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I S U M M A R Y )
ABSOLUTE SUMMiABlLlTY FACTORS OF
INFINITE SERIES
A THESIS SUBMITTED TO THE ALIGARH MUSLIM UNIVERSITY, ALIGARH
IN PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
IN HATH&MATICS
By NIRANJAN SINGH
DEPARTMENT OF MATHEMATICS AND STATISTICS ALIGARH MUSLIM UNIVERSITY, ALIGARH
1967
< 0 0
kut i^mmmm^ memm-rnimtmimmmm
mrmmt, mn mmmmsi
i^t iMt fCt) hB a periadle faisstteu with period
iattgralyiQ in tho smsQ of l^feaegae tet its
^yios be a eo
^ ^ + C^ SOS nf + gia at) 2 i ^
« S ^ t t ) ^
tti« series t«s thh F-mrler series tn
00 - • 0© -£ Cbjjj cos a* * sia nt) ~ Z
m w i t e ^^c^) =
fCt) « ftx-t>
e - fCX-t)|t
T- I t " ^•a.
Ca> » (log IJ}*^ f I ^ Of
^iiCt) Wi '.h similar a^ing^u
• g »
In CSsaptcf I I &f ttm pros^ thesis m hum prmrad
the fallowing ^htmrm eosit^emiiig factors of
Fouriey ssrScs,
JL
Z eCiifX) %(t>
f ** ^^ ^ Mf
^ lf>Cii)|aii » 0|t<i0g " I ' ) ! , i t - " © ) ^
this th©©rci3 gmeraliseg the mio^fiag of Pati £72
THmnm A* atf
t ! i « o<t)j a« t o t o - .
_ 1
sbS ^ ^ cofmfeie ^hat S Ciog td f^ ^ t
thim at the poiot t « % S ^Ct ) i « ^mhXe
"fbe Chapter l i t 1® to ^he ^ ^ of «
g^aomXiaatKsa theorem 1 jaeatioaed th»or«ffl
Sb as follows t
tll^B' H 2 « ^ [ W b© ft ss^^gg aich
non^egaflw in»«gml mlugs of ^ ^ Hi ^ ®
»
I 0 ^tilos , t o t f^y^v
m It ^ tir^iy % i s fe • i| • i ® »
1,4 Isi im^ l^ati [73 the foOlmiing ••hooro® on
sam^bjlitir factors F w i e r sertos •
8 ^ Jf he a eiriveix sequeace tuch that
< e®^ tii03i a nm^pruw^ aoi sufflcleftl; lamiitXm
t S |f(u)i<tlt e , o
i « mm
Z ^ - < &
In Gbup' sr IT w have gaieraiised tho theorem
of l ^ f i for smnsmbllltf" fey proving the follisutng thmrm^
TsmofiSK a. IX ^ ^ g ccmveiE c t umce gueh that
£ a aecostary and su f f i e i ^ t cggiditAoa for
S to eag-iaMQ |C,l|3c Ic 1 t
% & I I f<a)| an te oCt> , as t o
fe
n
4
I t fltny fee tfeat fheomi B ts c^m
of oar fhears® J!,
Ceasc ml ig isas-asate factors of
liifissits series l^I ^as proired the following two
C * I f ^ is a loaves se tueaee sach that
hk Z — < CO, aisd 2 is 'bcmdeS [B, log a^lj^ thm
% log a 1 tfe® gftrlss S i® gmssabl© —— [ ^
n
n eRKorJii » « t f a V av « ©<»). ana S is a
V SSQ »
\ mtth thmt n < t|i«n S l«sg a ^^ is H 1,11 1 II 1,111 •••••.Hf
I a
a n -
oa ftiearea D vas g.aeTalisje^ by Mdimtra {BJ for
M Gumv T V of tho pTjjv* *"ho f©3.1a«irisig thec r a
^idti gemmllLzm rescslti! amtioaed abo^a,
mmm 4. % ^ APr > o sM /^p^lo^sm
mm ^ ^ i ^ j E ^ s B ^ i^t t s l . n
Qiapt^r ITl dsals wi h h® s udy of sussaabillt|r
fmcfiors of infiai-^-e sc-rioi^. In this ehap^'^s^p ^he "^ollowlBg
thaoroB hais proved ifetch a sjjoeial
@ theorem ^f Srivas^av^^ eh tpa -ta imd Bass »
* B »
^mmmiBt % 1% P^h m
%
S^Qtim A of ehapi-.ei' ? I I is to
of almost c 'oryu'hcire sosrabSlity ^ of & ^-rigoao*
iietsi;0 E©rios, Iti this tha following ^hmrem ha® been
p t m ^ ^Sdl includes ae a special <sase, a rsmilt of tJl yanovM
ettd Ifmg £318] •
mcfa tliat y I i s CKmotoaic iner^agigig se^a^ce and
^^ ft ^
X
1^8 General isisc a rosat of B0S6Siq»i9t and Kyslop £13 for J
^on tiga^e of a Mas&ar £4 J provoa the
foJlowSfig thcoarea.
Tmomn © ir J »
* -.V 1 1 * <
tteca the ^r lea # » t « t »
* f
1>0<: >
la B of th® mmrn wt feaxf© f^i^tl
f^etar ^^ series S ^ ( x )
Jti A ef ^ liave
tlie folloir^g wtitcto a thcor^si of Pati £©3
^ ^ ^ 9 ^hmtm 0f fv^md ^ t t £6J «
fBHIB^ f , jii t \ h^] a cQgygx sQf^^Qg suc timt
»
» l^vf Z mmf^, m S .
» log II ^ Im £ Jji
S of mm is to ffat stu^
of l i t sasgmsMiiti factors of tn this
liaa ^ an proved irljids
SI Of f a t t^^ l «
• 7 •
Oajl
A
m i £ |%i a » <
H B F E H E W C E S
i tMiAim 5(1963), ss.68,
£3] « ils^r. Itoth^ m i ^ 319
£43 Ifti&ay, 8»}L t B^letlm Be la Kateasatlea
I S ] m^otm^ ^OC^ fapan Aeaa. 4ia9S«) , 40 • ^
fa t i , t
C83 a s . S.H* t Cak® Mftth. J,
C93 Siri^statm, r.P.p Hohapa s-at Mw and i%Ol Tatch«ll, J B* t 8913.054)
£113 tH^yaaov, § ilisaiss Surveys 19(19^4),
£183 Wiag, I M(10i l ) , 174-176 «
ABSOLUTE SUMMABILITY FACTORS OF
INFINITE SERIES
A THESIS SUBMITTED TO THE ALIGARH MUSLIM UNIVERSITY, ALIGARH
IN PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
IN MAFHEMATICS
By
NIRANJAN SINGH
DEPARTMENT OF MATHEMATICS AND STATISTICS ALIGARH MUSLIM UNIVERSITY, ALIGARH
1967
oO T700
P E B f A C B
i>res«at mtl^le^ ** iil»sciaut« SttaoftbUity Fftetor« of laftoite s^ri^s «»!»oat«s ths T«fuat of nr
tftiidi X doing niaet mth of llov«»1>or, 19$2 ttm iaigfts^ ^i faitw fho wafk bad
ddne imaer tli« galSaQoe of Dr» H»8o*f 1:1 of Mathfis^tios
fmA Statistleit itligafh ^ s l i a HaiiroTtitf^i ilH&i^
Ilio thosie o<xi8lst9 of o i ^ la t l » first
ehs^QT we giv« « T&gme of hittiofto Icnown i^loh h»vm Mmmtmetims with mm Introsf^liations^ Chaptes^ II
ootiedmedi vitli tho of at)so3.ttte Cesaro suumbilltf faotors of Fourier mvimf #tti« f ^&pmr XIX doaXs vltb a fUrtlJo^ goncfiXivstim of th® thsojees in the previous
Chapter X? i « (devotej! to the of * <
mmmhiX%%7 faetora of Foarlor soirias* ta Ghap -or V, wa prova a tli«or«a on abaoXuta emsisabilitir factors of InfSaita
emptor ecntaina a rasult on ^ | aimabUity factors of tnfliiita sarias, t^iXa Qimptwr fit ie oonoamad i^th ttie atud^ of abaoXuta ^i^ro loi mbSXitif factors of ti^goaaoetrlc aeriea aiMt ferias eoiijugat© to Poiirier a»ri«8« Iki Oisptar irXXXy iia stu^ tha abaoXuta Oaaaro giKasmbiXlty factors anS {li^ ^^ I giisHi iXiiar factor© of iaflalta gariaa, Tenarda tb® ea<It w iliva a fairly ajEtaoslTa "bibXiograi&f of varimig |«ibXicatl<m» i^idi hava bas® rafarr©^ to in tha praaaat thasig.
I t Slay B ^ t i e i ^ h^f^ that tlw pQw^im af
thi0 thesiBj prmmt&d is h® f^mu of fes*?*
ny pubi imtl^ in varlotts s s tb^ t i c a l |cii3!ml9«
Ob© of tiato l^s s lrc^^ l^m aceopte^i fo^ patjiicatioo ia
Avista di Kat^^ticsi o^tmTs b®^
at of laaiaa
aaa la^lt^ !:etea<2« Oosgr^sa
It is with mmm of gr^^ltsiae that I tak© tMe
oppos^nlty of teaefe^atoss to Haibar
for hlB mos ' valaaM© gui^aecip kind aaa wry
gmoaroas fbroaghsmt tti0 prapaim^ion of •bis
X m grat#fui al©^ to Praf, sia^itif of
th6 far his feimd csiOtmragmeat aaS helpw
riaally, t tha s^mcil Bcim^tfic laSus'-riaX
Ecseai^h of BjAIr, fa^ award of a ^tmior ^s^arcti Felioifghlp
ftm Sagiigt S2, to
April % 196?? sift^^ip smoa
c ^ntMwf0
mmB
wmmn f'wmfw stPtia 39
l i t oa 15IE ^ ^ l O T ms jm m ^ m m t m TAm'KB QT "PQUvsmi mmm ^^^^ m
w m jCt^tbi mmtiMt wm^h:} m mmwim ^sm ••• ^
V m JlBSOHfff
if^s^i Tmmm m XMPumn BfMmB ••• ^
f 11 m'^imu cEc.^ sm^iiBiWff • • r^cfoBi simiis im
ttn F C OaiS OF
nmumf^m
mmm * t
I S t E O B t J G T X O H
3U3. With ttie pijblica^lcm of C«aict:y*s msXym AlgstJrlqu®
ISSI fifed'e m thB BSfstaaSal sferi©s in
%tj© theoiy of ••fee ean^ergmoo of iafijait® ssriaii w»s placed on
foim^at it xjas s t i l l not pm&ihl^ *'o da al
satisfactorily wi h those series of ^Ich the sofa^^c^E of pftirtial gams oiscUlate, to ise®t this di^'^'ieul^ry, various nmt
isetho Ss, processes of irasssat ilitF^ wore developed towards thB eaa of th® las* emtury* ' Ii sb proc#sses extocidai' th« classical tjotic® of ^nu^ m the concept of
eaa^^f^enc© Tim to that of eiassability, so in ai ye
tises, tho nutlm of absolute ctjnveifg^ce l »d to the
formlatica of aisiother process s alXefi absolute suaaanfeUit^*
0£ven a series t s ^ ^^ ^ % ^n gas^ble in so««
tiihiie gcaisml, ^ % is itmlf aot so thea Is
taM tij a ©imsafetlity factor of t-h® t a ^ If th«
in q i e s t l ^ i s the faeti>F« natufally
(Sillftd ehs?>lut« ftusmhility "sstors*
Ihs pres«nt thesis is based aa certain lnv«s^-igotions of
th® attthoj* in* "> -hs theory axid at^plicatioas of ahsoXuts / /
s«5®»biaity factors of itsflait© series* Befor® giving a r»guj» of th® ©arlier ia th« light of whidi various r®siiit«
have obtaicied th© a'athor^ i t »««W8 €asirahls ta
hem th« d@fini*i«mg aiad UQ^&timB whit^ wi l l fe© rasiuired is
the s«c|U»l»
l^g^ SoB# Of the most fai^illar methodf of absolute
^Bpsabilityi tad witi5 ^icJi ve shall be «<Kie«m«d in th« #»iiuelt V thas« that are Imo^ m asthods @f Abel, C®saro and
9oT%m6» I t mt^ bomptrex*, b« mntlom^ that « I I thsm at thods
caa fee frao %vt> basic g«Bef«l p?0C«8s«» «hlch aye
as t
<1) f • prooess#s^ Cii) f * proc©s«««,
f - aethoSs are based m the foiwatlan of « i auxlXimty
[ defined fey the soqufsiee to tsetiii ncc trant-
farsiatiott »meh 'that
II '
t ex e Is th# &*th partial mm ef a given series Z
The Mitri* [|t|[ w ^ tihich i « the element ia the
a-th Ttm aod n- th cdlxmif is usaally oaXled the atttrlx ef
the fi « aethods ar@ f^sed txpm the forisatian of e fuaetloEial trmBformtim H%) defined by a seQu«tce • to» ftsactioa t3m&3forn» tian
tt .
Bore eenerally^ by the integrel tranaforsation.
o lehere x i « « contanttous paraiieter and the fmct lm ^ ( x )
£ or is defined over m mitafel® infeeriral of x
C or X end f 1 •
• 3 •
A stf ies £ or the ( S ij is said t0 bo
6oatfayg«iit to tlie a 2fa? 5 i f
llm ^ •/S.
Sly atialogy, a eerles S eaM to ®af«abl« by
f'^at^oii ioT f - aotho^) to th© mm • i f
l i s t - » » Cor tCas) w/SJt
s is 8
Imov that a series S % or the se aeRC© f Sjj] is
absoimtely emvcrgent if tfe© is
^ i % • •
DafloiB® 5iii:tiasay the nbs^lut© s«B»»bility df an
iaf ln i t « ssries,^ say that a S % is «b«e3.utiay
mmmhXe by « f • tietfeoS, sr eiisply sumable i f thm
corrospondiliig guatiXiaTy setmefnee boan le irarlatim,
^ 1 - I < CO
Absolut® stt^mbility a ^ •^•thod^ or mwmhility I f j , i » d«fiBed in tha saa© way %fith the that
ia this case th® corregpoaaiag foactian t (x ) i^ould b« «
tmctim of boand«d variaticei in «ii int«iiml of coitinuott®
w 4 *
Th9 leqaaacc: * to • trwasf arastl^m f is said to tHi yegalir If
11® % • S ««> ISa t - • S ,
ori^r ^nt f » » t h ^ b® ^©galar, it is accessarf ana gufflclisst that
Jo ^ S is ia«S«p«na«it of %
Cii) 11» ^ n • »t IS eo
and m it
( i l i ) l i s as Cmn *
t f ^ p l i t « proir«d this fcsoit foy triangiidar natsfix lakii^ Qxiimsim to g meir i e^ge is 300 to @t9lidiam8«
SSfflilarly the se<iuen©e- to- function tsansf'sniBtioo f is sale to he rr-g>Jl^T if
i ia « s Ila t<x) ® s »
•^BmMy^ IM}
ffoepXite^ |S®3
tsteiiOiattg,
• s *
fi«gularitF cdaSitl^s far «r« a&ilogmis to th©«o &t the T-tJmsfoiiB&tiaej,
x'im & emtSmxmfs p&mmt^t i^ieh to infinity
aSfS t { » ) » S % • fhm tb© m^ i«fflci«Ett
that t W t»a«ftn<i4 f o r n ^ 0 aiti
l is % » s « «> ilia tC3&> » a U"*
are
<1) S I Is emv^u^t fo? x 0
aM 2 I I < H.
f M t
iSior H it Iii4ip«nia«iit of « x «iitt
Cii) Xi» fn^si) » foi* «v«iry %
( m l i m CO
fhe ttmsformtim f Is »aia to ba absiautaX^ ragular
i f f is reg'iXar a.M tha trariatisii of J Sn Sffiplias the boimda variaticsi of 5 »
• e •
Th« proMtis of a^ssduts regularity ©f
m® f i rs t «if a l l stt^taS H^rs lattv m fey t + l^opp sM Sisiouc^ • Ifears proved thmt
fh« fiQcesfsary saf^iciGut for th« re clarity of t^jastboa, 40fliiL0«l lay aye thut
eo \ ^ t for a l l val««s of n»
AIA ' ^ for a l l ICi nhdre C Is m eomstant*
Subsequent rfisiJiltB Qf itoopp and Itormts, aad amoaehi l»ear close pairall^lsa to that of Hears, and ««y be c<»si6ered tlopllfied wrsliae of Htars* r®»tilts,
file pTdblm of abec^ut© regularity for Integrtl tra&sfons has b)s«n studied by fatehell ^ and Suncmdil aiid
4= fmebtrnrm •
•f Saopp, S and Lorusrta, £193
» f »
a. ABsoLSfs mwj* mmnmum^
A 6«ri#8 2 is saia to be 8U8SNiia« or
mmmWLm (Jk) to th® sm S i f ^ % ^^v i iTg^ la
0 < X < 1 lim fix} » s, f ( x ) Is the icei
of tlie s»ri#s £ au aethoa of suaraability* althoui^
lisaedl a^ftcr th« ©«l«l>rst®<l sikthtsatl^iim ^ ai«o soae tia«6 caXXedi PoSssaa? s a^thoa of sawaatiility f'ss- ttm
Tmsm tfeat Poissos studied its ftppllcntlm to F©urt«r s»ri«8» Jt caa ^ so fe© t3r»e«at thrtnig^ baek to ,
fli« of stjmibilit? ims introdueed bf ieilttal£«r » thus accoTiiag to feto a strios £ is said to Afedl or \A\ if £ a^ sP if
*
emergent ia 0 5 ae < 1 and it^ &m • fraction fix) is of i^riation in CO IK
The inclusion relatiots J C (A) is «vid«nt in v mt of the fact that if a fuactim f(x> is of variati©a In
any tntarval, theo the iiffiits fix • 0) axiirt at evary point •f*
of th« intanrai, Iftiittater also proved tba ra»ilt analosous to classical ^aori^ that saisaahility |A} is abse^utaly
regular i«a» any infinit© series ^ieh is ahsQltit«ly ©aisverfant
^ Ahal HI
laiittiOcar, CS23
- 8 «
Ss ale® iuamfeie tt ^as shorn tsy «hittaicer with the
fe^p of aua that m Foarlsr 8®ari«s mf ^mferg^ at a poiat wltliout TIDING SUASATBL© at that OB TH®
t ' ' kanSi c<m«tmc5t®a ^ eimspl® & Trntimr saries to
i^ow that a sories eaa be (4} at » point without
feeing eoniftrgsf^ at that point,
^ gtmmd •i t ad^a tnvtti T th® seop® of susssabillty
ixitTo&xaim ft j oeofdi g to tiSa * 8«ri*s s is gala to abaoiutdy (A^M aui8Babl« if ttio sorios S * is ^mvmgmt for all. poiitiv« x and
tBJffl- ftoetlcsn f<x> « % ^ is of boandaa '^'arlatiai in ( a positive mcmotonio Sner^asing
stquenc® t«iaing to tafinity 'tflth a.
itiBSOi;iJfE StWM&BllJfK ^
lo 1911 Fokst* d&fino^ th® absoliste Cesiro scuBaabUlty
for int0gr«l orders* lAttfr in 1926 Kogfeetltfi^* th i «
definitioa to fraoticcial aad iategativo ofdors* Aeeera^ng to
this (aeflnltlaa of Koghetlliffitat tfe© serioi £ tk vltfe partial
sua is siii€ to hB Ah$t)iuttlf- tff mftaxm of
order or eisply sisisaal e t^t^lt K > :|3L i f
t i rasad, S.S, £3?J
^Zjwsm^f iU
^ Fekata, % £103 4 KvvlvcLi-<udtv,E,G-. LHyi^j.
m 0
H l a
mi.
* a / iB
©.ftalJlistied thn eaaisisteaey thmr^m
the ab©olute C©s»to gmmeMXlty, naa!#ly fo* • t . '
m^vy ^ > » He aljBO that j dmn not n^cessarlSj?
iE^ly mmahUl^ (C, - e > sna that it p><>
m^ £ 1« l^f lU C f" ) tTamformA * » sarias of £ % Is •oCI *
+ tn timt for int©fyal vtlttes efoi o< f C_
\h\ l^t caa' QfisftX:?, lii collitmetlofi %ftth tlia
tWi^rts for |C| 4t f^llmis that
f l « t t definltiOQ of absolato
Cstairo ss*«abiXity W a wtafetr of |mras8«jt«rs
ff&k^f^ M. £3.13
4 Fl®tt, f . l i £231
ttms tteeosrdinf to l im m series ^ eat^ to be
I f < f rlii» V s r«al tsHRl y if k+feY-a o,; k
% I (Til • r < • Sy d^tkiUim^ U t ^ » the * '
sumaat^Hity |<?f ot # T Ij is th# s mt th« saamBSllty ' -X"
€arii«r fey him •
to tli« eth#r i f V® 0 aM It » 1 w®
taii^^Slit^ ^^ ds tlie suiasability
I t fellows tram certain r©salts of f l s t t that
tusfflBamity T FX IICY K > It saneatbility ar©
iMdepm^mt of eas^ dther*
imoimz nmump mmmtLm^
ff^Plima suCTafelllty» thoat^ QfigSnally ln i t lat « « te 190s
by WorooLi m& hs ving rea^ln©^ unknoua t i l l potatod tsy ,
Tsmtkixi Wfes iiiaai3«i3i«nt3Ly iatroSuc^d tif VtorlmsA
ia m^ i t has noif cuftsmry t » sss©eiat« i t vitli fei8 mm&» M 3.S37 tim concept of a^JSolut*
Sifluat sttsmhiltty, Dea^^lng by a se^a^se® of
eanst«nt®, rmX dr con^les^ wrttlag
Pjj « • — P a l Ki^p^i^O^
^ no t t , f.H, £12]
- u
M9 eali I th« irorlmd mmn of a | % j #
» Si tjy tiwi c^ffielsiit«[jpk55, » £ , ^o /
^ Pjj fh » fl©ri«» S i t saia to b© lUmaBX® to I f
Xim tft » S, a
«!id is ml^ to gu'mil^* IMP
in.p^U s V i t
Jt »af t>« reiiftriEwa thst 8'orlimcl aathoS is wot for nil tyiNn of e^sn^itioia
for abs'^lat* regularity e»n h9 deduced fras a th«or«» of Heart on Hitrix mmsmhillty. ^rnammrf ^m mf^Mmt cmditioRs, tit
stated l>y I'^fcrliifeofft for afeio late rtgalarity of ttit atthod mm t
Ci> l$M ^ • 0,
( U ) t I ^ J k i ^ I <
for tU positiv* inttgntl. iralti«« of # ,
4. C4S3
- IS -
k smmitmt similar r««a3.t has reesBtly obtained
) In tJt® spae ia l cas© i n ifhJcli » * » .u ....,-,.
^ > thm HorltiTid aeaa wducet t® th& faralliar ) iB«an ana mm a© stfimabOStf . Os tlm other hand, IL • , tli« Sforl-jna gut^blllty
to a thoS kamm as tli« h a r^ l e nuaasblltty,
2,6, mnnmitm | ¥ , t ^ I *
I»®t « of ra«l posit eoRst«at«
gueti that a
« s pv ***f a •
Iifift S »jj be fi givm infinite series with fi^ as Its
S i s sa id t o b<i s t rmgXy b m m M ( 1 or baan^td
i iJf ^ J • series £ % saW to b® (W,Pa{ If [ uJ ® s«<iU€fl<s« of boari<l0d T«rl«tloii,
13 ^
1 a Sv I f *,m,mmmm z mmm fg ^ s#miaRc« ©f b juBidad -^ariatlcm -ii^Cnn) v n ij
that lUM^bl* log&yX} , I f t«k« i • • I Pjj » ~ it is msf to see that log .
Mmmtf. sfmsiASjiiff ?4CfOEs of mmim smim mtf Its CMQGATS SIEISS,
Im% f (t ) « p^rioaie ftmetlact with ptrtoa audi lategsmfex© In the se&se of t#b@3gu« ( • « ^ v) , ' h* fmtl^T jB9Tim of fCt) is glir«ft hf
•ft ^ *** ^ • S C0® at • stn fit ) « £ il-Ct), S X o
and ttes eonjogat© s«tl«« of the Poarier striss of ftt) Is 09 CO S (b^ cos nt » ^ lit I
Mo vrit^
f ( t ) • • f {X*t ) • 2f(x)
!|'(t) « J f(x n> - f<x - t ^
^ ( t ) • f (t •ni ^ fCajAi ,
f o < t ) « f m f 4 *
* 14 -
«( i i ) » (log a ) , ^ ^ 0 #
5k (t) £« defined in the slsailar aaiiiiwr* f*
A seei^ieiie©^^"^ Is to b# csmv** i f 0,
to be if ^ ^^^ ^ ^t Ch «
lkmii«ralng the aiiaaRbilitf factors of a Fourier
s^riee, Pres'ii bad the foliating theoTMi t
"JKlDEBJttA • I f [ is «SF sue tb«
[ Urn i s )^ Cic^ lot ar^*^],
^Uog nf^Clog log ar^. (i^g log .••XogJ.j^)'^
(log ...log a > JO P
eo
th«a £ ^ ^ ms^^l® |4[ «t avery polat t »
vhere t J e t ) • X if(u)|«tt» 0(t)»
o aafi, th! refor«^ almost ,
^Pragaa, B H £383
• IS
tmml ana Svata ©attended theorem k in the form of
th© following
THEOFEH B , I f is a caave* sequeaco such that
Z ~ < eo, then S Xjj i^jlt) is suEsaabl© |A| at every
point t e where holds.
In i948f C h A ohtalBed a gensialtsation of Theory A In
the sense that he replaced sumiJabllity |A| by sujBiaablllty jC|,
B© pfcwed the follc^lng thsofem*
TUBOBEM C. I f f ^ ij] is any a© of the sequences7,1)
and (1,7,2) holds, thm S Jijj(x) is suansahle ^ot
evevfoi > 1,
In 1964, Pati proved the following theorem which
iscludes| as particular caseS| a l l these previous results,
THEOREM ©, I f ^ is a convex setia«nc© such that
£ < CO ^ and (1»7,2) holds, t h ^ S \ is
susmhle 1 Cf | for evei^ > 1 #
In 1947, Starting with the conditieaa
t I S * ) I \ f(u)|dii « 0 ftciog-r r l (t 0), p t o
I®s»i, S and Kvat^, [X«J t Cheng, Hif, £7] ^ Pati, T. [403
^dildi r««ltie«« to tb© etm^ltloa la tli® special c»8«
^OKISK S, tlB^iir the coR«iti€» Cl«7,d)» th« ••Ties
s g 6 > lit t » 3K, is swmbi* IC o - I
for «<> I. « t
Vmty T9emtlf l?iksbit obtalsea m ext«a«l<« of tli«or«ii t of ( b«nc« of 7h«oir«a & of ^ t l>
in the foTffi of ib© following tb«Qi'«tt»
WSDBM % I f [ is a ecaywni swiacRc* such that
•MM 1 »
th« 8eri«s £ is em^9Tgmtf th«n the s«ri*s 00 S « (a n ) at t » 3C» is «ua»bX» for 1
s. that holit*
fhQotm. S has baoB r«c« i t ly g«o#r«lis5«d by i^t i 4=
aiia Si»ba ins another airection by th® introduction ©f a niiir
coRc#pt of hfp^'cm^^ity^ Th«y obt»in«d th® fellowing
th€»oir«i&, mtsmmi l.«t b- be mn integer Z { Nil
ft nsn*iner«m»lng wh«ii h « and « byp«y
em'^m s«c|u«riea of ©ySer b > svteh tbiit
t Bik^it, 19} T aa<l giitia, B^K £48]
17-
K ( i ) £ < O©
n
im Sn^ <
fhani if ^ I IffeWlAi • ©<t) t o ' .
00
as t I ig |C, h • 8 j for
$ > 0 »
later on Ihmd obtained g<m«i«iiafttion of 7 in the same ^iraeti^ as fhesrss 0 s^arAllzes D
fHSOFJm H» li m int®g»r ^ o, let [ x^] «
«ich that 9saA Z < eo fhmn ^ tt
f f
0© as t 0, S ^Co is luaatia* |C,b |
f o r «Ttr2r 8 > 0 •
It is teowu tlwit s\i!iRabllity % doe© asjt lieeessarlly imply ra»«»t>ility Satur8llyjth«r«f«r« questicm arise® ^ssther ffeeore® S of Pati can b© fsn©rall«td
r plmeing the fC o l tti« suwmMlltF ^ i s T^mint stlXl lilthcwi^ Sa IMl , Chow hftiS prwea that i f ( i s a saoh tl^t
Z mm^ < thm S \ le eawMblt iXsott
Sat tti# of |»©lists ia Chew's theorsa is of polats at li^icfe
aaaa i t I© fe^m tfea.?- ao*; i»ply
litwa f<t) i© lat^^r^bl® Sa tjia s«n8« of
flms th# foilcmlnf bait T&min^ ytt iinsoXvaS* Xf (l^T^e) then ^ be suasBabln for
m^tt {oi» lea at ^cmmn ©®c|ti«fte« ^ ^
n
to pr^ fallowing
rtsuit, fmoMU I f [itj ^ m9 ctf th#
) — ^ J • y — • — I 7 f I ® j Jloga) H i m logn)*
t . C1 3
i©
thm Z ibC^) is |C,1| at m^ry puiss^ * • «t mt which feaias.
it-
ta the folXosflng form.
imiosEii I f
t
Ni i . and 5 JV a cmvm esc^m^ that S (log o) 2 < ee " a
then at the point t « £ JbCt) Is susmabl*
In camptai* I I of the present thesis » f»rtfe«r gon^rali*
sati^m of th® ikhovQ timavm o^ ^iM. has 1»#«ti obtained fo*
mmmmxitrrn »
C^nafning t^ilf sossmbiXtty^l^ti hts alno obtsi!i«d « neeessar? ana mf fMmt cmdltlon for tho sumsalsiiity |C,X| of £ «t a point" t « ae at whi*^ (l^T.S) Is sfttiflfiw! aad J artum il to l># a e myax nach that
^ < CO« H© fm9 th© fcaiowiag theoTM*
tHEO^ K, I f ^^^ he a convt* soq^aca »ieh that
£ — < m ^ th«a a nocossarjr m& m f i e i m ^ eaadittm tbat
S ^ iSjjCx) mm hX9 |C,lf ^mmmt <3.,7,g) hold* is that
3
[44]
• 20 -
U.7.6) n
J%(x) - f<x) [ < 0 0 ,
m CJhapter IT <sf the preset thesis the above th®oi«ii
has hem genemll.t«<i for smaraabllitjr t^flfj^t ^ > I in
tho followiag fom^
fKEOEBJI» l>e a segnence gach that
then, a a^eessayy and gufficlcat coriditicR S < CO, II *
for S \ Aa(3E) to be gUiSi bl© jC^lij^t ^ ^ 3L tAeneygr^
I 1 0<t>, t '•Oj o . ,
l& that
A Eifflillar situation arises
Sijfiha idsich is a gsaeraliisation
Prasad^ Iswmi and Kvata, Cheng,
as to idiet^er d can b@ replaced
in Chapter I I I the author has ccn
hy Ifflpari iBg certain adfiitionaX
theorem j cXudes ss a special a
thooram of the author proved in (
in th^oTma 0 of Patl and
of various results due to
^ t t and 0ikshlt stated ahove,
hy aero, Sa this ccsinectiim,
isidered suoDsability jC,h+l|
condition on His
ise for li « 0, th© previous
ftiapter I I ,
C-meerning th® ahsole^e Cejaro gumnstbility of the 1 ^ coajugate series of Fourier serii^s .Bo ua<iuet and SESf'slop
prcjvsd the fc^lowing theoreia.
^Bosanquet, I^S, and Hyslop, £4]
- m
mmmm i f ^ » o, & <p md *
the c aalttgat© at th« t « le
J for » # •
fi»««atly Ifeabar has obtained a g«aefalij8»ti©ii of tlii
abcrtr® thmrm. in the torn s
THS EIJKIf. ana
f t - ^ vJr^Ct) t <
thm S at p^tat t « x, 1»
euismbl© «
^ m^tim B of Chapteir Yll iroilta^ls outsat}Ultir
factor J has oMsine^ «o that S li^
amy |« It my t»e raaQrlcdd that th« »
th^oyas pro^ea In tJiis ctmpt^ ificlua«« th» following
tb«ore® o^ Sifi^ttava for th® ow^m
tHEOEiH toy ^ 0 l®t
Ci) »I*
+ Sriirastavsk, P £60]
• S2
thmn the S ^^ point t a % Is stiEiiabl* fji^m { is a such is nm»
iMCtmnnim^ th« series S ^ < ana f^Al^^l < «
SIF^^ii^ILm FACTO&S or fKIOO^OMinslC SSRISS.
Sofaraifig th« fiXiiiost saaaaRbility I^JfVin^^
of s tri$i3n«)«dtirle series Z { i^ coi ax ml' nx} = S A^Cx),
Wang in XSil pirw«(3 the foXXos#i:ig theor©!^
fHEORSK 0 . tt
eo S C%® ^ d m n r < C > 0)
corsrerges, theti tfe® trtg^soEetric «®rtes ^ ^
l t® ! t alfflost
H# also i yovaS that tMs pesislt i « flXi» for ^ « •
t ' l^ is feault tlang vas sul^s^tuently by it^o
proved ths followiiag tfeeoi?«s.
• ^ •
THSDBBH P I f
^ t thm U smmhU \ jw^]
is ft positive l^efeaiizm sst^ene® of mn^rs saelt I
thst < <».
It is elosr ttm th« ecsadltim of fimorm P that cannot !»« to log
In $«oticsi It of Vtl of tiie thesis « ^©oneai OB JC, ocl saassiMlity fmetor of trigononitrlc 8«rl»s
* *
h en obtsino i vbidi iitolud^s as » special cas« th9 dbov« thooress osiA •Iso tti« follcwing theoraa of VattS. »
Tlm^m q, it z Ci^^ * ) < «»,
tlWR
is suswablo IC^-Yl (<<> alfldtt •Tftryvtssre.
ABSOLOTE mJHKABimY FACf'ltf.e OF JSPBiltl S lSm.
Ceme ming th@ absolute haraonio suas^bilitsr factory of inf fait® lal prm^ th» following two thooroeis*
f im}
m •
tHWBM tf\ ^^ Is a eoBveai ©icti ttsat
hk 5 < gjgi S is lag then tli«
%i I series S t© I I ,
» a t j fi
fSmBm e, Xf £ « 0 ( a ) . ana Jlkg^isa VW3L L "J
^ % aX cottvesi ^cli that —»» < »»2 th^ £
&
IB rawsafele IR, mmimmm I ^
aine« eajBBiibliity |C,2.| n^t ir^ly 1 euamtsility ^ * cibtmim^ a suitable
wtaiaability motor ia <Hr< eir to mmro afesoliate har»3riic
•uraimtoility ftca suambility I!« pro -ed th« f^llmSng
fg^Bm ^^ ^ then an log a %
in pnafaafel® jH 1 • a n^t
It my he that th« alJOva th^otms of lal can ba Seduced as eos^laties frm fhaorwa f and tha following
raimJttg,
Bx t tf the s«il#s £ ia booadad CB. l i^ at l j ^nd j is a s«{|ti»aea that JC < m ^ tlieii S «,! ^n aimiKlilta |€,X{ «
aiGglit f , £^23.
i ^ ^ em mx. sttQuenos sueh tbat
s < eo, aa€ If^^l « thca the * *
£ ^ 1» rnxmsM^ Is the (C^I) msm of th« s^t^enee ^o j^ ] #
these are to Patt tihi^ special ease* of ffiore gm*TtX 7«sua.ts to l j bar .
O^erslielug theory S ^ if^rotza has fveeatl^r piroved the foXloirlsg tii^oreis' ^r aheidttte Vdrluzid mmmhilitf of litflfiit* «
fS^OMK % If f ^ « filca), «Eia ( i s a eoasre*
muiasee saeh tliait S < m ^ % ^ ©t /^Pn 0 an«
%Pa. Of series S is i innime
* *
tn Qmptmr ? of the presest thesis the following theoren for absolute lldrlmid saeaahil it: factors has been obtained
imerallsaa all the theorems tMtthioi d
fEEomii Pr I. 0» ^Pft ^ 0 and Ij Ha ©t
s la nasname tC^lU then S is misiiahle
i> *
f l la^art ^ ^ ^Hohrotia^ if iss JSS
B9mntly ^alshmsk^im h%9 prm^ thm f^jditmine
tb^orea m iog ajamswlssiitsr factors of Infinite
sso isio.jm I f z is log if
ml ®
a satisfies th© §
Ca> ^^ Is potiti^fif m^ smotcnie non incireaslRg
(to) E » flF2 a ioga
(e) 35
« s II
(d) £ a » I ^X^l e ©(X)
tfi«ii s oq. i s fBy log f it ! U
tHie Insult of Mk^TQiSatim wat «3Et«itd®d to - t sftii®iibilSty Mas ai Mho oMftinfKi the foXloiflng # •
th«or««» result of i « t!i« s|}«elal «ie®
p et wmmnrnm Qf thmvm^
- r I'kiH fHSOR^ tf S 4t %*i«re)—~ 1
is and saittsfiflns thm f^lfming conditlmtt
-X-Eolaliireththsi,
h}
e)
J , i \ i » m i , Sarg ,
SFB J' « •
IP'S Ujj '
V «
^uite rmmtly^ NoiiapetTa md obtained f<5llo«ing theomi leh ie^goiM^mlLisfttlm of fheorwi %
fHECm®! a t>maidea s»c|ix«ac« satisfying thsr ft^ovini cmdlticfts |
a^g n X&gn
(H) n log n^itgi^
m tt ^^ m^ is IJ^ log is,2], thm £
is Xog * *
It w fee that aAditlmX rwitrietian to a s^qomee in mp9rfXmms m* eaa^itiong (i)
m^ (1S.> ^^ b«ina«« vaylatlott sniS Gms^qvamtl it Is a hmx ^H
la Qmp%mr Vt 3 fmeth w gexiefi isfttl of Theox^ X lift® \mm flibfcal««a li pwlng trnvOM f ^
• #
for » th« ik ium tl»or«iiv a
la tha Section A of ^mptmr TlTt « tli®or«a e<iic«xning 10,1} fm^m of iDfiait© Ims bMo provti ^fM^h ifieludes irosu^t % of Bati C^rnadsr gtated) srA
f
flTOREH If is % Qfsmm saqnwace such that
S < c», Mia ^ I Sv • » , k i 0
« i a tii«ja £ (logCoi'l))**^ % 1»
In S«cti®R » of Chapfesy ITXIJ th« f^Xoiring timoTmm h«»
. » tBWBM ^ m£ mtfieimt eetaitims for
S jj to tee | J J £ a j Ig c gyergcnt
i l ) ^ <
+ l^rasi^, B.lt, muS Blmtt, t3S>J,
* m ^
It be tmstufke^ that th&Gtfem &s
m special caae, the f^lottim theomm of
Hccissafy im^ stitffiei«iit e3iiiftitiai« for ^ % ^ to be mm&Me i^mm^t £ ^ is eantreft«iit
/
f
<l> Z < eo^
iu) ^ -iSl < n
• I I
m tm jm%mt cmi^m mn mttm rm^m w ftmtm
t tm t h^ m g i i ^ ia f ta l t t
the £ % it to m' MsHn rnkt mmMm ik}^ or nis^Tf |4tf tf
o
is f ^ 0 < » < Jl « of t
iwiriaticm in t^a inft^^al »
ti(t ^^ ftfiS ^ iiwidl:* ti!« mmm ^
f < P> - i ) \n i ^ ]
% purttttl tt^ of S tm * % • f ! i » £ I s ml^ t^ lio ftlitolttttily Mttil&o CC,
or wmmWLm if th^ } i i of ••rJUitiQfit thiit i9 tQ i f t^© iiifiait« a«fl*s
f e ^ f tern T ^ i I < « •
n > t
It is wax Mmm t ^ t Ix^lss jr # ^ Piasiibllitjr t ^ is ciet neaosMtrily tt i » « «
IM tliftii tli® f^lXming v l X Imom id«Qtit i «»
f o r f > - a ;
lest ion ^ Hat vvx t mtta&iir^ ^.H, I^J
Itissntl^tt u i , CSJ
- I s a
P P /7 *
% - lit - crVl >
(2.1,8) tn ^s^ V t v . ^ rct> ft f^stioa l»t<ifs«bl* SB tli# MIIS9
fiVinUc df l^lNisfiw dirt? C * w ^T) iiBd \ wit^ ^riod 2w • M its
^ ^ ^ Cs • h^ ma nt >
m
1 r ^ fit} « { • 8f<3E)}^
cCa)
t a foari^r neries l^sai pTortd folloviiif
fKIPHH If, I fJ^"^ is mt <s? tlai
|aof o> I f Hlog u) Clog a> r ) Cldg r ) Ciog Im n) a©f
leg Idi^it) j t < « > 0) •
%hm S \ AnCt > is aawiafelQ | Al at potot t » IFl
» I J » m h
ftud A is th* t&m of th© f^aiowini thmwm^
fB l^ l^ t « If ^ ^ is « isaiiv i ins^ s is X AftCt)
t Cimng t^tvkiWLm^ * t fitti allcKti si of A
In ^mm ^ t tm m^m^^ mmmmh$Xt%r tiar iAaMs1>iIt«r |C|« Et pm^^ th9 f 2,<9itiiif th^wmt
^ twmi^ mm f^ im}
ISltlE©! fl^ 9m smt of the
and lic^^a, th«ri ^ ^ is mxmm^ I
©liOliiH % I f [ is & eo^v®* msh that
hi
is «saiifil4« I Ibr m&Tf <<> % ^ i t
M ^tth tlje eonditi m
o .. . ^teh to ths ooi^itim In th« vpmeM
f east ^ V Qy ana (D f^i^laese hy ^ettf otttftj cai th«
fallmtSiM
I» cJnaititSB tli«
jg t « > Of ^ * mammyim ISfYl
flMiewrai I of i fe®ne» of flui0?«» ID fa^tt}
kh fotm 0f t)i<d f OlXewing tb^afwn*
t rnimg^ l i f *
34 ^
flgiarjlf % If i « a e?r»v«3i ««qiiafie9 «ich that %
til® wl®® S ts sstKfergeal- timn thm II
ea S t » is Ibr
tfcat feiAfis.
le is wiil fenoia tiat sasm^titty X
u^ft mMmmtiilttf ^stofttUlf arises
^ttrnj? thmsTmm IS) etm ^ rsplftelftg th*
renins s t i l l ^pmrn Xn this dirsetion Psti hmw rmmtl^t
oMal&td tti» foUovlni
fBtlRfil I f
I IfCtDian « OCt> fts t o . -
thai St tti# poJtot t « i: ^ %<t) i «
fll« <tf this c!^pt«r i « to obt«t2i « trn^rnr f«»«raXl«atS3ti of tbm abovn of Fitl for
Iii f^llmt^ v » prmtf the f ^o i r l a f th«sr«%
^ftiti , f^ £44) .
mmm^ M{H\ Mm smm smmpp
JL Z Clog n^ Q < o© , j liffs IHg
f i o r l ^ e(ii^a) %Ct) afeiioluf l y mmm^^m |e,l| at
fikUloifiisf Xtiiais*
3 \BAm • I • mnilm a > T ^^
S < jj iyia w S^W ts « ae>gi*ti#gatlir» daca^iiag
» S a. »
4= f i r » t ijftift of th t « 'mmXt i s t o Qsow tlje
sa^mS to l^li » IMirtiooXar ctsoa of
t Qimi, B^q, £82
U S t z B s s ^ l M m m
^ 1 mtn p n
JL ^ n
latdgmitig by piyts tenirt
I i U H L « c ^ r • a. I t « 1 1 t8 » n n
• ©cai g a) > • oc I «t> n
m Q U m n/t • OC(I«g R)^
t.6, OF tfr. tmOEl^ tr« obwv* that th«
I a, I. i of S awM
Oraer 1 @f )
3 ? *
INiir
\in<*H
f l
f«l%> » S C^* S) eos ^Hf
tlioa tjy Afeol transf-^ffflti^ liiv© a f 3- c ^v^-fi
Cx) • ~ I fCt) ~ ) . ) f
^ s f
» 2 • Jf, •
s f i f ^ V fi ^
I • c X I I • Xi • V
SB s itutiag tbo orasf f fvCt) s e«t
%
« oc I I fCt)iat ^ t j ^
a n^t P « o ClogCv
• V ^ J # vm amivmn^^^
•" v«o
% ^ 1 it C ^ 2
Apilit ^ viitu® of UmmB B 4
n
^ ' t • <tt i>jv .a aof<v#3»P 5 ^
imtyvrn^ 1 t a
JL. I ^ n
« •iMliiiiiiiitiiiii-.if £ ,.ii,ii.iiii.«iiriiiii }
inm ilmt^mr^
Cleg tt)^^^ i^l ^ v»g . ^ PHI, 1. Willi WWMI l; mil III. -Ili.WJinlliff J[ ,
,111 ^ , ^ ^ mi £ i%ifi»>»->i < S mmmmm # ^ aiinwuiiili' a « a
CO t a
CO S lt • S ^ t VAXv f l I
ml ap v«i
^ ee Clof a) a iiyfi
^ ^ ml n^^ v « i
<w I « OC ^ J vftl iog^jj^ij
«© ®® 1 vmt mv^t #
• 40 •
to Xv ^ o Kv
to
^ isf Immi 3 ana fset
mfn a® i^m
Mlm
, . i . J l^JsEJtss . , «
jS V .IS
• iSJL g I fct) S at
% »
MritJtog X
H-H+AC-VC) • S V AVCIE).
a %<*> « Z
mesM. % n
Cti m V »
mm timmf^m
ftm Imstm % m hav*
tttid 1
iasu. ^ 7 Ci^SXlofCn^t)}
m *
z • fi ' '" "•""""•"'"•••••-f I • II»1 ft 11 1 i
• ©c I ^ a t * I.
a
Api ying vm NIT^ iigr vixtn* of
Jx ' -
' i M o ' W - '
» 1 1 - ««> J - om
••31 T A r ^ S ^ ml ®
» ©C « iKXog g I » i i Q ^ f ^ m r
w>l ^ ttog(a»s)) f
X i . a® MM €» O) • ©( « « A?^) •(9C s )
iPi ml a
3L
ly I sua tfi@ «f
% m
flm« m twf* fSailly
tip*! m
c^y^fid « ttt
m tm Ai^oi^i t^mmi mmwiLtfi rj^oBs OF fma?M mnmn
Xtm PWtmtZOm m S ag « infii^t® wtstlm with % «.« f^sh iJartial mm, tli« Z
9A14 t© or mmm^l^ l^^trlt If ] 4ft vai-intion, tl»t Is
« -<riit <
"t rnvm the o^ ©rdar , > L ^ ^ Of the ^ ,
If { t^ J sMli Ccsa^ a«aa of erdtr mf th«
^ ^ ^ ^ foilQviiig id«atit)r «
for ar^ ^ ^ • • ^ wit®
• % *
T ^ •IP*!
terict m the riiltt
• *
k » Of:ifSi
i It T*1 ^ ^ % A V
iili«r« ^ m%h C&mTQ pirn ©f k «f
I* f^l y putting f » St «t i«t
n ^ t II fe M i
4 Snqineace iaid e isvts if ^ %
ani it taid to ^ ^Bt^mmm of ordM* h i f
A \ ^ OP CH « } .
£«f, fCt> a fmetim with pas lo Bit •ad in tli« sm99 of av«r Vithtgat «ar loss of timmtsattf^my •ttiua* tfeat tti« c<««t«t t«x» in ttkn Fasiri«r ««ri«s of f(t> thiit
F fCt> • • w
• m
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- 1 • m tile
f Ct) » 1 MM a
f *
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Mri^s ^ to Pi^iiiiii Ist»i M i =1"
ass l^gliit la pftt^toas dsftptvr v m % PBti and S ^ a t] mm of tlie foilovU:^
^ riitSt f^ £#03
t, SSoha,
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iMpoalas om^itlms m th» ^
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following
LSmJU l!«i ^ ^ ^ ^ ^ amotetheto^
Cesatro^snas of K gotTesptnAinf to the series CO p O S
£ » i M S (fiinnt)h+X ^ 0)f i X respectiyeXy* thm
( i i> S^^ct) « (0 k>03
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am
(1)
(li>
(111) 1
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Oidfil' lE C O j ^ k C h ^ - t ) ^ series | G(n+l)(sin nt)h+l,
(h > 0),
t £141
4 i t e ^ un, m
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§ seguaaee meh th^ ^t i M S igp* <««t ^ fn
0
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S.1 £ ijj ft glirtm lBfi»it« "silth t«q*i«ac« of partial sua [ ^n • * ® By ^^ ftitd Tjj *» d«iiofc« n-th C«s»ro means of or^er ^ of th« se<|U«ac«a tad il^a]
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( s a a ) s I 4 • i <
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Condition cm also mritten as OL
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Mitb mx r giix s
C m^t} ' ItoiftaeDt of Matbew«tii;s h SSOtlSKC*
ijz^lim University, Aligsaffy,
