)(E,s)-summability mean of Fourier series

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1 Padha e al, Coge Mahemaic (06), 3: hp://dxdoiog/0080/ PURE MATHEMATICS RESEARCH ARTICLE Appoximaio of igal belogig o geealized Lipchiz cla uig (N, p )(E,)-ummabiliy mea of Fouie eie Receied: Sepembe 06 Acceped: 07 Ocobe 06 Fi Publihed: Ocobe 06 *Coepodig auho: Suaa Kuma Paikay, Depame of Mahemaic, VSS Uieiy of Techology, Bula 76808, Idia kpaikay_mah@uaci Reiewig edio: Hai M Siaaa, Uieiy of Vicoia, Caada Addiioal ifomaio i aailable a he ed of he aicle Tejawii Padha, Suaa Kuma Paikay * ad Umakaa Mia Abac: Degee of appoximaio of fucio of diffee clae ha bee udied by eeal eeache by diffee ummabiliy mehod I he popoed pape, we hae eablihed a ew heoem fo he appoximaio of a igal (fucio) belogig o he W(L, )-cla by (N, p )(E, )-poduc ummabiliy mea of a Fouie eie The eul obaied hee, geealize eeal kow heoem Subjec: Sciece; Mahemaic & Saiic; Adaced Mahemaic; Aalyi - Mahemaic; Fouie Aalyi Keywod: Degee of appoximaio; Fouie eie; weighed W (L, )-cla; (N, p )(E, )-mea; (N, p )(E, )-mea; Lebegue iegal AMS ubjec claificaio (00): Pimay 4A4; ecoday 4A5; 4B05; 4B08; 40C05 Ioducio The heoy of ummabiliy aoe fom he poce of ummaio of eie ad he igificace of he cocep of ummabiliy ha bee ighly demoaed i ayig coex, eg i Fouie aalyi, appoximaio heoy ad fixed poi heoy ad may ohe field The heoy of appoximaio of fucio ha bee oigiaed fom a well-kow heoem of Weiea, i ha become a exciig ABOUT THE AUTHORS Tejawii Padha i wokig a a eeach chola i he Depame of Mahemaic, Vee Sueda Sai Uieiy of Techology, Bula, Idia Cuely he i coiuig he PhD wok i he field of he Summabiliy heoy Suaa Kuma Paikay i cuely wokig a a aociae pofeo i he Depame of Mahemaic, Vee Sueda Sai Uieiy of Techology, Bula, Idia He ha publihed moe ha 35 eeach pape i aiou Naioal ad Ieaioal Joual of epue The eeach aea of Paikay i Summabiliy heoy, Fouie eie, Opeaio eeach ad Ieoy opimizaio Umakaa Mia i cuely wokig a a pofeo i he Depame of Mahemaic, Naioal Iiue of Sciece ad Techology, Behampu, Idia He ha publihed moe ha 30 eeach pape i aiou Naioal ad Ieaioal Joual of epue The eeach aea of Mia i Summabiliy heoy, Sequece pace, Fouie eie, Ieoy cool, mahemaical modelig ad Gaph Theoy PUBLIC INTEREST STATEMENT The heoy of ummabiliy i a wide field of mahemaic a egad o he udy of Aalyi ad Fucioal Aalyi I ha may applicaio, fo iace, i umeical aalyi (o peed of he ae of coegece), complex aalyi (fo aalyic coiuum), opeao heoy, he heoy of ohogoal eie, ad appoximaio heoy, ec; while he claical ummabiliy heoy deal wih he geealizaio of he coegece of equece o eie of eal o complex umbe Fuhe, i claical ummabiliy heoy, he ue of ifiie maice ha bee a igifica eeach aea i mahemaical aalyi a egad o he udy of ummabiliy of diege equece ad eie Recely, aiou ummabiliy mehod hae ieeig applicaio i appoximaio heoy The appoximaio of fucio by poiie liea opeao i a igifica eeach aea i mahemaical aalyi wih key eleace o udie of compue-aided geomeic deig ad oluio of diffeeial equaio 06 The Auho() Thi ope acce aicle i diibued ude a Ceaie Commo Aibuio (CC-BY) 40 licee Page of 9

2 Padha e al, Coge Mahemaic (06), 3: hp://dxdoiog/0080/ iedicipliay field of udy fo la 30 yea The appoximaio of fucio by geealized Fouie eie, baed o igoomeic polyomial i a cloely elaed opic i he ece deelopme of egieeig ad mahemaic The almo ummabiliy mehod ad aiical ummabiliy mehod ae ow a acie aea of eeach i ummabiliy heoy The eo appoximaio of peiodic fucio belogig o aiou Lipchiz clae hough ummabiliy mehod i alo a acie aea of eeach i he la decade The egiee ad ciei ue he popeie of appoximaio of fucio fo deigig digial file Paaki ad Mouakide (997) peeed a ew L -baed mehod fo deigig Fiie Impule Repoe digial file fo geig opimum appoximaio I imila mae, L p -pace, L -pace, ad L -pace play a impoa ole fo deigig digial file The appoximaio of fucio belogig o aiou Lipchiz clae, hough igoomeic Fouie appoximaio uig diffee ummabiliy mea ha bee poed by aiou ieigao, like Nigam (0), Lal (000), Paikay, Jai, Mia, ad Sahoo (0) ad may ohe Recely, by geealized Hölde iequaliy ad Mikowki iequaliy, Miha, Soaae, ad Miha (03) hae poed L appoximaio of igal belogig o W(L, )-cla by C N p -ummabiliy mea of cojugae eie of Fouie eie Miha ad Soaae (05) ha poed appoximaio of fucio belogig o he Lipchiz cla by poduc mea (N, p )(E,) of Fouie eie I a aemp o make a adace udy i hi diecio, i hi pape, we obai a heoem o he appoximaio of fucio belogig o W(L, ) by (N, p )(E, )-ummabiliy mea of Fouie eie which geealize eeal kow ad ukow eul Defiiio ad oaio Le u be a ifiie eie wih he equece of paial um Le p ad q be equece of poiie eal umbe uch ha, P = ad Q = ad le R = p 0 q + p q + + p q 0 ( 0), p = q = R = 0 The equece o equece afomaio (Miha, Palo, Padhy, Samaa, & Mia, 04), N = R p k k defie he equece N of he (N, p, q ) mea of he equece geeaed by he equece of coefficie p ad q Similaly, we defie he exeded Riez mea, N = R k () whee R = p 0 q 0 + p q + +p q ( 0), p = q = R = 0 If u, he he eie u i (N, p ) ummable o Aalogou o egulaiy codiio of Riez ummabiliy (Hady, 949), we hae (i) p k 0, fo each iege k 0 a ad R (ii) p q k k < CR, whee, C i ay poiie iege idepede of Page of 9

3 Padha e al, Coge Mahemaic (06), 3: hp://dxdoiog/0080/ The equece o equece afomaio (Hady, 949), E = ( + ) =0 ( ), () defie he equece E of (E, ) mea of he equece If E a, he u ummable o wih epec o (E, ) ummabiliy ad (E, ) mehod i egula (Hady, 949) T NE Now we defie, a ew compoie afomaio (N, p ) oe (E, ) of a If T NE = R E k = R a, he u i ummable o by (N, p )(E, ) mea ( + ) k =0 ( k ) k (3) Fuhe a (N, p ) ad (E, ) mea ae egula, o (N, p )(E, ) mea i alo egula Remak If we pu q = i Equaio () he (N, p )-ummabiliy mehod educe o (N, p )- ummabiliy ad fo p = i educe o (N )-ummabiliy Le f i a π peiodic fucio belogig o L [0, π],, wih he paial um (f ), he (f )= a 0 + (a k co kx + b k i kx) k= Hee, a egad o Lipchiz clae we may ecall ha, a igal (fucio) f Lip(α), if (4) f (x + ) f (x) ( α ) fo 0 <α, > 0, ad f Lip(α, ), fo 0 x π, if f (x + ) f (x) dx [0,π] Agai, f Lip(, ), if ( α ) fo 0 <α, > 0, f (x + ) f (x) = [0,π] f (x + ) f (x) dx (),, > 0, whee i a poiie iceaig fucio Similaly, f W(L, ), if [f (x + ) f (x)] i β x = Fuhe a egad o he om i L ad L -pace, we may ecall ha L -om of a fucio f :R R i defied by f = upf (x):x R [0,π] [f (x + ) f (x)] iβ x dx (), β 0 ad L -om of a fucio f :R R i defied by Page 3 of 9

4 Padha e al, Coge Mahemaic (06), 3: hp://dxdoiog/0080/ ( ) f = f (x) dx, [0,π] Nex, he degee of appoximaio of a fucio f :R R by a igoomeic polyomial of ode ude i defied by f (x) = up (x) f (x):x R, ad he degee of appoximaio of E (f ) of a fucio f L i gie by E (f )=mi f We ue he followig oaio houghou hi pape: φ() =f (x + )+f (x ) f (x) ad K () = p πr k ( + ) =0 ( k ) i( + ) k i( ) Remak If we ake β = 0, he W(L, )-cla coicide wih he cla Lip(, ); if = α he he cla Lip(, ) coicide wih Lip(α, )-cla ad if he Lip(α, )-cla educe o he Lip(α) 3 Kow heoem Dealig wih he poduc (C, )(E, q) mea, i Nigam (0) poed he followig heoem Theoem If f i a π peiodic fucio, Lebegue iegable o [0, π] ad belog o W(L, ) cla, he i degee of appoximaio i gie by [ ] C Eq f ( + ) β+ ξ, + (3) poided aifie he followig codiio: be a deceaig equece, (3) + 0 ad φ() i β d + (33) π ( δ ) φ() d + ( + ) δ, (34) whee δ i ay abiay umbe uch ha ( δ) > 0, + =, codiio (33) ad (34) hold uifomly i x ad C Eq i (C, )(E, q) mea of he Fouie eie (4) Nex, dealig wih degee of appoximaio, i Miha e al (04) poed he followig heoem Theoem Fo a poiie iceaig fucio ad a iege l >, if f i a π-peiodic fucio o he cla Lip(, l), he he degee of appoximaio by poduc (E, )(N, p )-ummabiliy mea of Fouie eie (4) i gie by Page 4 of 9

5 Padha e al, Coge Mahemaic (06), 3: hp://dxdoiog/0080/ τ f,0<α<, l, ( + ) α l whee τ i (E, )(N, p )-ummabiliy mea 4 Mai heoem Hee, ju by eplacig Nölud ummabiliy by exeded Riez ummabiliy ad akig he eee compoiio, we hae poed he followig heoem Theoem 3 Le f be a π peiodic fucio which i iegable i Lebegue ee i [0, π] If f W(L, ) cla, he i degee of appoximaio i gie by T NE f ( + ) β+ ξ, + (4) whee T NE i he (N, p, q )(E, ) afom of, poided aifie he followig codiio; be a deceaig equece, (4) [0, ad + ] ( φ() ) i β d + (43) [ +,π] ( δ φ() ) d ( + ) δ (44) To poe he heoem, we eed he followig lemma Lemma K () (), fo 0 + Poof Fo 0, a i i ; o we hae + K () = πr ( + ) k =0 ( + ) p πr k ( + ) k =0 ( + ) = πr () Lemma K () ( ), fo < π + ( k ( k [ + ]k k ( + ) i + ) k i ) k Poof Fo < π, a i (Joda lemma) ad i ; o + π Page 5 of 9

6 Padha e al, Coge Mahemaic (06), 3: hp://dxdoiog/0080/ K () = k i + πr k ( + ) k =0 i π p πr k k ( + ) k k =0 = p R k 5 Poof of mai heoem Uig Riema-Lebegue heoem, (f ) f(x) = φ() i( + ) d π i [0,π] Fuhe, E f (x) = π( + ) k T NE f (x) = πr = + [0, ] [ + +,π] = I + I (ay) [0,π] φ()k () d φ() i [0,π] ( φ() k i =0 ( k ) ( k i k + ), ad ) ( k i + ) d (5) Now, I [0, φ()k () d ] + A, φ(x, ) φ(x) f (u + x + ) f (u + x) + f (u x ) f (u x), o, by uig Mikowki iequaliy, [0,π] φ(x + ) φ(x) iβ x dx Fuhe f W(L, ) implie φ W(L, ), hu [0,π] + f (u + x + ) f (u + x) i β x dx [0,π] () f (u x ) f (u x) i β x dx I [0, ] + φ() i β K () i β d Now by Hölde iequaliy ad Lemma, we hae Page 6 of 9

7 Padha e al, Coge Mahemaic (06), 3: hp://dxdoiog/0080/ I [0, + ] lim ε 0 φ() i β [ε, ] + lim + ε 0 + lim ε 0 d K () d i β Alo, by d mea alue heoem, we hae [ε, [ε, ] + ] + ( O() i β fo ome 0 <ε< + ) d d i β by (43) I ()ξ + ( ξ ξ + ( + Nex, I π + [ε, Now by Hölde iequaliy ad Lemma, we hae ] + ( +β ) ( (β+ ) ) + ) ( + ) β+ 0 δ φ() i β ) d ice + = K () d δ i β (5) I [ +,π] δ φ() i β d ( + ) δ [ +,π] ( δ+β Agai by uig d mea alue heoem [,π] + ) d K () d δ i β by (44) = ( + ) δ ξ( ) y dy [,+] y δ β y by (4) π Page 7 of 9

8 Padha e al, Coge Mahemaic (06), 3: hp://dxdoiog/0080/ I ( + ) δ ξ + ( ( + ) δ ξ ( + ) δ ξ ( + ) β+ ξ dy y (δ β)+ [,+] π )( ( + ) (+β δ) π (+β δ)+ + ( + β δ) ( + ) β δ+ + ice + + = ) (53) Nex, by uig (5) ad (53), i (5) we hae T NE f ( + ) β+ ξ + T NE f = 0 Which complee he poof of heoem π ( O ( + ) β+ ξ + ( π ) ( + ) β+ ξ dx + 0 ( + ) β+ ξ + ) dx Coollay If we pu β = 0 i Theoem 3, he he geealized Lipchiz W(L, )-cla educe o Lip(, ), whee i ay poiie iceaig fucio ad l > If f i π-peiodic ad belogig o cla Lip(, ), he he degee of appoximaio by (N, p )(E, )-ummabiliy mea of Fouie eie i T NE f ( + ) ξ + Coollay If we pu β = 0 ad = α, 0 <, i Theoem 3, he geealized Lipchiz W(L, )-cla educe o Lip(α, ), he he degee of appoximaio of π peiodic fucio f belogig o cla Lip(α, ) by (N, p )(E, )-ummabiliy mea of Fouie eie i (54) T NE f ( + ) α,0<α<, l (55) Coollay 3 If we pu β = 0, = α, 0 <α ad he he geealized Lipchiz W(L, )-cla educe o Lip(α), he he degee of appoximaio of π peiodic fucio by (N, p )(E, )-ummabiliy mea of Fouie eie (f ) i T NE f ( + ) α, whee 0 <α< (56) Page 8 of 9

9 Padha e al, Coge Mahemaic (06), 3: hp://dxdoiog/0080/ Fudig The auho eceied o diec fudig fo hi eeach Auho deail Tejawii Padha ejawiibii@gmailcom Suaa Kuma Paikay kpaikay_mah@uaci Umakaa Mia umakaa_mia@yahoocom Depame of Mahemaic, VSS Uieiy of Techology, Bula 76808, Idia Depame of Mahemaic, NIST Pallu Hill, Golahaa 76008, Idia Ciaio ifomaio Cie hi aicle a: Appoximaio of igal belogig o geealized Lipchiz cla uig (N, p )(E, )- ummabiliy mea of Fouie eie, Tejawii Padha, Suaa Kuma Paikay & Umakaa Mia, Coge Mahemaic (06), 3: Refeece Hady, G H (949) Diege Seie ( ed) Oxfod: Oxfod Uieiy Pe Lal, S (000) O degee of appoximaio of cojugae of a fucio belogig o weighed (Lp, )cla by maix ummabiliy mea of cojugae eie of a Fouie eie Tamkag Joual of Mahemaic, 3, Miha, M, Palo, P, Padhy, B P, Samaa, P, & Mia, U K (04) Appoximaio of Fouie eie of a fucio of Lipchiz cla by poduc mea Joual of Adace i Mahemaic, 9, Miha, V N, & Sooae, V (05) Appoximaio of fucio of Lipchiz cla by (N, p ) (E, ) ummabiliy mea of cojugae eie of Fouie eie Joual of Claical Aalyi, 6, 37 5 Miha, V N, Soaae, V, & Miha, L N (03) L -Appoximaio of igal (fucio) belogig o weighed W (L, )-cla by (C N p )-ummabiliy mehod of cojugae eie of i Fouie eie Joual of Iequaliy ad Applicaio, Aicle ID, 440, 5 doi:086/09-4x Nigam, H K (0) Degee of appoximaio of a fucio belogig o weighed (L, ) cla by (C, ) (E, q) mea Tamkag Joual of Mahemaic, 4, 3 37 Paikay, S K, Jai, R K, Mia, U K, & Sahoo, N C (0) O degee of appoximaio of Fouie eie by poduc mea Geeal Mah Noe, 3, 30 Paaki, E Z, & Mouakide, G V (997) A L -baed mehod fo he deig of -D zeo phae FIR digial file IEEE Taacio o Cicui ad Syem, I44, The Auho() Thi ope acce aicle i diibued ude a Ceaie Commo Aibuio (CC-BY) 40 licee You ae fee o: Shae copy ad ediibue he maeial i ay medium o foma Adap emix, afom, ad build upo he maeial fo ay pupoe, ee commecially The liceo cao eoke hee feedom a log a you follow he licee em Ude he followig em: Aibuio You mu gie appopiae cedi, poide a lik o he licee, ad idicae if chage wee made You may do o i ay eaoable mae, bu o i ay way ha ugge he liceo edoe you o you ue No addiioal eicio You may o apply legal em o echological meaue ha legally eic ohe fom doig ayhig he licee pemi Coge Mahemaic (ISSN: ) i publihed by Coge OA, pa of Taylo & Faci Goup Publihig wih Coge OA eue: Immediae, uieal acce o you aicle o publicaio High iibiliy ad dicoeabiliy ia he Coge OA webie a well a Taylo & Faci Olie Dowload ad ciaio aiic fo you aicle Rapid olie publicaio Ipu fom, ad dialog wih, expe edio ad edioial boad Reeio of full copyigh of you aicle Guaaeed legacy peeaio of you aicle Dicou ad waie fo auho i deelopig egio Submi you maucip o a Coge OA joual a wwwcogeoacom Page 9 of 9