqenne BUBO)~aTC~ cae)~CTBna, co~ep)ratuae TO,HUe nepaaencTaa, roTopue ana~ornqau nepaBeHcTBau BepnmTe~Ha a Xap~n. </p><p>B ~aJibHefimeM nor 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. </p><p>w 1. OueHKH Hax.ayqmux npu6.anxexu~ aHa.aUTnqeCKUX B Kpyre ~yurdm~ </p><p>I'[yCTb f ( z ) - nporI3ao~buaa aI/adlriTHqecraa BI~JTprl e~nlmqrtoro rpyra ~yaxttvia, </p><p>f(z) = ~ Ck zk, Z = oe it, 0 ~_ O "&lt; 1. k=O </p><p>I'IocT~O 29 oxcra6pa 1974. </p></li><li><p>78 YI. B. Ta~fxoB </p><p>IIpocTpaHCTBO Hp (1 </p></li><li><p>HeroTop~,~e nepaae~tcTsa B Teopn~ npe6smTxen~a~ 79 ~ </p><p>I/IMe~ uesIbIO yCTarlOBHTb caa3b uannyqtunx np~6nmreHnfi artaJIrtTrtqecI~rix a rpyre qbynIglIrl~ </p><p>f (z) = ~ CkZ ~, z = ee ~', 0</p></li><li><p>80 JI. B. TafixoB </p><p>B CBI;I3It C 3THM AOCTaTOqttO AOKaBaTb HepaBeHCTBO </p><p>e , ( f ) </p></li><li><p>He~oTOpl~le HepaBeHcTBa B Teoprm npn6~mmerm~ 81 </p><p>O6~,e~mmeM otter~Kn (2) n (3) </p><p>l nl2n 1 nln / &lt; f o(F',t)dt. (4) E,( f ) ~_ [ IF ' ( t+x)-F ' ( t -x) l ldx = -4 o </p><p>TeopeMa 1 no~nocT~,m )xoKa3arra. </p><p>OTMeTnM, ~TO rmpa~ericTBo Trlna (4) 6biao iioay,~eno B 1961 r. H. I I . Kop- </p><p>Hefi'~yKOM [5] ~ Ia 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. </p><p>w 2. Crpy~rypmae cnoficTna ~m~@epemmpyeM~,ix ~ynlcm~fi </p><p>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~ </p><p>+~ </p><p>ilfll = { f If(Ol" at} p 1, </p><p>nsm 2~-nepHo~HqecKHe ~yHKUtm C I-tOpMO~ </p><p>I!711 = If(t)lPdt~ , -lz </p><p>HOBO Bcex c~yqaax cymecTBOBatnIe npoit3Bojltto~ qbyHKUm~ 03tta,taem, "iTO qbyineumo MO~HO IIpe}lcTaBHTt, HeonpeTleJIeHHLIM ttHTerpa~oM .rIr OT ee IIpOH3BO}IHOH. </p><p>B ~iaCTttOCTt~, npon3BO~tHaa qbyH~UU~ eCT~&gt; Heonpejlenetm~i~ nHTerpas~ J Ie6era er BTOpO~ IIpOHSBO,/~HOfi. 3aBI4CHMOCTb HOpMbl OT Be.IIHqI, IHbI p MbI He yKasbIBaeM, </p><p>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 </p><p>TeopeMa 2. ,ll~n mo6ofi 2zc-nepuoc)u,.wcnofi ~ymcguu c a6comomno nenpepbta- nofi npoussoOno?t u mo6oeo 2 &gt;0 cnpaseO~uso nepaeencmao </p><p>Hi'li : i </p></li><li><p>82 .rl. B. Ta~iroa </p><p>HMeeM </p><p>l l f ' l l ~ I I f ' -a(f ') l l + I I h ( f ' ) l l = </p><p>~ ~/2~ dt+ = f {--f'(x+t)+2f'(x)--f'(x--t)}cos2t 0 </p><p>~__ ~/2z cos 2t art + f {f'(x+t)+f'(x--t)} = o </p><p>1 nI2Z art + = 2f {f"(x+t)-f"(x-t) I{1-sinXt} </p><p>+ s ~2~ {f(x + t) - f (x - t)} sin 2t art 2{ " </p><p>Ta~HM o6pa3oM, nocae 3aMeHbI nepeMeHHO~ H nprIMeHeHHn HepaBeHCTBa MHH- KOBCI(OFO IIO.rlyqI4M </p><p>~---~ yz f"[ %}-f" ( - t ] ] {1-sint}dt+ (5) IIf"ll </p></li><li><p>HeKoTopbte nepaBencTaa B TeOpHrt nprt6nrt~e~Ha 83 </p><p>cnpaoeO~uao nepaeencmao </p><p>IlTm~)ll </p></li><li><p>84 51. B. Ta~oa </p><p>Hepaeencmeo ney~yuutaemoe, mo ecmb ~)na tca~cOoeo 2&gt;0 cyutecmeyem nocnec)o- </p><p>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. </p><p>or a3 aTe~ii, CTB O. OIIeHr~i I1 f(k)]l cor:IaCHO rrepaBeHCTBy (5) </p><p>llf(k)l] ~ x +~ x - - {1 -s in t}dt + = 22o, , [[ </p><p>2 ~/2 It t </p><p>HOpMN qbyHKI~n~ nO)I 3uaroM ~HTerpanoB 6y~IeM oIIeH~BaTb corJIaCHO uepa- BeHCTBy Xap~ [4] </p><p>lie(re)l[ &lt; !1~011 ("-~&gt;/m II~p(")ll m/", 0 _-&lt; m </p></li><li><p>HeKoTop/,Ie HepaBeHCTBa B TeoprlrI IlpH6JIrt,xem, la 85 </p><p>[4] F .F .Xap j Ia , )Ix.E..J-I/,ITTJIbBy,/~ H F. IIoJlrla, Hepa6eucmaa, HHocTparman anTepaTypa (Mocl~a, 1948). </p><p>[5] H. I I . KopHettqyK, O HaHJiymiieM paBnoMepnoM npn6Jm,xemm /mqbqbepenlmpyeMblX ~yHrtml~, ,lIotr AH CCCP, 141 (1960, 304---307. </p><p>[6] H. 17. Koprie~t~lylq To,man rOnCTaaTa B TeopeMe ~. ~meKcona o namlymlieM pamIoMepnoM Ilpn6Jmxemm rienpepl~mnl, IX nep~tojm~iecKax qb~, ,/[otcJt. AH CCCP, 154 (1962), 514---515. </p><p>[7] H. FI. Kone~y~, ~rcTpeMaJIbnBIe 3i-ia,-lem, Ia qbyrlKl~onaJIoB n Ha.BJIymJIee npn6mixenne na iolaccax nepno~m,teclmx qbynmm~, Plsa. AH CCCP, ceprm MaTeM., 35 (1971), 93--124. </p><p>[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. </p><p>[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. </p><p>[10] A. 3rlrMyr~t, TpueonoMempuuectr pm)bt. II, Map (Mocl~a, 1965). </p><p>Some exact inequalities in the theory of approximation of functions L. V. TAIKOV </p><p>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. </p><p>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. </p><p>As application, exact inequalities are deduced, analogous to the well-known Bernstein and Hardy inequalities. </p><p>JI. B. TAI~KOB CCCP, MBITHII]I~ 141 001 MOCKOBCKI41~ JIECOTEXHHqECKI41~I HHCTHTYT </p></li></ul>