Octavian Agratini and Ogün Doǧru 1. INTRODUCTION

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1 TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 4, pp , August 010 This paper is available olie at WEIGHTED APPROXIMATION BY q-szász-king TYPE OPERATORS Octavia Agratii ad Ogü Doǧru Abstract. By usig q-calculus, i the preset paper we costruct Szász type operators i Kig sese, this meaig the operators preserve the first ad the third test fuctio of Bohma-Korovki theorem. Rate of local ad global covergece is obtaied i the frame of weighted spaces. The statistical approximatio property of our operators is also revealed. 1. INTRODUCTION Durig the last decade, q-calculus was itesively used i the costructio for differet geeralizatios of may classical sequeces of liear positive operators. The pioeer work has bee made by A. Lupaş [18] ad G. M. Phillips [1] who proposed geeralizatios of Berstei polyomials based o the q-itegers. Numerous properties of these remarkable polyomials have bee exteded to their q-aalogues. We refer, for istace, to the results of S. Ostrovska [19], [0], M. M. Derrieic [7], V.S. Videskii [3], H. Wag [4]. Other importat classes of discrete operators have bee ivestigated by usig q-calculus. For example: q- Meyer-Köig ad Zeller operators appear i the papers of T. Trif [], O. Doǧru ad O. Duma [8]; q-bleima, Butzer ad Hah operators have bee studied by A. Aral ad O. Doǧru [4]; q-szász Mirakja operators are ivestigated by A. Aral ad V. Gupta [5]. It is also kow that J.P. Kig [15] has preseted a uexpected example of operators of Berstei type which preserve the test fuctios e 0 ad e of Bohma- Korovki theorem. We recall that the three test fuctios of this criterio, usually deoted by e k, are the moomials e k x) =x k, k =0, 1,. A geeral techique to costruct sequeces of operators of discrete type with the same property as i Received May 8, 008, accepted August 18, 008. Commuicated by Se-Ye Shaw. 000 Mathematics Subject Classificatio: 41A36, 41A5. Key words ad phrases: q-itegers, Positive liear operators, Statistical covergece, Weighted modulus of smoothess. 183

2 184 Octavia Agratii ad Ogü Doǧru Kig s example was preseted by the first author []. I the same paper particular classes such as Szász-Mirakja, Baskakov, Berstei-Chlodovsky operators, have bee modified i Kig s sese. The goal of this article is to costruct ad ivestigate a variat of Szász operators based o q-calculus ad Kig s model. The paper is orgaized as follows. Sectio cotais some basic facts regardig both q-calculus ad statistical covergece. The costructio of the aouced class of operators is also preseted. Sectio 3 deals with statistical approximatio property of our operators i a weighted certai space. Sectio 4 ceters aroud the rate of local ad global covergece of our sequece for fuctios with polyomial growth. The mai tool is a weighted modulus of smoothess.. NOTATION AND PRELIMINARIES For the reader s coveiece ad to make the expositio self-cotaied, we collect iformatio regardig two cocepts: q-calculus ad statistical approximatio. Set N 0 = {0} N ad R + =[0, ). Throughout the paper we shall assume that q 0, 1). Followig the defiitios ad otatios of [14, pp. 7-13], for ay real umber a ad x we set [a] q = 1 qa 1 q, 1 x)a q = j=0 1 q j x 1 q j+a x. As special case we cosider [a] 1 = a. For the iteger 1, the q-iteger is 1 [] q =1+q + + q 1. Oe also takes place 1 x) q = 1 q j x). The q-factorial [] q! of the elemet N meas [] q!=[1] q [] q...[] q. [ Set [0] q!=1. ] Also, the q-biomial or the Gaussia coefficiets are deoted by ad are k q defied by [ ] [] q! = k q [k] q![ k] q!, k =0, 1,...,. [ ] ) Clearly, [] 1!=! ad k represets, the ordiary biomial coefficiets. 1 k Two differet q-expasios amed E q ad ε q of the expoetial fuctio x e x are give as follows see [13, p. 9] or [14, pp ]).1) kk 1)/ xk E q x) = q [k] q!, x R,.) ε q x) = k=0 k=0 x k [k] q!, x < 1 1 q. j=0

3 q-szász-kig Type Operators 185 With the help of the otatio 1 a) q = 1 q j a), it is proved that j=0 E q x) =1+1 q)x) q, ε q x) = q)x) q ad, cosequetly, the followig relatio betwee q-expoetial fuctios E q x)ε q x) =1, x < 1 1 q, holds. We metio that these q-aalogues of the classical expoetial fuctios are valid for each q 1, 1). Moreover, lim E qx) = lim ε qx) =e x. q 1 q 1 Further o, let recall the cocept of statistical covergece. The desity of a set K N is defied by 1 δk) = lim χ K k), provided the limit exists, where χ K is the characteristic fuctio of K. Clearly, the sum of the right had side represets the cardiality of the set {k : k K}. Followig [10], a sequece x =x k ) k 1 is statistically coverget to a real umber L if, for every ε>0, k=1 δ{k N : x k L ε}) =0. I this case we write st lim x = L. It is kow that ay coverget sequece is statistically coverget, but ot coversely. Closely related to this otio is A- statistical covergece, where A =a,k ) is a ifiite summability matrix. For a give sequece x =x k ) k 1, the A-trasform of x deoted by Ax), N, is defied by Ax) = a,k x k, N, k=1 provided the series coverges for each. Suppose that A is o-egative regular summability matrix. The x is A-statistically coverget to the real umber L if, for every ε>0, oe has lim a,k =0, k Iε) where Iε) ={k N : x k L ε}. We write st A lim x = L, see e.g. [11], [16]. I approximatio theory by liear positive operators, the statistical covergece has bee examied for the first time by A.D. Gadjiev ad C. Orha [1].

4 186 Octavia Agratii ad Ogü Doǧru I order to itroduce a q-variat for Szász-Mirakja operators, right to the start we preset a costructio due to A. Aral [3] ad studied i deepess by A. Aral ad V. Gupta [5]. Let b ) 1 be a sequece of positive umbers such that lim b =. For each N, q 0, 1) ad f CR + ) the authors defied ).3) Sf)x) q x ) [k]q b []q x) k =E q [] q f b [] q [k] q!b k, k=0 where 0 x< b 1 q. The followig explicit expressios for Sq e k, k =0, 1,, have bee established [3, Eqs. 3.5)-3.7)] S q e 0 = e 0, S q e 1 = e 1, S q e = qe + b [] q e 1. I [] the classical Szász-Mirakja operators have bee modified i Kig s sese. Followig a similar route, we trasform the operators defied at.3) i order to preserve the quadratic fuctio e. Defiig the fuctios.4) v,q x) = 1 ) b + b q[] +4q[] qx, x 0, q we cosider the liear ad positive operators ).5) S,qf)x) v,q x) ) [k]q b []q v,q x)) k =E q [] q f b [] q [k] q!b k, k=0 [ ) b where x J q) := 0, 1 q. Lemma.1. The operators defied at.5) verify for each x J q) the followig idetities.6) S,qe 0 )x) =1, S,qe 1 )x)=v,q x), S,qe )x) =x,.7) S,qψ x)x)=xx v,q x)), where ψ x t) =t x, t 0. Sice the idetities are easily obtaied by direct computatio, we omit the proof. Examiig relatios.4),.6) ad based o Bohma-Korovki theorem, it is clear that S,q) 1 does ot form a approximatio process. The ext step is to

5 q-szász-kig Type Operators 187 trasform it for ejoyig of this property. For each N, the costat q will be replaced by a umber q 0, 1) such that lim q =1. At this stage we also eed a coectio betwee the ivolved sequeces b ) 1, q ) 1. Theorem.. Let q ) 1, 0 <q < 1, be a sequece ad let the operators S,q, N, be defied as i.5). If.8) lim q =1, lim b 1 q = ad lim b [] q =0, the for ay compact K R + ad for each f CR + ) oe has lim S,q f)x) =fx), uiformly i x K. Proof. The secod limit i.8) guaratees that J q )=R +. Cosequetly, the sequece of operators is proper defied, this meaig that it is suitable to approximate fuctios defied o R +. The third limit i.8) implies lim v,q x) =x uiformly i x K. The result follows from Bohma-Korovki criterio via Lemma.1. Remark.3. For removig ay doubt, we idicate pairs of sequeces q ) 1, b ) 1 which verify the plurality of requiremets imposed i Theorem.. 1 q 1 = 1 ad q = 1 ); b = λ for ay fixed λ 0, 1 ). q 1 = 1 ad q =1 1 ); b =[] q ) λ for ay fixed λ 0, 1). =1 3. WEIGHTED STATISTICAL APPROXIMATION PROPERTY A real valued fuctio ρ defied o R is usually called a weight fuctio if it is cotiuous o the domai satisfyig the coditios ρ e 0 ad lim ρx)=. x For example, the mappig x 1+x +λ, λ a o-egative parameter, is ofte used as weight fuctio. Let cosider the spaces B ρ R) ={f : R R a costat M f depedig o f exists such that f M f ρ}, C ρ R) ={f B ρ R) f cotiuous o R},

6 188 Octavia Agratii ad Ogü Doǧru edowed with the usual orm ρ, this meaig fx) f ρ =sup x R ρx). O. Duma ad C. Orha proved [9, Theorem 3] the followig weighted Korovki type theorem via A-statistical covergece. Theorem A. Let A =a,k ) be a o-egative regular summability matrix ad let ρ 1, ρ weight fuctios such that ρ 3.1) lim 1 x) x ρ x) =0. Assume that T ) 1 is a sequece of positive liear operators from C ρ 1 R) ito B ρ R). Oe has st A lim T f f ρ =0, f C ρ1 R), if ad oly if st A lim T F k F k ρ1 =0, k =0, 1,, where F k x) =x k ρ 1 x)/1 + x ). Clearly, all the above otatios ad results are still valid if we replace the domai R by R +. The mai result of this sectio is based o Theorem A. We choose the pair of weight fuctios ρ 0,ρ λ ), where 3.) ρ 0 x)=1+x, ρ λ x) =1+x +λ, x R +, λ>0 beig a fixed parameter. Relatio 3.1) is fulfilled ad B ρ0 R + ) B ρλ R + ). Moreover, takig A the Cesàro matrix of first order, Theorem A implies Corollary 3.1. For ay sequece T ) 1 of liear positive operators actig from C ρ0 R + ) ito C ρλ R + ), λ>0, oe has 3.3) st lim T f f ρλ =0, f C ρ0 R + ), if ad oly if 3.4) st lim T e k e k ρ0 =0, k =0, 1,. Next, we collect some elemetary properties of the fuctios defied by.4).

7 q-szász-kig Type Operators 189 Lemma 3.. Let v,q, N, be defied by.4), where q 0, 1) ad b > 0, N. The followig statemets are true. ) b i) v,q 0) = 0, v,q 1 q = b 1 q ; 3.5) ii) 0 v,q x) x, x [ ] b 0, 1 q ; 3.6) iii) x v,q x) x 0 v,q x 0 ) b q[] q, x 0, where x 0 = b [] q 1 q. Proof. Both i) ad ii) are obtaied by a straightforward calculatio. Because S,q is a positive operators, actually, the iequality v,qx) x sprigs from.7). For provig iii) we ca cosider the fuctio h :[0, ) R, hx) =x v,q x). The uique solutio of the equatio d dx hx) =0beig x 0, the mootoicity of h implies hx) hx 0 )= b 1 1 q) ad 3.6) follows. q[] q The mai result of this sectio will be read as follows. Theorem 3.3. Let the sequece q ) 1, 0 <q < 1, be give such that st lim q =1. Let the operators S,q, N, be defied as i.5). If 3.7) st lim b [] q =0, the, for each fuctio f C ρ0 R + ), oe has where λ>0. st lim S,q f f ρλ =0, Proof. Each fuctio S,q f, f C ρ0 R + ), is defied o J q ) see.5)). We exted it o R + i the classical maer. Let S,q be defied as follows S S,q f)x), x J q ),,q f)x)= fx), x b 1 q.

8 190 Octavia Agratii ad Ogü Doǧru For each N, the orm S,q f f ρλ coicides with the orm of the elemet S,q f f) i the space B ρλ J q )), for ay λ 0. Applyig Corollary 3.1 to the operators T S,q, the proof of Theorem 3.3 will be fiished. I this respect, it is sufficiet to prove that, uder our hypothesis, the operators verify the coditios give at 3.4). By the first ad the third idetity of relatio.6) it is clear that st lim S,q e k e k ρ0 =0 for k =0ad k =. The secod idetity of.6) ad Lemma 3. allow us to write sup x J q ) 1 ρ 0 x) S,q e 1 )x) x = x v,q x) sup x 0 1+x x v,q x) sup x J q ) 1+x b q [] q. Sice st lim q =1, based o 3.7) we get st lim S,q e 1 e 1 ρ0 =0ad the proof is completed. 4. RATE OF WEIGHTED APPROXIMATION The q-stirlig umbers of the secod kid deoted by S q m, k) m, k N 0, m k) are described by the recurrece formula S q m, k)=s q m 1,k 1) + [k] q S q m 1,k), m k 1, with S q 0, 0) = 1 ad S q m, 0) = 0 for m N. k>m. The closed form is the followig We agree S q m, k) =0for 4.1) S q m, k)= 1 [k] q!q kk 1) k 1) j q jj 1) j=0 [ ] k m [k j] m q, 1 k m. j q For q 1, S 1 m, k) represets the umber of ways of partitioig a set of m elemets ito k o-empty subsets [1, p. 84]. Lemma 4.1. Let the sequece q ) 1, 0 <q < 1, be give ad let the operators S,q, N, be defied as i.5). Oe has S,q mm 1) e m )x)=q v m,q x)+ m 1 k=1 b [] q ) m k kk 1) S qm, k)q v,q k x),

9 q-szász-kig Type Operators 191 x J q ), where S q m, k) are q -Stirlig umbers give by 4.1). Here e m stads for the moomial of m degree. Proof. Takig i view both.5) ad.1) ad usig the Cauchy rule or Mertes formula) for multiplicatio of two series, we ca write S,q e m )x) = = = k=0 i=0 k 1) i q k=0 m b k=0 ii 1) kk 1) S q m, k)[k] q!q [] q ) [k i]q b []q v,q x)) k e m [] q [i] q![k i] q!b k b [] q ) m k ) m k kk 1) S qm, k)q v,q k x) = S q m, m)q mm 1) v,q m x) + m 1 k=0 b [] q ) m k kk 1) S qm, k)q v,q k x). 1 [k] q! vk,q x) Kowig that S q m, m)=1ad S q m, 0) = 0 m 1), oe obtais 4.1). We metio that A. Aral proved a similar result [3, Lemma 1] for the operators give at.3). His proof is based o the forward q-differeces up to order m. Set A m ; q,b ):= ) m k m 1 k=1 S b q m, k). [] q b Uder the hypothesis lim q =1ad lim =0we get lims q m, k) = [] q S 1 m, k), 1 k m 1, ad a real costat B m depedig oly o m exists such that 4.) sup A m ; q,b )=B m. N Lemma 4.. Let the sequece q ) 1, 0 <q < 1, be give ad let S,q, N, be operators defied as i.5). If the coditios.8) are fulfilled, the oe has 4.3) i) S,q e m )x) 1 + B m )1 + x m ), x J q ), where B m is give at 4.).

10 19 Octavia Agratii ad Ogü Doǧru ii) For each N, the operator S,q maps the space B ρ λ ito B ρ λ, λ>0. Here λ represets the ceilig of umber λ. Settig µ x t) =1+x + t x ) + λ, t 0, the followig iequalities hold 4.4) iii) S,q µ x c λ 1 + e + λ ), 4.5) iv) S,q µ x c λ 1 + e + λ ), where c λ, c λ are costats idepedet o x ad. Proof. i) Based o Lemma 4.1, relatio 3.5) ad kowig that q 0, 1), for each x J q ) we ca write m 1 ) m k S,q e m )x) x m b + S q m, k) x k [] q k=1 x m + ma m ; q,b )1 + x m ) x m + mb m 1 + x m ), ad 4.3) follows. ii) If f B ρ λ, the f M f 1 + x + λ ). S,q mootoe. Relatio 4.3) implies our statemet. iii) For each t 0 ad x J q ) we get beig liear ad positive is µ x t) 1+x + t) + λ λ x) + λ + t + λ ). By usig 4.3) ad.6) we obtai 4.4). iv) Sice µ xt) 1 + x + t) 4+ λ ), the same relatios 4.3) ad.6) imply 4.5). We proceed with estimated of the errors S,q f f, N, ivolvig ubouded fuctios by usig a weighted modulus of smoothess associated to the space B ρλ. I this respect we use 4.6) Ω ρλ f; δ)= sup x 0 0<h δ fx + h) fx), δ > 0. 1+x + h) +λ

11 q-szász-kig Type Operators 193 This modulus was recetly cosidered by López-Moreo [17]. Clearly, Ω ρλ f; ) f ρλ for each f B ρλ. Amog some basic properties of this modulus, i [17] the followig is metioed: Ω ρλ f; mδ) mω ρλ f; δ) for ay positive iteger m. Sice Ω ρλ f; δ) is mootoically icreasig with respect to δ δ > 0) ad α<[α]+1 α +1 holds [α] gives the iteger part of α), the followig iequality 4.7) Ω ρλ f; αδ) α +1)Ω ρλ f; δ), δ > 0, is valid for ay α>0. Theorem 4.3. Let the sequece q ) 1, 0 <q < 1, be give ad let S,q, N, be operators defied as i.5) such that the coditios.8) are fulfilled. For each f B ρ λ the followig iequality ) S,q f)x) fx) k λ 1 + x 3+ λ b )Ω ρ λ f;, x J q ), [] q holds, where k λ is a costat idepedet of f ad. Proof. Let N ad f B ρ λ be fixed. For each t 0 ad δ>0, based both o defiitio 4.6) ad o property 4.7) with α := t x δ 1,weget ) t x ft) fx) 1 + x + t x ) + λ ) +1 Ω ρ λ f; δ) δ = µ x t)+ 1 ) δ µ xt) t x Ω ρ λ f; δ), where µ x was itroduced at Lemma 4.. Takig ito accout that S,q is liear positive operator preservig the costats, we ca write S,q f)x) fx) = S,q f fx); x) S,q f fx) ; x) S,q µ x + δ 1 µ x ψ x ; x)ω ρ λ f; δ) { = S,q µ x )x)+ 1 } δ S,q µ x ψ x )x) Ω ρ λ f; δ) { S,q µ x )x)+ 1 } S δ,q µ x S )x),q ψx )x) Ω ρ λ f; δ), where ψ x was itroduced at Lemma.1.

12 194 Octavia Agratii ad Ogü Doǧru The last icrease is based o Cauchy-Schwarz iequality frequet used for positive operators of discrete type. It was proved by Yua-Chua Li ad Se-Ye Shaw [6] that this classical iequality has great ad uexpected force. Relatios.7) ad 3.6) allow us to write S,q ψ x)x) b x [] q. Further o, by usig iequalities 4.4), 4.5) we get S,q f)x) fx) { c λ 1 + x + λ )+ c λ δ Choosig δ = c λ + c λ δ b [] q b [] q 1+x + λ ) x ) 1+x 3+ λ ) Ω ρ λ f; δ). } b Ω ρ λ f; δ) [] q ad settig k λ := c λ + c λ ), the coclusio follows. Corollary 4.4. Uder the assumptios of Theorem 4.3 the followig global estimate takes place ) S,q b f f ρ λ+1 k λ Ω ρ λ f;, f B ρ λ. [] q REFERENCES 1. M. Abramowitz ad I. A. Stegu eds.), Hadbook of Mathematical Fuctios with Formulas, Graphs ad Mathematical Tables, Natioal Bureau of Stadards Appplied Mathematics, Series 55, Issued Jue, O. Agratii, Liear operators that preserve some test fuctios, It. J. of Math. ad Math. Sci., Art. ID 94136, 006), A. Aral, A geeralizatio of Szász-Mirakya operators based o q-itegers, Mathematical ad Computer Modellig, ), A. Aral ad O. Doǧru, Bleima, Butzer ad Hah operators based o the q-itegers, J. Ieq. ad Appl., Art. ID 79410, 007), A. Aral ad V. Gupta, The q-derivative ad applicatios to q-szász Mirakya operators, Calcolo, ), Yua-Chua Li ad Se-Ye Shaw, A proof of Hölder s iequality usig the Cauchy- Schwarz iequality, Joural of Iequalities i Pure ad Applied Mathematics, 7 006), Issue, Article 6, 1-3.

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14 196 Octavia Agratii ad Ogü Doǧru Octavia Agratii Faculty of Mathematics ad Computer Sciece, Babeş-Bolyai Uiversity, Cluj-Napoca, Romaia Ogü Doǧru Departmet of Mathematics, Faculty of Scieces ad Arts, Gazi Uiversity, Tekikokullar 06500, Akara, Turkey