On the best approximation of functions of $n$ variables
We propose a new approach to the solution of the problem of the best approximation, by a certain subspace for functions ofn variables determined by restrictions imposed on the modulus of, continuity of certain partial derivatives. This approach is based on the duality theorem and on the representati...
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| Date: | 1999 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1999
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4734 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510899109888000 |
|---|---|
| author | Korneichuk, N. P. Корнійчук, М. П. |
| author_facet | Korneichuk, N. P. Корнійчук, М. П. |
| author_sort | Korneichuk, N. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We propose a new approach to the solution of the problem of the best approximation, by a certain subspace for functions ofn variables determined by restrictions imposed on the modulus of, continuity of certain partial derivatives. This approach is based on the duality theorem and on the representation of a function as a countable sum of simple functions. |
| first_indexed | 2026-03-24T03:04:19Z |
| format | Article |
| fulltext |
YJ2K 517.5; 519.65
H. 1I. Kopnef lqyK (Hn-T Ma'reMa'mxH HAH Yxpamlla. Kxel~)
O H A H J I Y q I I I E M H P H B J I H ) K E H I I H
tl)YHKI.IHI~I n I I E P E M E H H b I X
We propose a new approach to the solution of a problem of best approximation of functions of n
variables by certain subspace. The functions considered are determined by restrictions on modulus of
continuity of some partial derivatives. This approach is based on duality theorem and on the
presentation of function as a countable sum of simple functions
3anpononoz~azlo nozmfl niltxilt ]to po3B'a3alill~t 3a/taqi npo tlaflKpa~e Ha6~zH~ellll:,l ]I.e..qKHM ni]tnpoc-
TOpOM qbyHKItifl n aMimmx, ~o 3a/talo'rbc.'4 06MeaCeHHaMH na MollyJII, zmnepepmloc'ri /le~IKHX qac-
rummx noxi/umx. Heft niltxi]t t'pym3~eH,ca ua feopeMi l~l~oic'roc'ri 'ra na 3o6paa~ezmi qbynxnii aK
3qncJzezznoi cyMH npoc'rnx.
1. BBe~enHe r~ TeopeMa /:IBOflCTlZeHItOCTH. M~a He CTaHer4 H3.rlaFaTb TeopHIO BOn-
poca ~ ~TO 3araa~zo 6bz MHOFO MeCTa; Ha3OBeM ~[LUb HeCKOJIbKO (.tbaMHJIHIYl yqeHblX,
qbl, l pr II0 npa6~H~meHriio n MHOFOMepHoM cJzy,~ae x o p o m o H3BeCTHbI:
C. M. HrzKo~bcKma, B. H. Te~zaKOB, 5t. C. ByrpoB, M. K.I-IoTanOB, B. ~ . Ba6eHKo,
E). M. Fa~eeB, 2Imm 3yHr H ~p.
B HaCTOJRtleZ~ CTaThe npe~JzaraeTc~z HeKOTOp~ HOBbIIYI HO)~XO)~, FIO3BOJI~IIOIJAH~.[ B
pa~e c:xy'zaen ~ z nepHo~rz~zeCKHX ~yHKttrZ~ n nepe~eHm, tx no.nyqaT~ TOqHble pe-
3y:mTaTta rrpH otIeHKe HaH~zyqmero npH6~ZH>KeHHa nO/InpocTpaHCTBOM. HCXO/~HOi~
TOHKO[t JIBJL,ReTC~I c.neayzotuee yTBep~K/~eHH(~, KOTOpOe Ha3blBalOT TeopeMo~ ~BOIYlCT-
~enHocra/X~za HaHny,~mero npn6nrz~enHa [1 ] (cM. "raKa<e, HanpH~ep, [ 2, c. 1 13]).
T e o p e ~ a A. Flycmb X ~ /zune~noe nop~utpoeannoe npocmpancmeo. X * - -
npocmpancmeo, conp~ennoe c X, F ~ noOtwocmpancmao o X . ].Za~ .ato6oeo
x c X \ F
E(x, F) : = inf II x - u II = sup { f ( x ) : : ~ x*, Ilfll <-- 1, f ( , , ) = 0 ~ u ~ F ) . (~)
ucF
I'IycT~ n reopeMe A X eer~ npocrpaacaazo L,4,, 1 < p < **, 2rc-nepr~o~trz,aec~rtx
no z<a:c,Z~o)l nepeMeHHo~ dpyHKUHfl f ( ~ ) = f ( x l' X2 . . . . . X,) C O6~'~nOl:~ r~op~o~i:
II/IIp = ilflh".p = t l!lf(x)lPd~) ' l < p < - ;
/ sup vrai [ f(~)l, p = oo.
.u
Y,~rrnaBa~ o6tunfl BH~ zrme~Horo qbyH~UnoHa~a B npocTpattcTBe L,.t,, p > 1
(c~., HanpHMep, [3, c. 196]), a Taz<a~e npea~oa~eHne 1.4.2 H3 [4, C. 26 ] , XOTOpOe,
o~zesrz~Ho, cnpa~e~rtBO z~ B n-~epHoM c.ny,zae, y-r~ep~ztenrze Teope~4za A, T. e. COOT-
HomeHne (1), MO~HO ~anncaT~ ~Jz ,~(t) ~ L ,.p \ F c~e~yxomar, t o6pa3or,~:
(! ': t E(x, F)p = sup x(t)h(t)dt: h ~ L,.p,. tlh.llp,-< z, u(t)h(i)dt= 0 g u ( t ) ~ F , (2)
o
l<_p<~, 1 /p+ l / p ' = 1.
3a~e-m~, wro np~ p = ** paneHcT~ (2) cnpa~e~a:mBO, no KpallHefl ~epe, R.na KO-
rte,moMepmax no~npoerpaHern F [ 1, 4, c. 26].
2. Mo~ty~zz, Henpep~zr, Hoe'ma. B O~zo~epHo~ c~rzy,~ae Mo~ynr~ rzenpep~zBrmCra
toll, 8) qbymctOa~ f ( t ) r C[a, b] onpeae~taerca cooamoment4e~
Co(f, 8) = sup { I f (F) - f ( t " ) [ : t',t" r b], [ t ' - t" 1< 8 },
H. 11. KOPHEI;IqYK. 1999
I352 ISSN 0041-6053. Yxp. ~tum. ~.'ypn., 1999, m. 51, N ~ 10
0 HAHJ'IYqIIIEM I'IPHBJ'IH)KEHHH OYHKI.tHfl n IIEPEMEHHBIX 1353
KOTOpOC, ecJIH qbyaKLU4.a f ( t ) a6C0JIIOTHO HeripeplaBHa, MO~HO 3ailHCaTb TaK:
Ili [ } co(.f, $) = sup f ' ( t )d t : t ' ,t" e [a,b], I t ' - t " l < 6 . (3)
YcJtoBae (3), onpe~eJ1aIomee r,~o~ty~m HenpepEaSHOCTa, MOY, O~O nepeHec-ra Ha c.nyqma
di)yaKUnn f(-~) = f ( x I, x 2 . . . . . xn) n nepeMeHH~ax.
3a~a]InM B R n neKoropoe paccroaHne p(Y, y) M e ~ y rOqKablH ~ = { X 1'
X 2 . . . . . Xn} I4 y = {YI 'Y2 . . . . . Yn}. ~rrlM paccroanrleM a R n orlpe/zeylZeTcJz e/Zrl-
HHqHblI~I map Bp c 12eHTpOM B ayJ~e:
B o = { ~ : ~ R", p(~ ,O)<l}o
t lepea Bp(~, r) 6ygeM o6oaaaqaTb 3a~aBaeM~t~ paccToaHneM p map c UeHTpOM. B
TOqKe ~ E R n I4 pa~nycoM r:
Bp(~ , r ) = { x : x ~ R n, p ( ~ , ~ ) < r } .
EcJm ~ = 0, TO aMecro Bp(0, r) 6yaeM nncaT~ Bo(r) .
HaaOaeM MOgy~teM HenpeptaBHOCTn, COOTBeTCTByloII~HM qbyHKILtlH fi(.~), cyMMrl-
pyeMo~t a orparlrlqeHHO~ 3aMKHyTOt~ o6JIacTrl Q c.R'! , ae~anqHHy
r~e s epxHas rpam, M0~y~s m-rrerpa~a B ~ c J ~ s e r c a no SCeM mapaM B0(N, r) c
~eHTpOM S To~Ke N e Q , ~epa XoT0p~x He n p e ~ m a e T ~i.
IIpH ~aKC~pO~a~HO~ paCCTOSH~H p ~o~iy~b Henpep~sHoc'rH coO( f , $ ) ling
JIIO601R ~byHXttI,Irl f ( ~ ) ~ Q c R n mdeeT TaKHe CBOI~CTBa: 1) C0p(f, 0 ) = 0, 2)
qbynKtma C0p(f, ~5) Henpep~a~na H He y6r,,BaeT Z~zz 0 < $ < rues Q , 3) cop(f, ~5)
noayaaaHTrmaaa qbynKuHa Ha TO~ )Ke ~no)KecTBe. 3aaan qbyHKLtmO CO ($) C
TaKHMH CBO~CTBaMH, a TaK)Ke paccTo~Hrte p B n n, MU onpe]~e.rIrIM K.nacc H ~
cyM~npyemax HaMHOmeCTBe Q dpyaKmxi~ f ( ~ ) TaKHx, wro
c % ( f , $ ) < C0($), ~ e [0, m e s Q ] .
3. H p o c T ~ e dpyHKUHH H OCHOBHa$1 a e n ~ a . Fipn otteHKe n a n ~ y q m e r o npH6~H-
JKCHH$I O~byHKLIH~ O/~HO~I FlepeMeHHOI~ Ha KJlaCCe H o~, 3a~aBaeMOM BblrlyKYlldM BBepX
Mo~y.rteM HenpepblBHOCTl, l CO (~), KJnoqenylo poYlb rlrpaeT y'r~ep~K]~eHl, le [5, c. 190],
CM~aC.a KOTOpOro 3az~OqeH S paBeHcTBe
sup ~( t ) f ( t )d t : f e H~ ~] [ r ( ~ , t)m'(t)dt, (5)
o
r~Ie �9 (t) ~ n l~c ' r a s qbyHKUna [5, C. 1321, ~ " (t) = g (t), a r ( ~ , t) ~ y 6 ~ s a m m a s
nepecTanosxa qby~xaan ~P (t) .
B n-MepHOM cJvy'~ae nazoseM ~yHXtm~O ~0(~) = q)(X~, X2 . . . . . Xn) npoca'olt, eeJm
B~HO.~HeH~ cze~y~otaae yC.rlOBH~t:
1) ~0(~) a6coJnoTH0 Henpep~sHa B Rn;
2) ~0(~) > 0 aJL~ ~ e Q , rz~e Q - - o r p a m ~ e H H a s o6~tacTr~, T.e . nenycToe
oFpaHHtleHHOe oTKp~rroe CB~I3HOe MHO~Kec'rBo B Rn;
ISSN 0041 "6053. Yrp. .~tam. ~. pn.. 1999. m. 51, N" 10
1354 H. FI. KOPHEI~HYK
3) r = 0 ~.na ~ e Rn\ Q;
4) ecsm A = max r TO Ztna sno6oro y, 0 -< y < A, Hno>KeCTBO
X
M ( y ) = { x : x e Q, 9(2)>-y} (6)
eCTS o6Jmcrb, a MHO~KCCTBO
M ( A ) = { x : x ~ Q, ~(~)>_A}
o60~ac'mio He .qBJDICTCa.
3aMCTHH, qTO xapaKTepHCTHqeCKHM CB01~CTBOM IIpOCTOI~I OpyHKI.IHPI (D(.~) .,qBJLa-
eTCa csa3nocrb MHO~KCcTBa (6) ~l~a BCeX y e [ 0, A) s T0H cHucJle, wro J1~6~r ~se
TOqKH ~H0~KecTsa M(y) H 0 ~ o COe~HHHTb nenpepuaao~ KpHBOIt, ~cz~amet~
M(y). MHO3KefTBO Q 6yJlcM Ha3bIBaTb OCHOBaHtleM 17pOCTOt~ OpyHKRtIH (D(x)-
I'[pOCTO~I dpyHKII, rIr 6y/~CM Ha3~BaTb TaK;Kr 0TprllXaTC.rlbHyIO dpyHKI~HIO --(p(.~),
ecJra ~0(~)--npoczaa qbyHzuaa. IIpocTaa qbyHztma 9(~) C OCHOSaHneH Q 0~a-
HO3HaqHO Ol-IpejleJl.qe'rc.q MHO~KCCTBaMH ypOBH~ M ( y ) (6), rgc 0 _< y < A.
I'IycT~ paccroaHHeH p(2, y) S npocTpancTse R n 3a/IaH e~HHrlqHtafl map BO.
I'IpocTOI~t ~yHKID~tl ([3(2) C OCHOBaHrleM Q nOCTaBHM s COOTBeTCTBrle npocTym, txo
ca~MeTpa~rno y6uBalomyK) ~y~KttHm (p*(p, 2) c OCHOBamleH B p ( r ) , pa~r~yc XO-
Toporo r ma6pan na yc~o~Ha mesB~(r) = mes Q. OyHKUHa ~0*(p, 2) c OCHO~anH-
eM Bp(r ) o/moaaaqH 0 onpegea~eHa HHo~ecTsa~H ypo~na:
Mo(Y) = { x : x e Bo(ry), ~0*(p, 2 ) > y } ,
rz~e pa/xnyc ry nu6npaeTca Ha yCJIOBHJI
mesMp(y) = mesM(y) , O < y < A .
Ec~a mesM(A)= 0, TO HHo~ecrBo Mp(A) COCTOHT H30~nO~ TOaZn. B
c~yaae mes M(A) > 0 ShlrlOa'lHalOTC~l paBeHcr~a
mesMp(A) = mesBp(rA) = mesM(A).
IIycT~ 3a~aata paccTo.anHe p(~, y) H Ho/Xyam HenpcpraSHOC-Pa CO (5) . Ka]accy
H~(Q) onpe~eaeHHUX Ha ~Ho~Kecrse Q ~yrmttrItt conocram~H aZCTpeMa.rlbHylO
CrlHMeTprlqHO y6blsalOll~lO qbyHKu~m ~ ( 2 ) TaKylo, qTO (0p(j~p,~) = (.0 (5) IlprI
acex 0 < ~ - mes Q . ~eltcrmrrea~bHO, Ha zaa~oM yposne y > 0 Hra a~a6rxpaeM map
Bp(ry) c uerrrpo~ B Hyne H pa~Hyca ry TaK, qTO6bI Buno~HznOCS paBeHCTBO
mesBp(ry) = 5, (7)
a/Isis qbyrmtl~H f~p ~ pasenCTSO
SpCry)
COOTHOmVHH~H (7) H (8) ~yHK~I~ f~p (~) oIIpe~r O/I~IO3HaqHO.
.lIe~l~la L I lycmb t~(.~) - - n p o c m a ~ qbyttm4u~ c ocno~rmue~t Q u I-I~ (Q) - -
xnacc cy~atupye~tux Ha Q ~ y n x q u a f(.~) n nepe~teuta,tx, 3aOaoae~t~a o o6nac-
mu Q ycaooue~t
(op(f, 5) < (o(5), 0__.8<mesQ,
u paccmo~nue~ p. TozOa Ona n~o6oa Oyn~uu f ~ H~('~) cnpaoeOnuno nepa-
~btcrnao
ISSN 0041-6053. YKp. stam. "~. pn,.1999, m. M. N e 10
0 HAH~IYHIIIEM FIPHE;dIH;~KEHHH OYHKHHfl n I-IEPEMEHHbIX 1355
~f(~)~0(~)d~ < ~f~(~)~0*(~)d~, (9)
Q Bp(r)
e0e mes Bp( r ) = mes Q, q~*(.~) - - cu,~awmputmo y6btealou4aa nepecmano6ua ~yn-
Kt~uu ~0(~), f~ (.~) ~ arcmpezta.abnaa cuztzwmpu,mo y6bteWou~aa nepecmanoera e
~.aacce H ~ ( B p ( r ) ) .
OuenKa (9) TOqHa B TOM CMbICHC, qTO cymecTByeT npocTaa dibynKIIHJ] (p(.~) C
OCrIOBaHIaeM Q, /XH.q KOTOpO~ B (9) 6y~eT 3HaK paBeHCTBa.
]~OKa3aTeHbCTBO CBO~tlTC.q K rlpHMeHeHHIO TeOpeMbl O CHMMCTpHqHO y6MBalOLLIHX
nepecTaaoaKax. B ORHOMepHOM cHyqae 3Ta TeopeMa ~oKaaaHa s [6, c. 334], o6malt
cHyqaCt cM., nanpI~Mep, B [7].
CqHTaeM OpyHKLIHH f ( ~ ) u q~(~) rIOHO~KHTeHbHbIMH Ha Q. Tor~ta
~f(~)~p(~)d~ _< ~ f * ( ~ ) r m e s B o ( r ) = mesQ,
Q Bp(r)
r~e f*(~') rl ~0"(.~) --CI, IHMeTpHqHO y6t,mmomHe I'IepecTaHOBKH cl0yHKUI'II~ f(.~) 1,1
~0(.~). Ha xoro, '-tTO f ~ H ~ ( Q ) , rl rt3 orlpeZemHHa ~ynKtmrl ~ ( ~ ' ) CHe/XyeT, aT0
~.rl~I BCeX .~ E Bl~(r) BblnOYlH.,qerc~! HepaBeHCTBO f*(x) < fl~(.~). IIO3TOMy
Bp(r) Bp(r)
w r o . Tpe6osaHOCb ]~OKa3,aTb.
TOqrlOC~'b HepaBerlC'l~a (9) cHe~yr H3 TOr'O KbaKTa, qTO (p*(x) ---- TO~'<.C rlpOcTa~I
cI3yI-IKILH~q C OCltOBaHHCI~ B p(r) , mes B p(r) = rues Q.
3aMeTm, t, aTO yTBepxaetme mMMta 1 ~ KaaecTBeHHOM OTaOmeHHH cHa6ee OaHO-
MepHo~t ZeMMt~ 7.4.1 H3 [5, C. 190], r ae COOTHOmeHHe (5) a~-taeTcJt TO'amaH aH~
Kax~oIt KOHKpeTHO~t rIpOCTOlt ~yHKmm q~ (t), ecar! CO (5) ~ B~nyK.matt a~epx Ho-
ly.tit, Henpep~H0CT~L YleMMa me ! ~aeT oraenKy, Toqay~o Ha ~Hoxecame Bcex rlpo-
CTI~X qbyHKtIrt~, rlMelOll.~rtX O/~tty rl Ty ~Ke Cr~MMeTpWaHO y6~smomyao nepecTaHOBKy.
4. Pa3nomenHe OpyHKIIHH n nepeHeHHmx , a c'~eTnyto r npoc'r~,vr 06 -
m ~ i nHaH paccy~/~eHrd,~t aHanorrl~eH COOTBCTCTBylOIIAC~I rlpoLte/~ype laprt pa3~o-
~KeHnn d, byHKUml o~ao~ nepeMermo~ [8, 4]. OCHOaHaa Tpy~nocr~,, Koropaa aoarm-
KaeT B n-MepHOM cHyqae, CBJ/3alta c BO3MOJ'KHO~'~ CHO~KHOCTblO CTpOeHH~I MHOmeCTBa
m (y) = { .~ : f ( ~ ) = y } ~;a~Ke alia a6COHtOTHO Henpep~,mHOr'I ctbyHKmm f (~ ) , .~ =
= . . . . . x,,}.
FlycTb f(.~) onpe~eHena H a6COHIOTHO nenpepta~na n 3amaKaHmt Q o6HacwH
Q, coxpaHaeT 3rlaK ~nyTpn Q rI o6pamaeTca s aySP. Ha rpamme. 13y~ern Z~.na orrpe-
)XemHnocTH no.rlaraTr~, ~IT0 f ( ~ ) > 0 ~Ha .~ r Q. ~Ha HIO6OrO y > 0 sse~:eM
pacCMOTpeHne MHO~KeCTSO
m ( y , Q ) = { x : x ~ Q, f ( 2 ) = y } . (10)
~IcHo, HTO LI, H,q BCCX /~OCTaTOHHO MailbOX ,?. > 0 MHO,,~Kec'I'BO re(E;, Q) CC'l'~ CB,q3HOe
MHOYKec'I'BO.
ToqKy Yl > 0 Ha3OBeM TOqKO~'| BeTBJICHHX C13yHKLIHH f ( ~ ) , eCJIH ~H~ y e
(0, y l ) MH0meCTBO (10) a~HaeTca C~aa3HtaM, a ~;Ha y > Yi MHOmeCTBO (10)
CB$13HblM He ~IB.rI~ICTC,q. Ec.rm rlpH ~TOM mesm(y~, Q ) = 0, To f ( . ~ ) HBH.,qeTC~I
rlpOCTOfi dpyaxmmI, i c oeHo~aHrmM Q. Ecn~ me m e s m ( y l, Q ) > 0, To Maox~ec'r~o
m(y~, Q) co~aepmrrr ocHoBamta, no xpallae~t zepe, ~ y x npocTtaX dpyaKtm~ i~ T.~.
0603HaqaM qepr P MHOX<CCTBO BCeX TOqeK SeTB.neHHJ~ ~yHKarta f( .~).
ISSN 0041-6053..YI,'p. ,uam. ~ypn, 1999. m. 51. N '! !0
1356 H.H. KOPHE~IqYK
~CdlOBHe B, M n o m e c m e o P m o t t e u 6emenenua aScomomno nenpepbt6noa Ha
za~tbtrauuu Q o6nacmu Q clbynm4uu f (~ ) ne 6o,~ee ~e~t otemno.
Ha yCHOmia B c~le~yeT, qXO ec:m f (~) > 0 a o6Haca~a Q ri A = max { f (~ ) :
~ Q }, TO ~uomecrno [0, A ] \ f ( P ) oaxpuro a vtomeT 6~aT~ npencxaa~qeao ~ an-
Lie KOHeqHOA H.rlH cqeTHo~ CyMMbl nenepeceKmomHxca HHTepBaHOB:
[0, A]Xf(P) = U(ak, bk),
k
aTaK KaK m e s f ( P ) = 0, TO
~ ( b k - a D = A.
k
CONOCraBnM KaxLIoMy aaTepsa~y (a k, bt) qbyaKUH~
[ f ( .~) - -ak . at: < f ( x ) < bk;
fk(.~) = ~ b k - a k, f (YO>bk;
I
O, f ( ~ ) < a k.
EcJqH a k < y < bk., TO MHO~eCamO m ( y , Q) COCTOHT r13 He 60HeC qeM cYeTrloro qrlc-
.rta HenepeceKmomHxca o6~ac'reit, TaK qTO
fk(~) = ~ t p k , i ( ~ ) , (11)
i
rae ~k.i(~) ~ npocTl, le t.~yHKIAHH. CywMnpya qbynKuml (11) no k, noHyqaeM paa-
.rlO:~KeHHe f ( ~ ) Ha KOHeqHylO HJIH cqeTHylO cyMMy rlpocxhlX d~yHKLtrlt~ tp,n(~ ) C
OCHOBaHH,,qMH Qm c Q:
f (~) = ~ Cpk.iCX) = ~'~9,,,(X). (12)
k,i m
PaBeHcr~o (12) B~nOHHAeTca nowm ~c~/Iy a o6:lac'm Q.
5. K~accbi 2~-nepno~lnqecxHx ~yHKUHfl n nepeMem~blx. B ~a.nbHe~ItueM
�9 6y~teM paccMarpHaaTi, clJyHKI4rlH f(~) n NepeMeHH~X: .~ ---- { X I, X2 . . . . . Xn} B
npocTpaHCTae /--'n,l 2"/t-NepHOLIHqeCKHX HOKaHmIO CyMMHpyeM~x dpynKIJ, rffl. K~acc
H~ a aaLiaeTc.a Ha Ky6e [0, 27t ]n paccToJtHHe~,t la(~, .~) H MO~y.neM HenpeprcaaaocTa
o(~) :
o~p(f, $) <_. ~o(~), 0 _ < 5 < m e s [ 0 , x] n.
IIyca~ ~ r ( t ) ~ o~an0~epnaa qbynKtma 13epHyHaa [4, C. 107], r.e.
Br( t ) = ~ c o s ( k t - xr /2)
k r
k=l
Ec~m ~o(~)~ L~, l a
2n
I ~ x l , x2 . . . . . xk . . . . . x.)dx k = O,
o
TO 6y~teM roaopm~, wro qbyHKtlrl~l
r = l , 2 . . . . .
k -" 1 ,2 . . . . . n ,
ISSN 0041-6053. YKp. ~tam. ~. pu., 1999, m. 5 !, IV'-' IO
O HAHJ'IYqlI/EM I'IPHBJIH)K.EHHH OYHKIAH~ n FIEPEMEHHbIX 1357
. 2 x 2 ~ n
= ..... (13 )
'" 0 0 k=l
n
eCTb riCpHO/]HqeCKHI~I HHTeI~aJI Ilop~qLIKa rk, k = | , 2 . . . . . n, rio HepeMeHHOl~ t k OT
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3aTeM S pa3 rio rlepelVleHHOfi y nprl tlDHKCHpOBaltHOM X, rlO.rlyqttbI paseHCT80
r 3"tx, Y) .'x "dxd,: ~W~;F , E(f,F)p = sup Jo ~rxOSy gr.,( ,Y) ) gr, s rs .L
r/~e W y F J- eCTS ~ a c c qbyH~Uatl, n p e ~ c T a a n ~ x B BHae
2g 2n
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V$W;:SFJ'I$H~ 0 0
ISSN 0041-6053. Y~p. ~tam. ~.~pn., 1999. in. 51,1~ 10
1358 H. I'1. KOPHEI;IqYK
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ce I't~, zacganno~t na tuape Bp(r ) .
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~ f(x, y)v(x, y)axdy = j'[ f(x, Y)~ 'Pk(x, y)axdy =
K K k
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k K kQk
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g /= a~(r)
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Bp(r) k Bp(r)
I$5N 0041-6053, YKp. ~tam. ~.'vpu., ] 999. in. 51, N ~ ] 0
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1. HuKon~'Kuii C. 1t4. l'IpH6nU>Ke,Ue ~ylIKII~HI~t TPHI'OilOMeTpHqeCKHMH MnoroqaellaMH n cpe/IHCM
//HaB. AH CCCP. Cep. MaT. -- 1946 . - 10, N~3. - C . 2 0 7 - 2 5 6 .
2. Kopne~i,~yK H. 17. CnJtaflm,[ B TeOpHH NpH6JIH~Kr - M . : HayKa, 1984. - 352 c.
3. Kanrnopoew.t .]'L B., AKu.aotq F. 17. (~yllKllHOlladlbllhll~ allaJl143 B llOpMHpOBalllllx~X npocTpaHc'raax.
- M.: (DHaMa'tT~a, 1959. -- 684 C.
4. Koptteii~yx. H. 17. Tommlr Kouc'vama n "reopmt nprtOnrsacc,Hs. - M. : HayKa. 1987. - 424 c.
5. Kopne~i~yK H. 17. ~KCTpeM~|bnI,Ie 3altaqH B "reopHH n p H O s m ~ e . , a . - M. : Hayga, 1976. - 320 c.
6. XapOu F. F,, :lummm,oyD ,axc.E., 17oaua F. Hepanenc'rBa. - M. : Hzlt-ao mlocTp. JmT., 1948, --
456 c.
7. KonaDa B. H. Flepec'ra.onKH qbyHKlmfl H "reopeMl~ t~mo,,KenrL,'l//Yenexrl MaT. Hay'Ko -- 1989. -- 44,
man. 5. - C. 6 1 - 9 5 .
8. Kop,etl,~ytc H. 17. ~KCTpCMa.tll, tlble :maqenrt~ clJyllKItHOIlaJIoB H uarulytlmer npH6.nH~e,He na
K.uaccax neprlojtrtYecKrtx ~yumtHfl/ /Hm~. AH CCCP. Cep. MaT.- 1971.--35. N~I. - C . 9 3 - 1 2 4 .
IIoayqc.o 25.05.99
ISSN 0041-6053. YKp. ~tam. .~yp,., 1999, m, 51, N ~ 10
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| id | umjimathkievua-article-4734 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:04:19Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ec/2becb0d481516d786e0d83fe991f5aec.pdf |
| spelling | umjimathkievua-article-47342020-03-18T21:12:54Z On the best approximation of functions of $n$ variables О наилучшем приближении функций $n$ переменных Korneichuk, N. P. Корнійчук, М. П. We propose a new approach to the solution of the problem of the best approximation, by a certain subspace for functions ofn variables determined by restrictions imposed on the modulus of, continuity of certain partial derivatives. This approach is based on the duality theorem and on the representation of a function as a countable sum of simple functions. Запропоновано поний підхід до розв'язання задачі про найкраще наближення деяким підпростором функцій $n$ змінних, що задаються обмеженнями на модуль неперервності деяких частинних похідних. Цей підхід грунтується на теоремі двоїстості та на зображенні функції як зчисленної суми простих. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4734 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1352–1359 Український математичний журнал; Том 51 № 10 (1999); 1352–1359 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4734/6169 https://umj.imath.kiev.ua/index.php/umj/article/view/4734/6170 Copyright (c) 1999 Korneichuk N. P. |
| spellingShingle | Korneichuk, N. P. Корнійчук, М. П. On the best approximation of functions of $n$ variables |
| title | On the best approximation of functions of $n$ variables |
| title_alt | О наилучшем приближении функций $n$ переменных |
| title_full | On the best approximation of functions of $n$ variables |
| title_fullStr | On the best approximation of functions of $n$ variables |
| title_full_unstemmed | On the best approximation of functions of $n$ variables |
| title_short | On the best approximation of functions of $n$ variables |
| title_sort | on the best approximation of functions of $n$ variables |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4734 |
| work_keys_str_mv | AT korneichuknp onthebestapproximationoffunctionsofnvariables AT korníjčukmp onthebestapproximationoffunctionsofnvariables AT korneichuknp onailučšempribliženiifunkcijnperemennyh AT korníjčukmp onailučšempribliženiifunkcijnperemennyh |