International Journal of Mathematical Archive-3(1), 2012, Page: Available online through

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1 eril Jrl f Mhemicl rchive-3 Pge: vilble lie hrgh wwwijmif NTE N UNFRM MTRX SUMMBLTY Shym Ll Mrdl Veer Sigh d Srbh Prwl 3* Deprme f Mhemics Fcly f Sciece Brs Hid Uiversiy Vrsi UP - ND E-mil: shym_ll@rediffmilcm Deprme f Mhemics Lvely Prfessil Uiversiy Pjb ND E-mil: mrdlsigh@gmilcm 3 Deprme f Mhemics UET CSJM Uiversiy pr-84 UP ND E-mil: srbhjcb@rediffmilcm Received : 7--; cceped : 3-- BSTRCT The prpse f he prese pper is esblish ew resl ccerig ifrm mri smmbiliy f cjge series f Frier series Relev cecis f he resls preseed herewih vris w resls re briefly idiced ey Wrds: Uifrm riglr mri smmbiliy Cjge series f Frier series Frier cefficies Nörld smmbiliy MS Mhemics Sbjec Clssifici: 44 4C5 NTRDUCTN ND PRELMNRES: Le f be peridic Lebesge iegrble fci wih Frier series give by f cs b si The cjge series f series is give by Le i ii si b cs B T be ifiie lwer riglr mri sisfyig Silverm-Töepliz [9] cdiis f reglriy ie s fr > d iii M where M is fiie cs Le be ifiie series defied i [ b] [ ] The h pril sm f he series is give by S ν [ b] ν Srbh Prwl 3* srbhjcb@rediffmilcm eril Jrl f Mhemicl rchive- 3 J 33

2 f here eiss bded fci S sch h ifrmly [ b] sm S Priclr Cses: { } S S s he we sy h he series is smmble T ifrmly i b JM ll Righs Reserved 34 he Severl hrs sch s []-[4] see ls [5] sdied he mri smmbiliy mehd d bied my ieresig resls The impr priclr cses f he riglr mri mes re: i Cesàr me f rder r C me if ii Hrmic mes whe lg iii C mes whe iv H p mes whe v Nörld mes whe p q lg p lg vi Riesz mes N p whe p q P P where p p P vii Geerlised Nörld Mes N p q whe P We dee S he h pril sm f he series Le p q P R where R p q R f f c d 3 f lim f 4 f f 5 Ψ d 6 where 7 iegrl pr f 8

3 cs d 9 si Se [6] discssed he ifrm hrmic smmbiliy f cjge series f Frier series i he fllwig frm: Therem: f Ψ d ifrmly i se E s lg he he series is smmble by hrmic mes ifrmly i E he sm f prvided he limi 4 eiss ifrmly i E Triphi d Sigh [8] eeded he bve resl he cse f ifrm Nörld smmbiliy i he fllwig frm: Therem: f he seqece { } h q is rel -egive d mic -icresig seqece f cefficies sch Q s d he fci λ λ Q [ Q ] s he if d λ icrese miclly wih d λ q Ψ d Q he he series is smmble N q ifrmly i E he sm f prvided he limi 4 eiss ifrmly i E ifrmly i se E s he pi MN THEREM: The prpse f his pper is geerlize he resl f Se [6] d Triphi d Sigh [8] fr ifrm mri smmbiliy mehd fc we prve he fllwig ieresig resl Therem: Le T be ifiie riglr mri sch h he elemes -decresig wih d if Ψ d lg s ifrmly i se E [ b] where is psiive fci f sch h lg he he cjge series f Frier series is smmble T ifrmly i E f c d prvided he limi 4 eis ifrmly i E [ b] re -egive d T prve r mi herem we reqire he fllwig lemms 3 LEMMS: Lemm: 3 f is -egive d -decresig wih he cs fr < < < s Prf: cs cs cs JM ll Righs Reserved 35

4 r m cs cs r by bel s Lemm Nw cs si r si sice [ ] 3 Therefre 3 By 3 d 3 we hve cs Lemm: 3 f is -egive d -decresig wih d fr < < < < si cs si Prf: Sice fr < < We hve si [ ] cs frm Lemm 3 Hece he lemm is prved 4 PRF F THE MN THEREM: is give by 9 he is well w h he iegrl frml fr he h pril sm f cjge series f Frier series is give by see [7]: cs cs S d si JM ll Righs Reserved 36

5 JM ll Righs Reserved 37 d f S si cs Nw { } d f S si cs d si cs d 4 S i rder prve r mi herem we hve shw h d ifrmly i E 4 We se d d d d 3 sy 43 Sice limi 4 eiss ifrmly i E s d ifrmly i E 44 ls fr < < si cs cs si si si si si si Sice si si fr < < 45

6 JM ll Righs Reserved 38 Hece d d d c si cs cs d ifrmly i E sig 44 d 45 d ifrmly i E lg ifrmly i E by cdii lg ifrmly i E s ifrmly i E by hyphesis f herem 46 Nw d d by sig Lemm 3 Ψ Ψ d d d d d lg lg lg [ ] [ ] [ ] d d lg lg lg lg [ ] [ ] [ ] [ ] lg lg lg pplyig he Me Vle Therem fr iegrls lg s ifrmly i E 47

7 ls by he vire f Riem Lebesge herem d reglriy f mehd f smmi we hve 3 s 48 Ths frm d 48 we hve { S f } This cmplees he prf f he herem Remr: f we p resl f Se [6] s ifrmly i E lg [ b] E he we bi he crrespdig Remr: The resl f Triphi d Sigh [8] is priclr cse f r herem if λ q lg d [ b] E α Q q Q Q q REFERENCES: [] Shym Ll he degree f pprimi f cjge f fci belgig weighed W L P ξ clss by mri smmbiliy mes f cjge series f Frier series Tmg J Mh 7-6 [] Shym Ll d J shwh pprimi f cjge f fcis belgig he geerlized Lipschiz clss by lwer riglr mri mes J Mh l [3] Shym Ll d Prim Ydv Mri Smmbiliy f he cjge series f derived Frier series Tmg J Mh [4] ML Mil d R mr mri smmbiliy f Frier series d is cjge Series Bll Cl Mh Sc [5] Qreshi he degree f pprimi f peridic fci by lms Nörld mes f is Frier series Tmg J Mh [6] sh Se ifrm hrmic smmbiliy f Frier series d is cjge series Prc N s Sci di Pr [7] E C Tichmrsh Thery f Fcis Secd edii frd Press 939 [8] L M Triphi d N Sigh he ifrm Nörld smmbiliy f Frier Series d is cjge series Bll Cl Mh Sc [9] Töepliz Überllgemeiee liere Miel bildger PMF [] Zygmd Trigmeric series Cmbridge Uiversiy Press 977 ******************* JM ll Righs Reserved 39