ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF FOURIER SERIES

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1 M aheaical I equaliies & A pplicaios Volue 19, Nube 1 (216), doi:1.7153/ia ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF FOURIER SERIES W. ŁENSKI AND B. SZAL (Couicaed by I. Peić) Absac. The esuls coespodig o soe heoes of S. Lal Appl. Mah. ad Copu. 29 (29), ] ad he esuls of he auhos Baach Cee Publ., 95, (211), ] ae show. The sae ad soeies bee degees of poiwise appoxiaio as i eioed papes by weake assupios o cosideed fucios ad exaied suabiliy ehods ae obaied. Fo peseed poiwise esuls he esiaio o o appoxiaio ae deived. Soe special cases as coollaies fo ieaio of he Nölud o he Riesz ehod wih he Eule oe ae also foulaed. 1. Ioducio Le L p (1 p < ) especively L ] be he class of all 2 peiodic eal valued fucios iegable i he Lebesgue sese wih p h powe esseially bouded] ove Q =, ] wih he o ( 1 ) 1/p 2 f := f ( ) L p = f () p d whe 1 p <, Q esssup f () whe p = Q ad coside he igooeic Fouie seies Sf(x) := a ( f ) 2 + ν=1 (a ν ( f )cosνx + b ν ( f )si νx) wih he paial sus S k f 8, Th.(3.1)IV]. Le A := ( a,k ) ad B := ( b,k ) be ifiie lowe iagula aices of eal ubes such ha a,k adb,k whek =,1,2,...,, a,k = adb,k = whek >, a,k = 1ad b,k = 1, whee =,1,2,..., Maheaics subjec classificaio (21): 42A24. Keywods ad phases: Degee of appoxiaio, Fouie seies. c D l,zageb Pape MIA

2 288 W. ŁENSKI AND B. SZAL ad le, fo =,1,2,...,, A, = a,k ad A, = Le he AB asfoaio of (S k f ) be give by T,A,B f (x) := = k= a,k. a, b,k S k f (x) ( =,1,2,...). We defie wo classes of sequeces. Followig by L. Leidle 2], a 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 = 1 c c +1 K (c) + 1 /2 fo all posiive iege. Aalogously as i 4], a sequece c :=(c ) of oegaive ubes will be called he Mea Head Bouded Vaiaio Sequece, o biefly c MHBVS, if i has he popey 1 = 1 c c +1 K (c) + 1 c, = fo all posiive ieges <, whee he sequece c has oly fiie ozeo es ad he las ozeo e is c. Cosequely, we assue ha he sequece (K (α )) = is bouded, ha is, ha hee exiss a cosa K such ha K (α ) K holds fo all, wheek (α ) deoe he cosas appeaig i he befoe iequaliies fo he sequeces α =(a, ) =, =,1,2,... Now we ca give he codiios o be used lae o. We assue ha fo all ad < ad 1 a, a,+1 K 1 k= a, a,+1 K 1 = + 1 /2 a, = hold if (a, ) = belogs o MRBVS ad MHBVS,fo = 1,2,..., especively. As a easue of appoxiaio of f by T,A,B f we use he poiwise oduli of coiuiy of f i he space L p defied by he foulas { 1 δ wx p f (δ) = δ ϕ x (u)si u 2 p 1 p du} whe 1 p <, ess sup ϕ x (u)si u 2 whe p =, <u δ c, a,

3 ON POINTWISE APPROXIMATION OF FUNCTIONS BY MATRIX MEANS OF FOURIER SERIES 289 wx p f (δ) = sup < δ { 1 ϕ x (u)si 2 u p 1 p du} whe 1 p <, ess sup ϕ x (u)si 2 u whe p =, <u δ ad he classical oes ω f (δ) L p = sup si < δ 2 ϕ () L p, whee ϕ x () := f (x + )+ f (x ) 2 f (x). The deviaio T,A,B f f wih he lowe iagula ifiie aix A, defied by a, = 1 +1 whe =,1,2,..., ad a, = whe >, ad wih he lowe iagula ifiie aix B, defied by b,k = p k / p ν whe k =,1,2,..., ad b,k = ν= whe k >, was esiaed by S. Lal 1, Theoe 2]. The deviaio T,A,B f f i geeal fo was esiaed a he poi as well as i he o of L p i 3]. The poiwise esiaes fo his pape ae followig: THEOREM. Le f L p (1 < p ), ad le a odulus ype fucio ω saisfy { } 1 ϕx () p si p p ω () 2 d = O x ( + 1) 2 p,whe 1 < p <, +1 ess sup ϕ x () ω () si 2 = O x (1), whe p = ad +1, ] { +1 ϕx () p si p } 1 p ω () 2 d = Ox ( + 1) 1 p, whe 1 < p <, ess sup ϕ x () ω () si 2 = O x (1), whe p =,, +1 ] wih < 1 1 p. If he eies of ou aices saisfy codiios a, ad a, b, l a,+1 b +1,+1 l a, ( + 1) 2 fo l 1, he T,A,B f (x) f (x) ( 1 = Ox a, = () ω ) () ω

4 29 W. ŁENSKI AND B. SZAL ad, i he case < < 1 1 p, T,A,B f (x) f (x) = O x (( + 1) ω ( + 1) a,k () ]), 1 fo cosideed x. I his pape we shall coside he deviaio T,A,B f f wihou ay special assupios o odulus of coiuiy ad aohe assupios o suabiliy ehod obaiig he sae degees of appoxiaio as above. Fially, we also give soe esuls o o appoxiaio ad coollay geealizig he esuls of H. K. Niga, A. Shaa 6] ad H. K. Niga, K. Shaa 7]. We shall wie I 1 I 2 if hee exiss a posiive cosa K, soeies depedig o soe paaees, such ha I 1 KI 2. Le foulae ou ai esuls. 2. Saee of he esuls THEOREM 1. Le f L p wih 1 < p ad le < < 1 1 p.if a,k MRBVS ad ν (2) b,k si 2 τ (1) =μ fo μ ν wih τ = ],whee <,he T,A,B f (x) f (x) ( = Ox ( + 1) ]) a,k wx p f + w 1 x + 1 f, fo alos all cosideed x. THEOREM 2. Le f L p wih 1 < p ad le < < 1 1 p.if( a,k ) MHBVS ad he eies of aix B saisfy he codiio (1) fo μ ν, he T,A,B f (x) f (x) ( = Ox fo alos all cosideed x. ( + 1) a, k w x p f ]) + w 1 x f, + 1 Nex, we foulae he esuls o esiaes of L p o of he cosideed deviaio.

5 ON POINTWISE APPROXIMATION OF FUNCTIONS BY MATRIX MEANS OF FOURIER SERIES 291 THEOREM 3. Le f L p wih 1 < p. If a,k MRBVS ad he eies of aix B saisfy he codiio (1) fo μ ν, he whee < < 1 1 p. T,A,B f ( ) f ( ) L p = O ( ( + 1) a,k ω f ) L p THEOREM 4. Le f L p wih 1 < p. If a,k MHBVS ad he eies of aix B saisfy he codiio (1) fo μ ν, he T,A,B f ( ) f ( ) L p = O whee < < 1 1 p. ( ( + 1) a, k ω f ) L p Fially, we give a applicaio of ou esuls as a coollay ad eaks. Takig a, = p / ν= p ν (o a, = p / ν= p ν ), whe =,1,2,..., ad a, =, whe > wih p ν >, p ν p ν+1,adb,k = ( k)γ k, whe k =,1,2,..., ad (1+γ) b, = whek > wih γ >, Theoe 1 ad Theoe 2 iply: ad COROLLARY 1. If f L p wih 1 < p ad < < 1 1 p, he 1 p p ν = (1 + γ) γ k S k f (x) f (x) k ν= = O x ( + 1) ] p k wx p f + w 1 x p ν + 1 f, ν= 1 p p ν = (1 + γ) γ k S k f (x) f (x) k ν= = O x ( + 1) ] p k wx p f + w 1 x p ν + 1 f, ν= wih p ν >, p ν p ν+1 ad γ >, fo alos all cosideed x.

6 292 W. ŁENSKI AND B. SZAL REMARK 1. I special case p = p = 1wehaveesiaes = (1 + γ) γ k S k f (x) f (x) k = O x (( + 1) 1 wx p f + w 1 x f + 1 ]). REMARK 2. If we ake = i he above cosideaios he we have o esiae he quaiies I 2 ad I 2 L p usig he Hölde iequaliy aalogously as i esiae of I 1. Thus we obai i he all above esiaes ( + 1) 1/p isead of ( + 1). 3. Auxiliay esuls We begi his secio by soe oaios followig A. Zygud 8, Secio 5 of Chape II]. I is clea ha S k f (x)= 1 f (x + )D k ()d ad whee T,A,B f (x)= 1 f (x + ) = D k ()= 1 k 2 + cosν = ν=1 a, b,k D k ()d, si (2k+1) 2 2si 2 Hece T,A,B f (x) f (x)= 1 ϕ x () a, b,k D k ()d. = Now, we foulae soe esiaes fo he Diichle keel. LEMMA 1. (see 8]) If < /2, he D k () 2. ad fo ay eal we have D k (). LEMMA 2. Le b,k be such ha he codiio (1) holds fo μ ν. If a,k MRBVS, he a, b,k D k () τa,τ =

7 ON POINTWISE APPROXIMATION OF FUNCTIONS BY MATRIX MEANS OF FOURIER SERIES 293 ad if a,k MHBVS, he a, b,k D k () τa, τ = wih τ =/] ad +1,],fo=,1,2,... Poof. The poof is aalogous o he poof of 5, Lea 11]. 4. Poofs of he esuls 4.1. Poof of Theoe 1 We sa wih he obvious elaios T,A,B f (x) f (x) = = I 1 + I 2 +1 ϕ x () ϕ x () = = a, b,k D k ()d a, b,k D k () ad T,A f (x) f (x) I 1 + I 2. By he Hölde iequaliy 1p + 1 q = 1 ad Lea 1, fo < 1 1 p,wehave I 1 ( + 1) { ( + 1) wx p f + 1 Usig Lea 2 we obai I 2 = +1 ϕ x () d ( + 1) ϕ x () k=1 ϕ x () si 2 { ( + 1) τ a,k a,k a,k d = ϕ x () si 2 si 2 d ] p d } 1/p { ( + 1) +1 } 1/q q d ( + 1) w p ϕ x () d ϕ x () d + ϕ x () τ a,k d (a, + a, ) si q 2 d } 1/q x f + 1 ϕ x () d.

8 294 W. ŁENSKI AND B. SZAL = ( a,k k=1 ( a,k k=1 + ( a,k k=1 =k ) ϕ x () +a, a, d + ){ ] } ϕx () si 2 d +a, =k a, { +a, =k ϕ x () si 2 1+ } ] ] 1 d /2 si ){ ] } ϕx () si 2 d 1+ { } + a, ( + 1) + 1 ϕ x () si ] d 2 { k+1 a,k 1 d } ϕ x (u) si u ]d d +1 2 du +( + 1) a, w 1 x f + 1 a,k { (1 + ) +( + 1) k+1 +1 ϕ x (u) si u 2 du ] k a, w 1 x f a,k { () 1+ k+1 +1 } ϕ x (u) si u ]d 2 du + 1 ϕ x (u) si u ] 2 du ] } k+1 1 +(1 + ) 1+ w1 x f () d +1 +( + 1) a, w 1 x f + 1 { ] a,k () w 1 x f ] } k+1 1 +(1 + ) 1+ w1 x f () d +1 +( + 1) a, w 1 x f + 1

9 ON POINTWISE APPROXIMATION OF FUNCTIONS BY MATRIX MEANS OF FOURIER SERIES 295 a,k { +1 +( + 1) k+1 1 ( ) ] } 1 w1 x f d a, w 1 x f + 1 Sice wx p f (δ) is odeceasig ajoa of wx p f (δ) we have wih > +1 a,k w 1 1 x f d k+1 1 +( + 1) a, w 1 x f + 1 ( + 1) a,k w 1 x f. Collecig hese esiaes we obai he desied esul Poof of Theoe 2 Le as usual T,A,B f (x) f (x)=i 1 + I 2 ad T,A,B f (x) f (x) I 1 + I 2. The e I 1 we ca esiae by he sae way as i he poof of Theoe 1. Theefoe I 1 ( + 1) wx p f. + 1 Aalogouslyo he above, by he Hölde iequaliy 1p + 1 q = 1 ad Lea 2 I 2 +1 ϕ x () k= τ a,k d = ϕ x () d ϕ x () a, k ( + 1) a, k w 1 x f. Collecig hese esiaes we obai he desied esul Poofs of Theoes 3-4 τ a, k d The poofs ae siila o hese above ad follows fo he evide iequaliy w p L f (δ) ω p f (δ) L p, because odulus of coiuiy ω f (δ) L p is odeceasig fucio of δ.

10 296 W. ŁENSKI AND B. SZAL 4.4. Poof of coollay 1 The poof is aalogous o he poof of 5, Coollay 6]. REFERENCES 1] S. LAL, Appoxiaio of fucios belogig o he geealized Lipschiz Class by C 1 N p suabiliy ehod of Fouie seies, Appl. Mah. ad Copu., 29 (29) ] L. LEINDLER, Iegabiliy codiios peaiig o Olicz space, J. Iequal. Pue ad Appl. Mah., 8 (2) (27), A. 38, 6 pp. 3] W. ŁENSKI AND B. SZAL, Appoxiaio of fucios fo L p (ω) by geeal liea opeaos of hei Fouie seies, Baach Cee Publ. 95, (211), ] W. ŁENSKI AND B. SZAL, Appoxiaio of fucios fo L p ( w) by liea opeaos of cojugae Fouie seies, Baach Cee Publ. 92 (211), ] W. ŁENSKI AND B. SZAL, O poiwise appoxiaio of cojugae fucios by soe Hup aix eas of cojugae fouie seies, Joual of Fucio Spaces, Volue 215 (215), Aicle ID ] H. K. NIGAM AND A. SHARMA, O (N, p,q)(e,1) Suabiliy of Fouie Seies, Iea. J. Mah. Mah. Sci., Volue 29, Aicle ID , 8 pages. 7] H. K. NIGAM AND K. SHARMA, Degee of appoxiaio of a class of fucios by (C,1)(E,q) eas of Fouie seies, I. J. Appl. Mah. 41:2, IJAM (211). 8] A. ZYGMUND, Tigooeic seies, Cabidge, 22. (Received Apil 24, 215) W. Łeski Uivesiy of Zieloa Góa Faculy of Maheaics, Copue Sciece ad Ecooeics Zieloa Góa, ul. Szafaa 4a, Polad e-ail: W.Leski@wie.uz.zgoa.pl B. Szal Faculy of Maheaics, Copue Sciece ad Ecooeics Zieloa Góa, ul. Szafaa 4a, Polad e-ail: B.Szal@wie.uz.zgoa.pl Maheaical Iequaliies & Applicaios ia@ele-ah.co