К обобщению одной теоремы Харди-Литтльвуда
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| Date: | 1953 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
1953
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7735 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512718956527616 |
|---|---|
| author | Щеглов, М. П. Щеглов, М. П. |
| author_facet | Щеглов, М. П. Щеглов, М. П. |
| author_sort | Щеглов, М. П. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T10:12:02Z |
| description | - |
| first_indexed | 2026-03-24T03:33:15Z |
| format | Article |
| fulltext |
1953 YKPAHHCKI111 MATEMATI14ECKI1Yf )KYPHA.Jl
HHCTHTYT MATEMATHKH
K o6o6D..J:eHHIO o.n.uoif TeopeMhl Xap.n.u-JIHTTJibBY.li.a
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T. V, N~ 3
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1)
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2) rp(u)= - ~ sve u -8,2)
U v=O u-+OO
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Hoe qHcJio. Tor.u.a
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a,.= + 1 ~ Sv- .
n v =O n-+oo
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<qJ ( u) --7 S 4). CcpopMyJIHpoBaHHa5I TeopeMa X.- JI. B o6meM BH.U.e He HMeeT
MeCTa IIpH S = oo, TaK KaK MQ)KHO IIOCTpOHTb IIOCJie.U.OBaTeJibHOCTb { Sn},
r .u.e sn>O H .U.JI5I KOTopoil lim <p (u) =Dm rp (u) _:.__ oo(u--?oo),liman=O,
lim an= oo (n--? oo) s) . H acT05Illl.35I 3aMeTim cJiyMHT HeKOTopbiM06o6me
H.HeM TeopeMbi X ap.u.u-JlHTTJibBY.U.a.
0 rr p e .u. e JI e H u e. CKa)!(eM, qTo rroCJie.u.oBaTeJibHOCTb { s,.} E P, eCJIH
.S11 > 0. BBe.u.eM qHcJia (KOHeqHbie HJIH 6ecKol!eqHbre)
lima,. = D, limrp(u) = D'.
n-+- c. u- oo
l..J:HCJia D H D' (.U.JI5I MHOMeCTBa P) CB5I3aHbi orrpe.u.eJieHHOH 3aBHCHMOCTbJO.
11TaK, rrycTb rrocJie.u.oBaTeJibHOCTb { s,.} E P, HMeJOT MecTo CJie.u.yJOlll.He
Teop eMbi:
I. D H D' - o.u.HoBpeMeHHO KOHeqHbie HJIH 6ecKoHeqHbie ( oo) qHCJia .
I) CM. T3KLKe [2], CTp. 197, TeopeMa 93.
2 ) BhlpaLKeHHe, onpe.n:eJIHiow:ee cp(u), npe.n:cTaBJIHeT OAHY H3 <j:JopM MeTo.n:a cyMMHpO·
DaHHH ITyaccoHa (ronopHT TaK)I(e - A 6eJIH).
3) an - onpe.n:eJIHeT npocTeiiumii MeTO.!I. cyMMHpoaaHHll 4 e3apo.
4 ) CM. [2], CTp. 140, TeopeMa 55.
S) HcnoJib30BaH npHMep, npHBe)leHHblii a pa6oTe [3], cTp . 277, c1!y'laii XIX.
299
II. flycTb, KpoMe Toro, 0 < D' < oo, npn 3THX yCJioBHHX:
a) inf (D-D') = 0, 6) sup (D- D') = oo,
(P) (P)
B) . f D r)
D
In D' =1, sup D' = e,
(P ) (P)
r')
D+a
0 < a = const < oo. sup D'+ = e, r.n.e
(P) (X
III. ECJIH lim sn = D, Tor.n.a D' =D.
IV. CymecTsyeT noCJie.n.osaTe.TibHOCTh { s,.} E P, .U.JIH KoTopoii D = D' <
<Iimsn .
.D. o K a 3 a T e JI b c T B o. Beer .n.a
D'-<. D 11. (1)
KpoMe Toro, JierKo noKa3aTh, liTO .U.JIH noCJie.n.osaTeJibHOCTH { s,. }e P HMeeM
On<. e<p(n+ 1). (2)
0TCIO,ll.a CJie,ll.yeT
D <. eD'. (2')
J.b (1) H (2'), olleBH,li.HO, BbiTeKaeT TeopeMa I. CJiyllaH a) .H a) TeopeMbi II
llOJiyllaJOTCSI H3 COOTHOilleHHH ( 1) , YliHTbiBaH TOT !lJaKT, liTO JierKO o6pa30-
BaTb noCJie.n.osaTe.TibHOCTb {s,.}e P, ,li.JIH KOTopoii D = D'.
PaccMoTpHM CJiyllaH 6), r) H r'). Bo3bMeM nocJie.n.osaTeJihHOCTb
s,.= {~> O
r.n.e
(k = 0, 1, 2 ... ),
npn n2k+l <. n < n2k+2
HeTpy.n.Ho noKa3aTh, liTO ,li.JIH :.noii noCJie.n.osaTeJibHOCTH
D = M ( 1- } ) , D' =MI.- <~t ( 1 - } ) .
0TCIO,ll.a
(3)
D _ t _
_ =J.l-t-e. (4)
D' <.-.t
113 (3), OlleBH,li.HO, CJie.n.yeT 6). yJ:g ( 4), npHHHMaSI BO BHHMaHHe (2'), BbiTe
KaeT r). AHaJiorHl!HO noJiyllaeTcH r'). Dpn .n.oKa3aTeJibCTBe TeopeMbi III sse
.n.eM llOHHTHe ,llJIOTHOCTH'1•
Bo3bMeM MHO)I{eCTBO
N= {n}, (n=O, 1, 2, ... )
H llOCJie,li.OBaTe.TibHOCTb cerMeHTOB
r.n.e
I) CM. [4], CTp . 50.
300
DycTb ,n:aHo MHOMeCTBO N<m> C N. 06o3Ha'!HM qepe3 Nk<m> nopi.I,HIO ,n:aHHoro
MHOMeCTBa, npHHa,n:JieMaiUyiO . ,k"-TOMY cerMeHTy, nk(m) - 'IHCJIO 3JieMeHTOB
B nopi.I,HH .,k". CocTaBHM OTHOllleHHe
n~m)
-=--_:_:_-:--o- = (l~m) •
n 1.'- nk + 1
lime~m>=e111, r,n:e O<e(m>< l,
k-> 00
6y,n:eT CJIYMHTb xapaKTepHCTHKOH .,nJIOTHOCTH" ,n:aHHoro MHOMeCTBa N<m>·
DO 3a,n:aHHOH DOCJie,n:oBaTeJibHOCTH cerMeHTOB.
0epeXOtJ:HM K tJ:OKa3aTeJibCTBy.
ECJIH D = =, Tor,n:a D' = = (B CHJIY TeopeMbi I). ECJIH D = 0, Tor,n:a,
oqesH,n:Ho, D' = 0. DoJIORmM 0 < D < =· 3a,n:a,n:HM npoH3BOJibHO MaJioe·
'IHCJIO t:, 0 < t: < D H KaK yro,n:Ho 6oJibllloe '!HCJIO M > 0. Pa3o6heM MHO
MeCTBO
N= {n}, (n = 0, 1, 2, .)
Ha TPH no,n:MHOMeCTBa CJie,n:yroruHM o6pa3oM:
N(I)[sn:>-D+ ~]· N(2)[sn<D-s]
N<3>[D- t:< sn < D+ ~].
QqesH,n:Ho, N<l) - KOHe'IHoe MHOMeCTBO, TaK KaK lim s11 =D. Bo3bMelvt
DOCJie,n:oBaTeJibHOCTb (o) H TaKyiO, 'ITO
n ,
__!!__- oo,
nk k--><x•
06pa3yeM DOpi.I,HH MHO}I{eCTB
N1ml (m= 1, 2, 3)
H HM COOTBeTCTBYIOLUHe 'IHCJIOBble xapaKTepHCTHKH:
nim)' eim)' (>(m) 1).
YcTaHOBJieHo, Hail:,n:eTCH '!HCJio k0 TaKoe, 'ITO
Npl = 0 2 l npH k> k0
n, cJie,n:oaaTeJibHO,
npH k > ko.
DoKaMeM, 'ITO e<2> = 0, e<3> = 1.
DoJIOMHM, k > k 0 • l1MeeM
I) CM. Bhllllecromi.\He ¢opMyJihl (crp. 300).
2) 0yCTOe MHO)I{eCTBO.
3or
'OTKy.rr.a
c D + f]( 3) ~ D- E(j(2) + _ n (3) + f]( 2) 1)
" ~ .. , M"* it ,
HJIH
E
c.n(2) ~ _ 0 (3) + TJ('l) "* ~ M'k .,* .
flepexo.rr.H K npe.rr.eJiy npH k _., oo, CIJHTaH
(i= 1, 2, 3, 4),
1
(1(2)-< - Q(3).
M
0Tcio.rr.a, npHHHMaH BO BHHMaHHe, 'ITO M KaK yro.rr.Ho 60JiblliOe IJHCJIO
> 0, CJie.rr.yer Harne yrBep1K.LI.eHHe.
Ha OCHOB3HHH ;;noro cpaKTa H COOTHOllieHHH ( 1) HeTpy.LI.HO .li.OK333Tb
D-e :r;;;D' -<. D.
l-:13 3Toro BbiTeKaeT HCKOMoe paBeHCTBO D' = D.
T.eopeMy IV MOJKHO onpaB.rr.aTb c noMOlll.biO Bblllie onpe.rr.eJieHHOH noCJie
. .rr.oBaTeJibHOCTH, npHHHl\1a51 }. = },k _., 1 (k-+ oo ).
B 3TOM CJiyiJae
D = D' = 0, lim Sn = M > 0.
0 n p e .rr. e JI e H H e. CKa.iKeM, 'ITO nocJie.rr.oBaTeJibHOCTb { s,.} E P0 , ecJIH
{sn}E P, D=D'=O
{s,.}EP00 , eCJIH {s11 }EP, D=D'= oo .
Teo p eM a V. :YhreeT MecTo
l . an 1' CJ',. sup 1m ) = sup 1m = e.
, ,, , , .... oo rp(n+l ( I'ocl , .... oo p(n+1)
.UoKa3aTeJibCTBo anaJiorHIJHO reopeMe II, cJiyiJai-i r) (Heo6xo.rr.HMO HcnOJib-
30BaTb COOTHOllieHHe (2)) .
0 n p e .rr. e JI e H H e. CKmKeM, 'ITO nocJie.rr.oBaTeJibHOCTb { s,} E P[x, yJ>
CCJIH
0 < X :::;;; Sn -< !J < 00.
BBe.rr.eM cpyHKU.HIO
D
fl(x,y)= sup D''
(P[x, y])
YKaiKeM Ha HeKOTOpbie CBOHCTBa ::noi'! cpymm.HH:
a) J.L(X, x) = 1, 6) 1 :::;;; fl(X, y) -< e,
B) J.L(kx, fly) = f!.(X, y), r.rr.e 0 < fl < oo,
y y'
r) J.L(X, !J) :r;;;fl-(X', y'), eCJIH - -<: --, .
X X
Tipe.LI.CTaBJIHeT mnepec rJiy6iKe H3YIJHTb ¢YHKU.HIO fL (x, y) .
. 302
Jll1TEP ATYP A
1. G. H a r d y an d I. Lit t 1 e wood, Tauberian theorems concerhing power
:.series and Dirichlet's series whose coeficients are positive, Proc. Lond. Math. Soc.
•(2), 13 (19131), 174L,_191.
2. r . X a p !1. H, Pacxo.n.HmHeCl! pH.l\bl, 1951.
3. M. n. ill er Jl 0 B, K o6o6meHHIO TeopeM Tay6epa, MaT. c6., T. 28(70); 2 (1951),
:'245-282.
4. A. 3 H r My H .11., TpHrOHOMeTpH'!ecKHe p.R.ll.hl, M.-JI., 1939.
DoJiy'leHa 18 OKTH6pH 1952 r .
MocKsa
0299
0300
0301
0302
0303
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| id | umjimathkievua-article-7735 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:15Z |
| publishDate | 1953 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/42/859c7716173322fbbd77b1324225d842.pdf |
| spelling | umjimathkievua-article-77352023-08-15T10:12:02Z К обобщению одной теоремы Харди-Литтльвуда К обобщению одной теоремы Харди-Литтльвуда Щеглов, М. П. Щеглов, М. П. - - Institute of Mathematics, NAS of Ukraine 1953-08-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7735 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 3 (1953); 299-303 Український математичний журнал; Том 5 № 3 (1953); 299-303 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7735/9432 Copyright (c) 1953 М. П. Щеглов |
| spellingShingle | Щеглов, М. П. Щеглов, М. П. К обобщению одной теоремы Харди-Литтльвуда |
| title | К обобщению одной теоремы Харди-Литтльвуда |
| title_alt | К обобщению одной теоремы Харди-Литтльвуда |
| title_full | К обобщению одной теоремы Харди-Литтльвуда |
| title_fullStr | К обобщению одной теоремы Харди-Литтльвуда |
| title_full_unstemmed | К обобщению одной теоремы Харди-Литтльвуда |
| title_short | К обобщению одной теоремы Харди-Литтльвуда |
| title_sort | к обобщению одной теоремы харди-литтльвуда |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7735 |
| work_keys_str_mv | AT ŝeglovmp kobobŝeniûodnojteoremyhardilittlʹvuda AT ŝeglovmp kobobŝeniûodnojteoremyhardilittlʹvuda |