Research Article Korovkin-Type Theorems in Weighted L p -Spaces via Summation Process

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1 Hidawi Publishig Corporatio The Scietific World Joural Volume 203, Article ID , 6 pages esearch Article Korovi-Type Theorems i Weighted L p -Spaces via Summatio Process Tucer Acar ad Fadime Diri 2 Kırıale Uiversity, Faculty of Sciece ad Arts, Departmet of Mathematics, Yahşiha, Kırıale, Turey 2 Siop Uiversity, Faculty of Scieces ad Arts, Departmet of Mathematics, Siop, Turey Correspodece should be addressed to Tucer Acar; tuceracar@ymail.com eceived 27 August 203; Accepted 27 September 203 Academic Editors: I. Beg, G. Dai, F. J. Garcia-Pacheco, H. Iidua, ad S. A. Mohiuddie Copyright 203 T. Acar ad F. Diri. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited. Korovi-type theorem which is oe of the fudametal methods i approximatio theory to describe uiform covergece of ay sequece of positive liear operators is discussed o weighted L p spaces, p< for uivariate ad multivariate fuctios, respectively. Furthermore, we obtai these types of approximatio theorems by meas of A-summability which is a stroger covergece method tha ordiary covergece.. Itroductio The fudametal theorem of Korovi []o approximatio of cotiuous fuctios o a compact iterval gives coditios i order to decide whether a sequece of positive liear operators coverges to idetity operator. This theorem has bee exteded i several directios. Oe of the most importat papers o these extesios is [2] that where the author obtaied Korovi-type theorem o ubouded sets for the weighted cotiuous fuctios o semireal axis. Korovi-type theorems were also studied o L p -spaces (see [3, 4]). The extesio of Korovi s theorem from compact itervals to ubouded itervals for fuctios that belog to L p - spaces was obtaied by Gadjiev ad Aral [5]. We recall some otatios preseted i that paper. Let deote the set of real umbers. The fuctio ω iscalledaweightfuctioifitis positive cotiuous fuctio o the whole real axis ad, for a fixed p [, ), satisfyig the coditio Let L p,ω () ( p < ) deote the liear space of measurable, p-absolutely itegrable fuctios o with respecttoweightfuctioω;thatis, L p,ω () ={f: ; f p,ω =( f (t) p /p ω (t) dt) < }. (2) The aalogues of () ad(2) i multidimesioal space are give as follows. Let Ω be a positive cotiuous fuctio i, satisfyig the coditio ad for p< oe has L p,ω ( ) t 2p Ω (t) dt <, (3) t 2p ω (t) dt <. () ={f: ; f p,ω =( f (t) p /p Ω (t) dt) < }. (4)

2 2 The Scietific World Joural The authors obtaied Korovi-type theorems for the fuctios i L p,ω () ad also i L p,ω ( ).Theaforemetioed results are the extesios of Korovi s theorem o ubouded sets ad more geeral fuctios spaces by ordiary covergece. O the other had, most of the classical operators ted to coverge to the value of the fuctio beig approximated. At the poits of discotiuity, they ofte coverge to the average oftheleftadrightitsofthefuctio.however,thereare exceptios which do ot coverge at poits of discotiuity (see [6]). I this case matrix summability methods of Cesáro type are strog eough to correct the lac of covergece [7]. ) be a sequece of ifiite matrices with o-egative real etries. For a sequece x=(x j ),the double sequece Let A := {A }=(a () j defied by Ax :={(Ax) :, N}, (5) a () j= (Ax) := j x j, (6) is called A-trasformof x wheever the series that coverges for all,,adx is said to be A-summable to l if a () j= j x j =l, (7) uiformly i ([8, 9]). If A =Afor some matrices A, the A-summability is the ordiary matrix summability by A,ad if a () j =/,for j +( =, 2, 3,...), ad a () j = 0 otherwise, the A-summability reduces to almost covergece [0]. eplacig the ordiary covergece by A- summability some approximatio results have bee studied i [ 3] adithespecialcases[4, 5]. Also, Korovi-type theorems i weighted space via A-summability have bee studied i [6, 7]. Our purpose i the preset paper is to obtai Korovitype theorems o weighted L p spaces i uivariate ad multivariate case via A-summatio process. More precisely, asequece{l j } of positive liear operators from L p,ω ito L p,ω is called a A-summatio process o L p,ω if (L j (f)) is A-summable to f for every f L p,ω ;thatis, a () j= j L j (f) f p,ω =0, uiformly i, (8) where it is assumed that the series i(8) coverges for each,, adf. Cosiderig this fact we exted (8) tospace of sequeces of liear positive operators to approximate the fuctios that belog to L p,ω spaces via matrix summability method. 2. Mai esult Throughout this sectio we will use the followig otatios: A () (f; x) is the double sequece: A () (f; x) = a () j= j L j (f (t) ;x), =,2,..., (9) ad miimum ad maximum values of the weight fuctio ω o fiite itervals will be deoted by ω mi ad ω max, respectively. Now we preset the followig mai result. Theorem. Let A ={A } be a sequece of ifiite matrices with oegative real etries ad let {L j } be a sequece of positive liear operators from L p,ω ito L p,ω. Assume that If A(), L p,ω L p,ω <. (0) A() (ti ;x) x i =0, p,ω i=0,,2, () the for ay fuctio f L p,ω (),oehas A() f f =0. p,ω (2) Proof. Let χ A (t) be the characteristic fuctio of the iterval [ A, A] ad χ A 2 (t) = χa (t) for ay A 0.Wecachoose asufficietlargea such that for every ε>0 fχa 2 <ε. (3) p,ω Usig the assumptio of the covergece of the series (9) for each,, adf ad the liearity of the operators L j,we get A() f f p,ω = + = A() A() A() I + (χa +χa 2 )f (χa +χa 2 )f p,ω (χa f) χa f p,ω (χa 2 f) χa 2 f p,ω I. (4) By coditio (0), there exists a costat K>0such that A(), Hece, from (3), we compute I A() p,ω K. (5) χa 2 f p,ω + χa 2 f p,ω (K+) χa 2 f p,ω < (K+) ε. (6)

3 The Scietific World Joural 3 For every fuctio f L p,ω () the iequality χa 2 f p ω /p mi f p,ω (7) implies that L p,ω ( A, A) L p ( A, A). Sice the space of cotiuous fuctios is dese i L p ( A, A),givef L p,ω (), for each ε > 0, there exists a cotiuous fuctio φ o [ A, A] satisfyig the coditio φ(x) = 0 for x > A such that (f φ) χa <. p (K+) ωmax /p (8) Usigtheiequalities (5) ad(8), we get I = A() (χa f) χa f p,ω A() (f φ) χa p,ω + A() (φχa ) φχa + p,ω (f φ) χa p,ω A() (φχa ) φχa p,ω +ε. ε (9) O the other had, sice χ A 2 χa φ=0for some A >A, we get the iequality A() (φχa ) φχa p,ω = (χa +χ A 2 )A() (φχa ) (χa +χ A 2 )φχa p,ω [A() (φχa ) φχa ]χa p,ω + χa 2 A() (φχa ). p,ω (20) Now, by posig that M φ = max t φ(t) χ A (t),weget χa 2 A() = ( M φ ( (φχa ) p,ω A() t >A A() t >A +M φ ( χ A 2 /p. ω (t) dt) /p (φχa ;t) p ω (t) dt) /p (; t) p ω (t) dt) (2) Sice ω L (),wecachoosetheumbera such that ( χ A /p 2 ω (t) dt) < ε. (22) M φ Usig this iequality, we have χa 2 A() (φχa ) M p,ω φ A() (; x) p,ω +ε. (23) As a corollary, we get the followig iequality for I : I 2ε +M φ A() (; x) p,ω (24) + [A() (φχa ) φχa ]χa. p,ω Sice φχ A is a cotiuous fuctio o [ A, A], foray give ε >0, there exists a δ>0such that φ (t) χa (t) φ(x) χa (x) <ε +2M φ (t x) 2. (25) Furthermore, this iequality also holds i the case that t [ A, A] ad that x [ A, A) (A, A ] sice φ(x)χ A (x) = 0 ad φ(t)χ A (t) are cotiuous. So, we have [A() (φχa ) φχa ]χa p,ω [A() ( φ (t) χa (t) φ(x) χa (x) ;x)]χa (x) p,ω + φ (x) χa (ε + 2M φ + 4M φ + 2M φ A (x) (A() (; x) ) p,ω A2 +M φ ) A() A() (t; x) x p,ω (; x) p,ω +ε A() (t2 ;x) x 2. p,ω (26) Usig (24)ad(26), we ca write I 3ε +(ε + 2M φ + 4M φ + 2M φ A A() A2 +2M φ ) (t; x) x p,ω A() (; x) p,ω A() (t2 ;x) x 2. p,ω (27)

4 4 The Scietific World Joural Theweobtaithefollowigequalityfor(4): A() f f p,ω 3ε + (K+) ε+c{ A() + (; x) p,ω A() (t; x) x p,ω + A() (t2 ;x) x 2 }, p,ω (28) where C := max{ε +(2M φ / )A 2 +2M φ,(4m φ / )A, (2M φ / )}. By the hypothesis of theorem ad arbitrarity of ε ad ε, A () f f 0as which is desired p,ω result. Nowwegiveaexampleofasequeceofpositiveliear operators which satisfies the coditios of Theorem i weighted space L p,ω (). Example 2. We choose ω(x) = e x. Note that this selectio of ω satisfies coditio (). Also ote that for p< is, L p,ω () ={f: :e x f (x) L p ()}. (29) Also A =Cfor each where C is the Cesáro matrix; that c j = { {, j, { 0, otherwise. (30) The Katorovich variat of the Szász-Miraya operators [8]byreplacigf(sb /) with a itegral mea of f(x) over the iterval [(s + )b /, sb /] is as follows: S (f; x) := (s+)b / P b,s (x) f (t) dt, s=0 sb / N, x [0,b ), (3) where (b ) is a sequece of positive real umbers satisfyig the coditio b =0, b =, P,s (x) := e x/b (x)s s!b s, s=0,,2,... It is ow that S (; x) =, S (t; x) =x+ b 2, S (t 2 ;x)=x 2 + 2b x+ b (32) (33) Furthermore by simple calculatios, we obtai Also, A() (; x) =0, p,ω A() (t; x) x = p,ω 2 p,ω A() 2 x p,ω (t2 ;x) x 2 p,ω j= A() L, p,ω L p,ω =, f p,ω = j + A() j= 3 p,ω j 2. j= b 2 j j, (34) (f; x) p,ω <. (35) Hece, coditios (0), () are provided which meas that for ay fuctio f L p,ω (),wehave A() f f =0. p,ω (36) Also, aalogue of Theorem forthespaceoffuctio of several variables ca be obtaied. Now, we establish this theorem. For the sae of coveiet otatio, we preset our secod results o L p,ω ( m ), m N, isteadofl p,ω ( ) to avoidaycofusioabouttheidicesofa ={A }. Theorem 3. Let A ={A } be a sequece of ifiite matrices with oegative real etries ad let {L j } be a sequece of positive liear operators from L p,ω ( m ) ito L p,ω ( m ). Assume that If A(), L p,ω L p,ω <. (37) A() (; x) =0, p,ω A() (t i;x) x i p,ω =0, i=,2,...,m, A() ( t 2 ;x) x 2 =0, p,ω the for ay fuctio f L p,ω ( m ),oehas (38) A() f f =0. p,ω (39) Proof. Cosiderig the characteristic fuctio χ A of the ball x A ad χ A 2 (t) = χa (t), itispossibletochoosea sufficiet large A such that fχa 2 (t) p,ω <ε. (40)

5 The Scietific World Joural 5 O the other had, by coditio (37) there exists a positive costat K such that A () p,ω K,adso,forgive ε > 0, there exists a cotiuous fuctio θ o x A satisfyigthe coditioθ(x) = 0 for x > A such that (f θ) χa p,ω < ε (K+) (max t A Ω (t)) /p. (4) Keepig i mid the fact that the series (9) isacovergece for each,,adf, ad usig the liearity of the operators L j, which meas the liearity of A (),weget A() f f p,ω Let A >A,sowealsohave A() A() + (K+) ε+ε. (χa θ) χa θ p,ω (χa θ) χa θ p,ω [A() (χa θ) χa θ] χa +M θ A() +M θ χa 2 p,ω, () p,ω p,ω (42) (43) where M θ := max t m θ(t) χ A (t). Furthermore,weca choose A such that χ A 2 p,ω <ε /M θ,adforsufficietly large, weestimate A () () p,ω < ε /M θ.usig these estimatios i (42), we obtai A() f f p,ω Sice [A() (χa θ) χa θ] χa + (K+) ε+3ε. χa (t) θ (t) χa (x) θ (x) <ε +2M θ t x 2 we ca write A() f f p,ω (K+) ε+4ε K Ω /P +2M θ (+A) 2 { + A() m A() l=0 ( t 2 ;x) x 2 p,ω p,ω (44), (45) (t i;x) x i p,ω + A() () }. p,ω (46) Usig the coditios of theorem, we have A() f f p,ω (K+) ε+4ε K Ω /P +2M θ (+A) 2 ε [ 2M θ (+A) 2 +m ε 2 2M θ (+A) 2 + ε 3 2M θ (+A) 2 ] = (K+) ε+4ε K Ω /P +ε +mε 2 +ε 3, (47) which meas that A() f f =0. p,ω (48) Now we give the followig example. Example 4. We choose Ω(x, y) = e x y.notethatthisselectio of Ω satisfies coditio (3). Also ote that, for p<, is; L p,ω ( 2 )={f: 2 :Ω(x,y)f(x) L p ( 2 )}. (49) Also A =Cfor each where C is the Cesáro matrix, that c j = { {, j, { 0 otherwise. (50) The Katorovich variat of the Szász-Miraya operators [8] byreplacigf(tb /, sb /) with a itegral mea of f(x, y) over the iterval [(t + )b /, tb /] [(s + )b /, sb /] is as follows: S (f; x, y) := 2 b 2 P,t,s (x, y) t=0s=0 (t+)b / tb / (s+)b / f (u, V) du dv, sb / N, x,y [0,b ), (5) where (b ) is a sequece of positive real umbers satisfyig the coditio b =0, b =, P,t,s (x, y) := e ((x+y)/b (x) t (y) s ) t!s!b t+s, t, s = 0,, 2,.... (52)

6 6 The Scietific World Joural It is ow that S (; x, y) =, S (u;x,y)=x+ b 2, S (V;x,y)=y+ b 2, S (u 2 + V 2 ;x,y) Furthermore we obtai Also, =x 2 +y 2 + 2b (x+y)+ 2b A() (; x, y) =0, p,ω A() (u;x,y) x = p,ω 2 p,ω A() (V;x,y) y = p,ω 2 p,ω, A() j= j= (u2 + V 2 ;x,y) (x 2 +y 2 ) p,ω 2 x+y p,ω A() L p,ω L p,ω =, f p,ω = j= A() j + 2 j, j, 3 p,ω j 2. j= b 2 j (53) (54) (f; x, y) p,ω <. (55) Hece, coditios (0)ad()are provided which meas that for ay fuctio f L p,ω ( 2 ),wehave Coflict of Iterests A() f f =0. p,ω (56) The authors declare that there is o coflict of iterests regardig the publicatio of this paper. [3] P. C. Curtis Jr., The degree of approximatio by positive covolutio operators, Michiga Mathematical Joural,vol.2,o. 2, pp , 965. [4] V. K. Dzjady, Approximatio of fuctios by positive liear operators ad sigular itegrals, Matematichesii Sbori,vol. 2,o.70,pp ,966(ussia). [5] A. D. Gadjiev ad A. Aral, Weighted L p -approximatio with positive liear operators o ubouded sets, Applied Mathematics Letters, vol. 20, o. 0, pp , [6].BojaicadF.Cheg, Estimatesfortherateofapproximatio of fuctios of bouded variatio by Hermite-Fejer polyomials, i Proceedigs of the Caadia Mathematical Society Coferece,vol.3,pp.5 7,983. [7]. Bojaic ad M. K. Kha, Summability of Hermite-Fejer iterpolatio for fuctios of bouded variatio, Joural of Natural Scieces ad Mathematics,vol.32,pp.5 0,992. [8] H. T. Bell, Order summability ad almost covergece, Proceedigs of the America Mathematical Society,vol.38,pp , 973. [9] M. Stieglitz, Eie verallgemeierug des begriffs festovergez, Mathematica Japoica,vol.8,pp.53 70,973. [0] G. G. Loretz, A cotributio to the theory of diverget sequeces, Acta Mathematica,vol.80,o.,pp.67 90,960. [] T. Nishishiraho, Quatitative theorems o liear approximatio processes of covolutio operators i Baach spaces, Tôhou Mathematical Joural,vol.33,pp.09 26,98. [2] T. Nishishiraho, Covergece of positive liear approximatio processes, Tôhou Mathematical Joural, vol. 35, pp , 983. [3]J.J.Swetits, Osummabilityadpositiveliearoperators, Joural of Approximatio Theory, vol. 25, o. 2, pp , 979. [4] S. A. Mohiuddie, A applicatio of almost covergece i approximatio theorems, Applied Mathematics Letters, vol. 24, o., pp , 20. [5] S. A. Mohiuddie ad A. Alotaibi, Korovi secod theorem via statistical summability (C, ), Joural of Iequalities ad Applicatios,vol.203,p.49,203. [6] Ö. G. Atliha ad C. Orha, Matrix summability ad positive liear operators, Positivity,vol.,o.3,pp , [7] Ö. G. Atliha ad C. Orha, Summatio process of positive liear operators, Computers ad Mathematics with Applicatios, vol. 56, o. 5, pp , [8] O. Szász, Geeralizatio of S. Berstei s polyomials to ifiite iterval, Joural of esearch of the Natioal Bureau of Stadards,vol.45,pp ,959. Acowledgmet The authors are thaful to referee(s) for maig valuable suggestios leadig to the better presetatio of the paper. efereces [] P. P. Korovi, Liear Operators ad Theory of Approximatio, Hidusta, Delhi, Idia, 960. [2] A.D.Gadjiev, OP.P.Korovitypetheorems, Matematichesie Zameti,vol.20,pp ,976(ussia).

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