APPROXIMATION PROPERTIES OF STANCU TYPE MEYER- KÖNIG AND ZELLER OPERATORS

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1 Hacettepe Joural of Mathematics ad Statistics Volume 42 (2 (2013, APPROXIMATION PROPERTIES OF STANCU TYPE MEYER- KÖNIG AND ZELLER OPERATORS Mediha Örkcü Received 02 : 03 : 2011 : Accepted 26 : 04 : 2012 Abstract I this paper, we itroduce a Stacu type modificatio of the - Meyer- Köig ad Zeller operators ad ivestigate the Korovki type statistical approximatio properties of this modificatio via A statistical covergece. We also compute rate of covergece of the defied operators by meas of modulus of cotiuity. Furthermore, we give a rth order geeralizatio of our operators ad obtai approximatio results of them. Keywords: Meyer-Köig ad Zeller operators, Stacu type operators, rth order geeralizatio, statistical covergece, modulus of cotiuity 2000 AMS Classificatio: 41A25, 41A36 1. Itroductio The classical Meyer-Köig ad Zeller (MKZ operators defied o C [0, 1] were itroduced i 1960 (see [17]. I order to give the mootoicity properties, Cheey ad Sharma [5] modified these operators as follows: M (f; x = (1 x +1 ( f k +k (+k k x k, x [0, 1 f (1, x = 1. Durig the last decade, Calculus was itesively used for the costructio of geeralizatios of may classical approximatio processes of positive type. The first researches have bee achieved by Lupaş [15] ad Phillips [20]. Phillips itroduced the type geeralizatio of the classical Berstei operator ad obtaied the rate of covergece ad the Vorooskaja type asymptotic formula for these operators. Later, Trif [22] defied the Departmet of Mathematics, Gazi Uiversity, Faculty of Scieces, Tekik-okullar, Akara, Turkey (M. Örkcü medihaakcay@gazi.edu.tr

2 140 M. Örkcü MKZ operators based o the itegers. Also, i order to give some explicit formulae for secod momet of the MKZ operators based o the itegers, Doğru ad Duma [6] have preseted the followig MKZ operators for (0, 1]: M (f; x = (1 s x f ( [k] [+k ] [+k] k xk, x [0, 1 f (1, x = 1. Now we recall some defiitios about Calculus. Let > 0. For ay N 0 = N {0}, the iteger of the umber ad the factorial are respectively defied by [] = { 1 { if 1 1 [1][2]...[] if = 1, [] = if = 1, 2,... 1 if = 0. The biomial coefficiets are defied as [ k ] = [] [ k], k = 0, 1,...,. It is obvious that for = 1 oe has [] 1 =, [] 1 = ad [ k ]1 = ( k, the ordiary biomial coefficiets. Details of Calculus ca be foud i [4]. Recetly, Nowak [18] itroduced a type geeralizatio of Stacu s operators [21]. Beside, a other two aalogues of Stacu operators earlier itroduced by Lupaş [16]. Agratii [1] preseted approximatio properties of the metioed class of operators. I [23], a Stacu type geeralizatio of Baskakov-Durrmeyer operators was costructed ad some approximatio properties were obtaied by Verma, Gupta ad Agrawal. I this paper, we itroduce a Stacu type modificatio of the MKZ operators ad ivestigate the Korovki type statistical approximatio properties of this modificatio via A statistical covergece. Now we recall the cocepts of regularity of a summability matrix ad A statistical covergece. Let A := (a k be a ifiite summability matrix. For a give seuece x := (x k, the A trasform of x, deoted by Ax := ((Ax, is defied as (Ax := a k x k provided the series coverges for each. A is said to be regular if lim (Ax = L wheever lim x = L [11]. Suppose that A is o-egative regular summability matrix. The x is A statistically coverget to L if for every ε > 0, lim a k = 0 ad we write k: x k L ε st A lim x = L [9, 12]. Actually, x is A statistically coverget to L if ad oly if, for every ε > 0, δ A (k N : x k L ε = 0, where δ A (K deotes the A desity of subset K of the atural umbers ad is give by δ A (K := lim k=1 a kχ K(k provided the limit exists, where χ K is the charateristic fuctio of K. If A = C 1, the Cesáro matrix of order oe, the A statistically covergece reduces to the statistical covergece [8]. Also, takig A = I, the idetity matrix, A statistical covergece coicides with the ordiary covergece. The rest of this paper is orgaized as follows. I Sectio 2, we preset a Stacu type geeralizatio of MKZ operators. Furthermore, we obtai statistical Korovki-type approximatio result of the defied operators ad compute their rate of covergece by meas of modulus of cotiuity. I Sectio 3, we give a rth order geeralizatio of our operators ad get approximatio results of them. k=1 2. Costructio of the Operators I this sectio, we preset a Stacu type geeralizatio of MKZ operators ad obtai statistical Korovki-type approximatio result.

3 Approximatio properties of Stacu type Meyer-Köig ad Zeller operators 141 where For f C [0, 1], N ad (0, 1] M,α (f; x = [ ] (2.1 m,α,k (x = + k k ( m,α,k (x f [k], x [0, 1 [+k] f(1, x = 1, k 1 +k. As usual, we use the test fuctios e i (x = x i for i = 0, 1, Lemma. For all x [0, 1 ad N, we have (2.2 (2.3 ( α M,α (e 0; x = 1, M,α (e 1; x = x, x (x + α M,α (e 2; x x (x + α + x. 1 + α [] Proof. Item (2.2 easily follows by m,α,k (x = 1. A direct computatio yields M,α (e 1; x = [ + k 1] k=1 [k 1] [] k 1 +k x k (x + α [i] [ + k] = [] +k+1 [ + k] = [] = x, x k 1 (x + α [i + 1] +k (1 + α [i + 1] which guaratees (2.3. Now we will prove (2.4. We immediately see M,α (e 2; x = 2 [ + k 1] k=1 [k 1] [] k 1 +k [ + k].

4 142 M. Örkcü Sice = [k 1] + 1 for k 1, we get ad hece M,α (e 2; x = 2+1 [ + k 1] [ + k] + 2 [ + k 2] k=2 [k 2] [] [ + k 1] k=1 [k 1] [] ( k 1 ( k 1 x + α [i] +k x + α [i] +k [ + k] (2.5 M,α (e 2; x = ( k+1 [ + k] 2+1 x + α [i] [ + k + 1] [k] [] +k+2 (1 + α [i] [ + k + 2] ( k [ + k] + 2 x + α [i] [k] [] +k+1 [ + k + 1] Sice (0, 1, x [0, 1 ad by usig [ + k + 1] = [+k+2] 1 ad [ + k + 1] < [ + k + 2], we obtai (2.6 M,α (e 2; x = 2 2 [ + k] [] [ + k] [] k+1 k+1 +k+2 +k+2 [ + k + 2] k (x + α [i] [ + k] + 2 [k] [] +k+1 [ + k + 1] 2 x (x + α 1 + α α x (x + α α x (x + α = 2 x (x + α. 1 + α [ + k] [] [ + k] [] k+1 i=2 +k+2 i=2 [ + k + 2] k +k+1 [ + k + 2]

5 Approximatio properties of Stacu type Meyer-Köig ad Zeller operators 143 O the other had, usig the ieualities [ + k + 1] < [ + k + 2], [ + k + 1] > [] for all k = 0, 1, 2,..., N ad from (2.5 it follows M,α (e 2; x α x (x + α [ + k] [k] [] + 2 [] x which guaratees [ + k] [] k+1 i=2 +k+2 i=2 k +k+1, + (2.7 M,α (e 2; x x (x + α + x. 1 + α [] The, by combiig (2.6 ad (2.7, the proof is completed. The well-kow Korovki theorem (see [3, 13] was improved via the cocept of statistical covergece by Gadjiev ad Orha i [10]. This theorem ca be stated as the followig: 2.2. Theorem. If the seuece of positive liear operators L : C [a, b] B [a, b] satisfies the coditios st lim L e j e j = 0, where j {0, 1, 2}, the, for ay fuctio f C [a, b], we have st lim L f f = 0. I other words, the seuece of fuctios (L f 1 is statistically uiform coverget to f o C [a, b]. Here B [a, b] stads for the space of all real valued bouded fuctios defied o [a, b], edowed with the sup-orm. Theorem 2.2 is true for A statistical covergece, where A is o-egative regular summability matrix [7]. Now, we replace ad α i the defiitio of M,α, by seueces (, 0 < 1, ad (α, α 0, respectively, so that (2.8 st A lim = 1, st A lim 1 [] = 0 ad st A lim α = 0. For example, take A = C 1, the Cesáro matrix of order oe, ad defie ( ad (α seueces by { 1 2 =, if = m2 (m = 1, 2, e, if m2. { e, if = m 2 (m = 1, 2, 3... α = 0, if m 2.. Sice the C 1 desity (or atural desity of the set of all suares is zero, st A lim = 1, st A lim 1 [] = 0 ad st A lim α = 0. It is observed that ( ad (α satisfy the eualities i (2.8 but they do ot coverge i the ordiary case.

6 144 M. Örkcü 2.3. Theorem. Let A = (a k be a o-egative regular summability matrix ad let ( ad (α be two seueces satisfyig (2.8. The, for f C [0, 1], the seuece {M,α (f;.} is A statistically uiform coverget to f o the iterval [0, 1]. Proof. By (2.2 ad the defiitio of the operators M,α clear that st A lim M,α e 0 e 0 = 0. Takig ito accout the case x = 1 ad from (2.3, (2.9 M,α e 1 e 1 1. For a give ε > 0, we defie the followig sets; U := { : M,α e 1 e 1 ε}, U := { : 1 ε}. The by (2.9, we ca see U U. So, for all N, 0 a k a k. k U k U Lettig ad usig (2.8, we coclude that lim st A lim M,α e 1 e 1 = 0. i the case of x = 1, it is a k = 0, which gives Fially, by (2.4 ad the defiitio of the operators M,α i the case of x = 1, we get M,α (e 2; x e 2 x (x + α + x x α [] ( (1 2+1 x 2 α α 1 + α + 2 x [] So, we ca write [] 1 α 1 + α 2+1 (2.10 M,α e 2 e α α + 2. [] k U Sice, for all N, 0 < 1, oe ca get st A lim that ( st A lim 1 1 α = st A lim = α [] We defie the followig sets; D := { : M,α e 2 e 2 ε}, { D 1 := : 1 1 } { α ε ad D 2 := : 1 + α [] The, by (2.10, we get D D 1 D 2. Hece, for all N a k a k + a k. k D k D 1 k D 2 Takig limit as j, we get st A lim M,α e 2 e 2 = 0. = 1. So, by (2.8 observe ε }.

7 Approximatio properties of Stacu type Meyer-Köig ad Zeller operators 145 So, from Theorem 2.2, the proof is completed. Now we will give a theorem o a degree of approximatio of cotiuous fuctio f by the seuece of M,α (f; x. For this aim, we will use the modulus of cotiuity of fuctio f C [0, 1] defied by ω (f; δ = sup { f (t f (x : t, x [0, 1], t x δ} for ay positive umber δ Theorem. Let x [0, 1] ad f be a cotiuous fuctio defied o [0, 1]. The, for N, the followig ieualities holds: where M,α (2.11 δ,α (x = (f; x f (x 2ω (f, δ,α (x { ( ( x 2 α x} 1 + α 1 + α + 2. [] Proof. Sice the case of x = 1 is obvious, assume that x [0, 1 ad f C [0, 1]. By the kow properties of modulus of cotiuity, we ca write for ay δ > 0, that ( t x f (t f (x ω (f, δ 1 +. δ Therefore, by the liearity ad mootoicity of the operators M,α, we obtai for ay δ > 0, M,α (f; x f (x M,α ( f (t f (x ; x By the Cauchy-Schwarz ieuality we have { (2.12 M,α (f; x f (x ω (f, δ δ The, we ca write from Lemma 2.1 ω (f, δ M,α { } δ M,α ( t x ; x. ( (t x 2 ; x }. M,α ( (t x 2 ; x α x (x + α + (1 2 x x [] I the ieuality (2.12, takig δ = δ,α (x = = ( x α ( α α + 2 [] { ( ( x 2 α x} 1 + α 1 + α + 2, [] x. the proof is completed Remark. Let ( ad (α be two seueces satisfyig (2.8. If we take = ad α = α i Theorem 2.4, the, we obtai the rate of A statistical covergece of our operators to the fuctio f beig approximated.

8 146 M. Örkcü 3. A rth order geeralizatio of M,α Kirov ad Popova [14] proposed a geeralizatio of the rth order of positive liear operators such that it keeps the liearity property but loose the positivity. Also, this geeralizatio is sesitive to the degree of smoothess of the fuctio f as approximatios to f. I [19], usig the similar method itroduced by Kirov ad Popova, Özarsla ad Duma cosider a geeralizatio of the MKZ-type operators o the itegers. Recetly, Agratii [2] itroduced a geeralizatio of the rth order of Stacu type operators. For every iteger 1, L : C [a, b] C [a, b] be the operators defied by (L f(x = p,k (x f (x,k, x [a, b] k J where J N 0 := {0} N ad a et o the compact [a, b] amely (x,k k J ad p,k (x = 1. For r N 0 by f C r [a, b] we mea the space of all fuctios f k J for which their rth derivative are cotiuous o the iterval [a, b] ad (T rf (x,k ;. be Taylor s polyomial of r degree associated to the fuctio f o the poit x,k, k J. Agratii [2] cosidered the liear operators L,r : C r [a, b] C [a, b], (L,rf (x = (T rf (x,k ;. p,k (x k J = r k J f (i (x,k i (x x,k i p,k (x, x [a, b]. Now, we itroduce a geeralizatio of the positive liear operators M,α, by usig the Agratii s method. (3.1 M,α,r (f; x = (T rf (x,k ;. m,α,k (x = r f (i (x,k i (x x,k i m,α,k (x, where m,α,k (x is give by (2.1, x,k =, f C r [0, 1] ad x [0, 1. If x = 1 we [+k] defie that M,r,α (f; 1 = f(1 as stated before. Clearly, M,α,0 = M,α, N for every f C [0, 1], x [0, 1], (0, 1] ad N. We have the followig approximatio theorem for the operators M,r,α Theorem. Let A = (a k be a o-egative regular summability matrix ad let ( ad (α be two seueces satisfyig (2.8. Let r N ad x [0, 1]. The, for all f C r [0, 1] such that f (r Lip M (α ad for ay N, we have st A M,α,r f f = 0. Proof. For x [0, 1, we obtai from (3.1 that (3.2 f (x M,r,α (f; x = m,α,k (x {f (x (Trf (x,k; x}. Usig the Taylor s formula, we ca write 1 f (x (T rf (x,k ; x = (x x,k r (1 t r 1 (r 1 0 [ ] f (r (x,k + t (x x,k f (r (x,k dt.

9 Approximatio properties of Stacu type Meyer-Köig ad Zeller operators 147 Sice f (r Lip M (α, we obtai (3.3 f (x (T rf (x,k ; x M x x,k r+α B (r, α + 1 (r 1 B (r, α + 1 = M ϕ r+α x (x,k, (r 1 where ϕ x (t = x t ad B (r, α + 1 = r j=1 I view of relatios (3.2 ad (3.3, we get f (x M,α,r (f; x M B (r, α + 1 (r 1 Sice the case of x = 1 is clear, we deduce (3.4 M,α,r f f B (r, α + 1 M M,α (r 1 (α + j 1, B beig Beta fuctio. ϕ r+α x m,α,k (x ϕr+α x (x,k. Firstly, we replace ad α by two seueces ( ad (α respectively, such that (2.8 holds. The, for ε > 0, we defie the followig sets, U := { N : M,α,r f f } ε ad U := { N : M,α ϕ r+α } x ε. From (3.4, we have U U. So, for all N, that a k a k. k U k U Because ε is arbitrary ad ϕ r+α x C [0, 1], lim a k = 0 from Theorem 2.3. So, the k U proof is completed. Sice ϕ r+α x (t = x t r+α C [0, 1] ad ϕ r+α x (x = 0, we ca give the followig result Corollary. Let x [0, 1] ad r N. The, for all f C r [0, 1] such that f (r Lip M (α ad for ay N, we have M,α,r (f; x f (x MB (r, α (r 1 where δ,α (x is give by ( Ackowledgemets. ω ( ϕ r+α x, δ,α (x The author would like to thak the referee for his/her valuable suggestios which improved the paper cosiderably. Refereces [1] Agratii O., O a aalogue of Stacu operators, Cet Eur J Math, 8(1, , [2] Agratii O., Statistical covergece of a o-positive approximatio process, Chaos Solitos Fractals, 44, , [3] Altomare F, Campiti M., Korovki-type approximatio theory ad its applicatios, Walter de Gruyter studies i math. Berli:de Gruyter&Co, [4] Adrews GE, Askey R. Roy R., Special fuctios, Cambridge Uiversity Press, [5] Cheey EW, Sharma A., Berstei power series, Ca J Math, 16, , [6] Doğru O., Duma O., Statistical approximatio of Meyer-Köig ad Zeller operators based o the itegers, Publ Math Debrece, 68, , 2006.

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