Research Article On Pointwise Approximation of Conjugate Functions by Some Hump Matrix Means of Conjugate Fourier Series

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1 Hidawi Publishig Copoaio Joual of Fucio Spaces Volue 5, Aicle ID 475, 9 pages hp://dx.doi.og/.55/5/475 Reseach Aicle O Poiwise Appoxiaio of Cojugae Fucios by Soe Hup Maix Meas of Cojugae Fouie Seies W. Aesi ad B. Szal Faculy of Maheaics, Copue Sciece ad Ecooeics, Uivesiy of Zieloa Góa, Ulica Szafaa 4a, Zieloa Góa, Polad Coespodece should be addessed o B. Szal; b.szal@wie.uz.zgoa.pl Received Ocobe 4; Revised 7 Jauay 5; Acceped Jauay 5 Acadeic Edio: Rodolfo H. Toes Copyigh 5 W. Łesi ad B. Szal. This is a ope access aicle disibued ude he Ceaive Coos Aibuio Licese, which peis uesiced use, disibuio, ad epoducio i ay ediu, povided he oigial wo is popely cied. The esuls geealizig soe heoes o (N, p )(E, γ) suabiliy ae show. The sae degees of poiwise appoxiaio as i ealie papes by weae assupios o cosideed fucios ad exaied suabiliy ehods ae obaied. Fo peseed poiwise esuls, he esiaio o o appoxiaio is deived. Soe special cases as coollaies ae also foulaed.. Ioducio Le L p ( p< )(esp., L )beheclassofall-peiodic eal-valued fucios iegable i he Lebesgue sese wih ph powe (esseially bouded) ove Q [, wih he o f : f ( ) L p ( f () p /p d) Q ess sup f () Q whe p<, whe p ad coside he cojugae igooeic Fouie seies Sf (x) : () (a (f) si x b (f) cos x) () wih he paial sus S f.weowhaiff L he whee f (x) : f (x, ε) : ψ x () co d li f (x, ε), (3) ε + ε ψ x () co d, (4) wih ψ x () : f (x+) f(x ), (5) exiss fo alos all x [, Th.(3.)IV. Le A:(a, ) ad B:(b, ) be ifiie lowe iagula aices of eal ubes such ha a,, b,, whe,,,...,, a,, b,, whe >, a,, b,, whee,,,..., (6)

2 Joual of Fucio Spaces ad le, fo,,,...,, A, A, a,, a,. Le he AB-asfoaio of S f be give by (7) Cosequely, we assue ha he sequece (K(α )) is bouded, ha is, ha hee exiss a cosa K such ha K(α ) K (4) holds fo all, wheek(α ) deoe he cosas fo he sequeces α (a, ),,,,... appeaig i he iequaliies () ad () as K(c). Now we ca give he codiios o be used lae o. We assue ha fo all ad < T,A,B f (x) : a, b, S f (x) (,,,...). (8) We defie wo classes of sequeces (see [). Sequece c : (c ) of oegaive ubes edig o zeo is called he Res Bouded Vaiaio Sequece,obiefly c RBVS, if i has he popey c c + K(c) c, (9) fo all posiive iege,wheek(c) is a cosa depedig oly o c. Sequece c:(c ) of oegaive ubes will be called he Head Bouded Vaiaio Sequece, obieflyc HBVS, if i has he popey c c + K(c) c, () fo all posiive iege,o oly fo all if he sequece c has oly fiie ozeo es ad he las ozeo e is c. Now, we defie he ohe classes of sequeces. Followig Leidle [3, sequece c:(c ) of oegaive ubes edig o zeo is called he Mea Res Bouded Vaiaio Sequece, o biefly c MRBVS, if i has he popey c c + K(c) + / c, () fo all posiive iege. Aalogously as i [4, sequece c:(c ) of oegaive ubes will be called he Mea Head Bouded Vaiaio Sequece,o biefly c MHBVS, if i has he popey c c + K(c) + c, () fo all posiive ieges <,wheehesequecec has oly fiieozeoesadhelasozeoeisc.iisclea ha (see [5) RBVS MRBVS, HBVS MHBVS. (3) a, a,+ K + a, a,+ K + / a,, a, (5) hold if (a, ) belogs o MRBVS ad MHBVS, fo,,...,especively. We also defie wo hup aices i he followig way: a lowe iagula aix C(c, ) is called a axial hup aix if, fo each, hee exiss iege (),suchha (c, ) is odeceasig fo < ad (c, ) is oiceasig fo,buohewisewewillhavea iial hup aix. The hup aices wee defied ad cosideed i [6, 7. As a easue of appoxiaio of f by T,A,B f,weusehe poiwise odulus of coiuiy of f i he space L p defied by he foula w p x f (δ) δ δ ess sup <u δ ad he classical oe ψ x (u) si u p /p du ψ x (u) si u ω f (δ) L p whe p<, whe p (6) sup < δ si ψ () L p. (7) The deviaio T,A f f T,A,B f f,wihb, ad ohewise, was esiaed a he poi as well as i he o of L p by Queshi [8 ad Lal ad Niga [9. These esuls wee geealized by Queshi [. The ex geealizaio was obaied by Lal [. I he case a, +, a,, whe,,,...,, whe >, b, ( )γ ( +γ), whe,,,...,, b,, whe >wih γ>, (8) he deviaio T,A,B f f was esiaed by Soe ad Sigh [ as follows.

3 Joual of Fucio Spaces 3 Theoe. Le f(x) be a -peiodic, Lebesgue iegable fucio which belogs o he Lip(α, p)-class wih p ad αp. Thehedegeeofappoxiaioof f(x), he cojugae of f(x) by (C, ) (E,γ)eas of seies (), isgive by +( +γ) ( )γ S f ( ) f ( ) L p O((+) α+/p ), povided /(+) ( ψ p /p x () α ) d O((+) ), ( δ ψ p /p x () /(+) α ) d O((+) δ ), (9) () whee δ is a abiay posiive ube wih (α + δ)q + < ad p +q, p>. I his pape we will coside he deviaios T,A,B f( ) f( ) ad T,A,B f( ) f(, /( + )) i geeal fo. I he heoes we foulae he geeal codiios fo he fucios ad he odulus of coiuiy obaiig he sae degees of appoxiaio as above ad soeies esseially bee oe. Fially, we also give soe esuls o o appoxiaio wih esseially bee degees of appoxiaio. The obaied esuls geealize he esuls fo [4, 9. We will wie I I if hee exiss posiive cosa K, soeies depedig o soe paaees, such ha I KI.. Saee of he Resuls Le L p ( w x ) f L p : w p x f (δ) w x (δ), () whee w x is a posiive, wih w x (), ad odeceasig coiuous fucio. Wecaowfoulaeouaiesuls.Ahebegiig, we foulae he esuls o he degees of poiwise suabiliy of cojugae seies. Theoe. Le f L.IfaixA is a axial o iial hup aix wih O( ) such ha (a, ) MHBVS MRBVS ad b, cos μ fo μ,he (+) () T,A,B f (x) f(x, + ) O((+) [ w x + f( + ) ), fo alos all cosideed x,whee [,. (3) Theoe 3. Le f L p ( w x ) wih <p<,adle w x saisfy /(+) p/(p ) ( w x () si / ) d O((+) +/p w x ( + )), /(+) ( ψ p /p x () w x () ) si p d O((+) /p ) (p )/p (4) (5) wih soe.ifheeiesofaixb saisfy codiio () fo μ ad if aix A is a axial o iial hup aix wih O( ) such ha (a, ) MHBVS MRBVS,he T,A,B f (x) f (x) O((+) [ w + x ( (6) + )), fo alos all cosideed x such ha f(x) exiss. Nex, we foulae he esuls o esiaes of L p o of he deviaio cosideed above. I case of he deviaio T,A,B f( ) f( ),le L p ( ω) f L p : ω f (δ) L p ω (δ), (7) whee ω is posiive, wih ω(), ad alos odeceasig coiuous fucio. Theoe 4. Le f L p ( p ).Ifheeiesofaix B saisfy codiio () fo μ ad if aix A is a axial o iial hup aix wih O( ) such ha (a, ) MHBVS MRBVS,he T,A,B f ( ) f(, + ) L p O((+) [ ω + f( + ) ), L p whee. (8) Theoe 5. Le f L p ( ω) wih < p <,whee ω isead of w x saisfies (4) wih soe. Ifheeiesof aix B saisfy codiio () fo μ ad if aix A is

4 4 Joual of Fucio Spaces aaxialoiialhupaixwih O( ) such ha (a, ) MHBVS MRBVS,he T,A,B f ( ) f ( ) L p O((+) [ ω( (9) + + )). Fially, we give coollay ad eas as a applicaio of ou esuls. Taig a, p / p whe,,,..., ad a, whe > wih p >, p p + ad b, ( )γ /( +γ) whe,,,...,ad b, whe> wih γ>, Theoe (Theoe 3 aalogously) iplies he followig. Coollay 6. If f L,he p p ( +γ) ( )γ S f (x) f(x, + ) O((+) [ w x + f( + ) ), p p ( +γ) ( )γ S f (x) f(x, + ) O((+) [ w x + f( + ) ), (3) fo alos all cosideed x,whee, p >, p p +, ad γ>. Rea 7. I special case, if p p, he +( +γ) ( )γ S f (x) f(x, + ) O((+) [ w x + f( + ) ), ad if f Lip(α, p),he (3) +( +γ) ( )γ S f (x) f(x, + ) (3) O((+) α ) whe <α< O((+) log (+)) whe α, fo alos all cosideed x,whee ad γ>. Rea 8. Taig, we have, by Theoe 3 wih w x (δ) δ α,fo < α < adαp >,heesiaeliei[ wih he bee ode of appoxiaio wihou ay addiioal assupios. Rea 9. Aalyzig he poofs of Theoes 5, we ca deduce ha, aig he assupio (a, ) RBVS o (a, ) HBVS isead of (a,) MRBVS o (a,) MHBVS, especively, we obai he esuls lie ha fo [3. 3. Auxiliay Resuls We begi his secio by soe oaios followig A. Zygud [, Secio 5 of Chape II. I is clea ha S f (x) f (x+) D () d, T,A,B f (x) f (x+) a, b, D () d, whee Hece whee D () si T,A,B f (x) f(x, (33) cos (/) cos ((+) /). (34) si(/) + ) /(+) ψ x () a, b, D () d + /(+) T,A,B f (x) f (x) ψ x () a, b, D () d, ψ x () a, b, D () d, D () co si (35) cos ((+) /). (36) si(/) Now, we foulae soe esiaes fo he cojugae Diichle eels. Lea (see [). If < /,he ad fo ay eal oe has D () (37) D () (+), D () +. (38)

5 Joual of Fucio Spaces 5 Lea. Le (b, ) be such ha () fo μ holds. If (a, ) MRBVS,he a, b, D () A,, (39) ad if (a, ) MHBVS,he a, b, D () A, (4) wih [/ad [/(+),,fo,,,... Poof. Le K () : a, b, cos The elaio (a, ) MRBVS iplies s a,s a, a, a,s a, a,+ (+). (4) we obai K () + ( a, l l+ +a, b, + b l, cos l+ +( + +[ [ [ + l + + b l, cos / a, / / )a, b, cos a, a,+ (+) a, + + (a, + + a, / / (+) A, / l / (a, + / + a, / (+) a, ) A, a,l ) A, l /4 a,l ) A, (45) a, a,+ + / a,, ( <s ) (4) + + +[ + / / (a, + / + a, + + / l /4 a,l ) A, a, + a +,l l whece a,s a, + + / a,, ( <s ), (43) A, +3 l a,l 4A,. The elaio (a, ) MHBVS iplies ad hus, by ou assupio, l+ l b l, cos (+), (44) s a, a,s a, a,s a, a,+ a, a,+ a +,, ( <s ) (46)

6 6 Joual of Fucio Spaces whece a, a,s + a +,, ad hus, by ou assupio, we ge K () ( l b l, cos l + A, + l b l, cos l +a, +[ + +[ + l l A, +[ / A, + A,. ( <s ), (47) (+), (48) )a, a, a,+ (+) b l, cos b, cos (+) (+) A, a, +a, A, a, + + a, (a, + + a, + Now, ou poof is coplee. / / + a, a, ) a, (49) Lea. If aix A is a axial o iial hup aix wih O( ),he o + a +, O() (5) + a +, O() (5) fo,,,...,especively. Poof. Siceheabovefoulasaesiila,wepovehefis oe oly. If <,he a, a, + a (+), a, (a (+), a, ), (5) whece (/( + )) a, odeceases wih espec o ad heefoe a +, a +, + Bu if,he + O( ). a, a +, a +, + ad ou poof is coplee. + O( ) a, Lea 3. If f L p ( p )ad,he w p x f( + ) [ w p x + f( + ) holds fo evey aual ad all eal x. Poof. The poof follows by he easy accou [ w p x f( + ) + p /(+) (+)(+) + (+) + + p /p ψ x (u) p (+) du (+)(+) /(+) ψ x (u) p du /(+) ψ x (u) p du [ w p x + f( + ) Now, ou poof is coplee. p. (53) (54) (55) (56)

7 Joual of Fucio Spaces 7 4. Poofs of he Resuls whece 4.. Poof of Theoe. We sa wih he obvious elaios T,A,B f (x) f(x, + ) /(+) ψ x () a, b, D () d + /(+) T,A f (x) f(x, ψ x () a, b, D () I + I, + ) I + I. (57) I + /(+) (+) + (+) + d /(+) d [ [ ψ x () si si (/) d /(+) [ ψ x () si d ψ u x (u) si du d ( + ) [ ψ u x (u) si du /(+) By Leas ad 3,wehave /(+) I (+) ψ x () d (+) /(+) ψ x () si si d (+) /(+) (+) w x f( + ) ψ x () si d ( ) + (58) + /(+) + ) 3 [ ψ u x (u) si du d ( [ w x f () ( + ) w x f( + ) + w x f () d (+) /(+) w x f( + ) (6) (+) w x f( + ), + + w x f( ) d (+) fo [,. Usig Leas ad we obai w x f( + ). I /(+) /(+) ψ x () ψ x () a, d + d (59) Collecig hese esiaes, we obai he desied esul. 4.. Poof of Theoe 3. We sa wih he obvious elaios o T,A f (x) f (x) I /(+) /(+) /(+) ψ x () ψ x () ψ x () a, d a, d + d, (6) /(+) ψ x () a, b, D () d + /(+) ψ x () a, b, D () d I + I, T,A f (x) f (x) I + I. (6)

8 8 Joual of Fucio Spaces By he Hölde iequaliy (/p+/q ), Lea, (5), ad (4), I /(+) ψ x () d /(+) [ ψ p /p x () w x () si d /(+) q w [ x () si (/) /q d (+) w x ( + )(+) w x ( + ) (+) w x ( + ). (63) We ca esiae he e I byhesaewaylieihe poof of Theoe : I /(+) /(+) /(+) (+) (+) ψ x () ψ x () ψ x () a, d a, d + d w x f( + ) w x ( + ). Collecig hese esiaes, we obai he desied esul. (64) 4.3. Poofs of Theoes 4-5. The poofs ae siila o hese above ad follow fo he evide iequaliy wp f (δ) L p ω f (δ) L p (65) ad addiioally i case of Theoe 5 fo he esiae fo f L p ( ω). /(+) ( ψ p () ω () ) /(+) /(+) ( ψ si L p ω () si p d/plp /p )d ( ω p /p f () L p ) d ω () /(+) /p d O x ((+) /p ) (66) 4.4. Poof of Coollay 6. Fis of all we oe ha ou aix A is he hup aix wih. Nex, we have o veify he assupios of Lea. Fo he fis oe, we oe ha ay odeceasig sequece belogs o he class MRBVS ad ay oiceasig sequece belogsoheclassmhbvs.thesecodoefollowsfohe followig calculaios: (+) ( +γ) ( μ )γ cos Re ( +γ) ( μ )γ exp (+) i Re ( +γ) exp i ( μ )γ exp i Re ( +γ) exp i μ ( +γexp i) Re [exp i +γexp i ( ) Re [exp i μ +γ ( μ +γexp i ) +γ (( +γexp i) / ( +γ)) μ+ ( +γexp i) / ( +γ) (( +γexp i) / ( +γ)) μ+ ( +γexp i) / ( +γ) ( +γexp i) / ( +γ) ( +γ) ( +γ) +γ γexp i γ exp i ( +γ) ( +γ) γ ( cos ) + si γ cos ( +γ) +γ γ si γ (/) si (/) +γ γ. Thus poof is coplee. Coflic of Ieess (67) The auhos declae ha hee is o coflic of ieess egadig he publicaio of his pape.

9 Joual of Fucio Spaces 9 Refeeces [ A. Zygud,Tigooeic Seies, Cabidge, UK,. [ L. Leidle, O he degee of appoxiaio of coiuous fucios, Aca Maheaica Hugaica, vol.4,o.-,pp. 5 3, 4. [3 L. Leidle, Iegabiliy codiios peaiig o Olicz space, Joual of Iequaliies i Pue ad Applied Maheaics, vol.8, o.,aicle38,6pages,7. [4 W. Lesi ad B. Szal, Appoxiaio of fucios fo L p ( w) by liea opeaos of cojugae Fouie seies, Baach Cee Publicaios,vol.9,pp.37 47,. [5 B. Szal, A oe o he uifo covegece ad boudedess a geealized class of sie seies, Coeaioes Maheaicae, vol.48,o.,pp.85 94,8. [6 V. N. Misha ad L. N. Misha, Tigooeic appoxiaio of sigals (fucios) i L p -o, Ieaioal Joual of Coepoay Maheaical Scieces,vol.7,o.9,pp.99 98,. [7 M. L. Mial, B. E. Rhoades, V. N. Misha, ad U. Sigh, Usig ifiie aices o appoxiae fucios of class Lip(α, p) usig igooeic polyoials, Joual of Maheaical Aalysis ad Applicaios,vol.36,o.,pp ,7. [8 K. Queshi, O he degee of appoxiaio of fucios belogig o he Lipschiz class by eas of a cojugae seies, Idia Joual of Pue ad Applied Maheaics, vol.,o.9, pp. 3, 98. [9 S.LaladH.K.Niga, Degeeofappoxiaioofcojugae of a fucio belogig o (ξ(), p) class by aix suabiliy eas of cojugae Fouie seies, Ieaioal Joual of Maheaics ad Maheaical Scieces,vol.7,o.9,pp ,. [ K. Queshi, O he degee of appoxiaio of fucios belogig o he class Lip (α, p) by eas of a cojugae seies, Idia Joual of Pue ad Applied Maheaics, vol.3,o.5, pp ,98. [ S. Lal, O he degee of appoxiaio of cojugae of a fucio belogig o weighed W(L p, ξ()) class by aixsuabiliy eas of cojugae seies of a Fouie seies, Taag Joual of Maheaics, vol. 3, o. 4, pp ,. [ S. Soe ad U. Sigh, Degee of appoxiaio of he cojugae of sigals (fucios) belogig o Lip (α, )-class by (C, )(E, q) eas of cojugae igooeic Fouie seies, Joual of Iequaliies ad Applicaios, vol.,aicle78,. [3 W. Lesi ad B. Szal, Appoxiaio of fucios belogig o he class L p (ω) by liea opeaos, Aca e Coeaioes Uivesiais Tauesis de Maheaica,vol.3,pp. 4,9.

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