Research Article On q-bleimann, Butzer, and Hahn-Type Operators

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1 Abstract ad Applied Aalysis Volume 205, Article ID , 7 pages Research Article O q-bleima, Butzer, ad Hah-Type Operators Dilek Söylemez Akara Uiversity, Elmadag Vocatioal School, Departmet of Computer Programmig, Akara, Turkey Correspodece should be addressed to Dilek Söylemez; dsoylemez@akara.edu.tr Received 6 October 205; Revised 4 December 205; Accepted 5 December 205 Academic Editor: Simeo Reich Copyright 205 Dilek Söylemez. 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 work is properly cited. Sequece of q-bleima, Butzer, ad Hah operators which is based o a cotiuously differetiable fuctio τ o R +,withτ(0) 0, if τ (x), has bee cosidered. Uiform approximatio by such a sequece has bee studied ad degree of approximatio by the operators has bee obtaied. Moreover, shape preservig properties of the sequece of operators have bee ivestigated.. Itroductio I 987, Lupaş ] itroducedthe first q-aalogue of Berstei operators ad ivestigated its approximatio ad shape preservig properties. Aother q-geeralizatio of the classical Berstei polyomials is due to Phillips 2]. After that may geeralizatios of positive liear operators based o q-itegers were itroduced ad studied by several authors. Some are i 3 3]. Bleima et al. 4] proposed a sequece of positive liear operators L defied by k L (f; x) (+x) f( k+ k )xk, x 0, N for f C0, ),wherec0, ) deotethespaceofall cotiuous ad real valued fuctios defied o 0, ). Also the authors proved that L (f; x) f(x) as poitwise o 0, ) for ay f C B 0, ), wherec B 0, ) deote the space of all bouded fuctios from C0, ).Itis well kow that () f C B sup f (x) (2) x 0 defies a orm o C B 0, ). Moreover, they showed that the covergece is uiform o each compact subset of 0, ). I 5], usig test fuctios (t/( + t)) V for V 0,, 2, Gadjiev ad Çakar stated a Korovki-type theorem for the uiform approximatio of fuctios belogig to some suitable fuctio spaces by some liear positive operators. As a applicatio of this result, they proved uiform approximatio by Bleima, Butzer, ad Hah operators. Further results cocerig such a sequece of operators ad its geeralizatios may be foud i 6 9]. Now, we recall some otatios from q-aalysis 20, 2]. The q-iteger ] ad the q-factorial ]! are defied by 0] 0,ad ] fl ] q { q { q, q {, q, for N (3) ]! fl { ] q 2] q ] q,,2,... {, 0, { (4) for N, 0]!, respectively, where q > 0. For itegers r 0 the qbiomial coefficiet is defied as r ] ] q! r] q q! r] q!. (5) Moreover, Euler idetity is give by s0 (+q s x) q k(k )/2 k ] x k. (6) q

2 2 Abstract ad Applied Aalysis Aral ad Doğru 22] costructed the q-bleima, Butzer, ad Hah operators as where L,q (f; x) k] f( l (x) k+] q k )qk(k )/2 k ]xk, l (x) s0 (7) ( + q s x) (8) ad f is defied o the semiaxis 0, ). The authors studied Korovki-type approximatio properties by usig the test fuctios (t/( + t)) V for V 0,,2. Moreover, they obtaied rate of covergece of the operators ad proved that rate of the q-bleima, Butzer, ad Hah operators is better tha the classical oe. A geeralizatio of the q-bleima, Butzer, ad Hah operators was itroduced by Agratii ad Nowak i 23]. I this paper, the authors gave represetatio of the operators i terms of q-differeces ad ivestigated some approximatio properties. A Voroovskaja-type result ad mootoicity properties of these operators are ivestigated i 24]. I 25], the authors itroduced a ew geeralizatio of Berstei polyomials deoted by B τ ad defied as B τ (f; x) fl B (f τ ;τ(x)) ( k )τk (x τ(x)) k (f τ k ), (9) where B is the th Berstei polyomial, f C0, ], x 0, ], adτ is a fuctio that is cotiuously differetiable of ifiite order o 0, ] such that τ(0) 0, τ(), ad τ (x) > 0 for x 0,].Also,theauthorsstudiedsome shape preservig ad covergece properties cocerig the geeralized Berstei operators B τ (f; x). I 26], Aral et al. costructed sequeces of Szasz- Mirakya operators which are based o a fuctio ρ. They studied weighted approximatio properties ad Voroovskaja-type results for these operators. They also showed that the sequece of the geeralized Szász-Mirakya operators is mootoically oicreasig uder the ρcovexity of the origial fuctio. A similar geeralizatio for Bleima, Butzer, ad Hah operators is studied by Söylemez 27]. Also the class H τ ω was defied, a Korovkitype theorem was give for the fuctios i this class, ad uiform covergece of the geeralized Bleima, Butzer, ad Hah operators was obtaied 27]. Moreover, the mootoicity properties of the operators were ivestigated. Now we recall the defiitio of H τ ω that is a subspace of C B 0, ) 27]. Let ω be a geeral modulus of cotiuity, satisfyig the followig properties: (a) ω is cotiuous, oegative, ad icreasig fuctio o 0, ), (b) ω(δ +δ 2 ) ω(δ )+ω(δ 2 ), (c) lim δ 0 ω(δ) 0. The space of all real valued fuctios f defied o 0, ) satisfyig f (x) f(y) ω +τ(x) τ(y) +τ(y) for all x, y 0, ) is deoted by H τ ω. It is clear from coditio (b) that we have (0) ω (δ) ω(δ), N () ad oe ca get from the coditio (a) that for ay λ>0 ω (λδ) ω((+ λ ]) λ) +λω(δ), (2) where λ ] deotes the greatest iteger that is ot greater tha λ. Now we defie a ew geeralizatio of q-bleima, Butzer, ad Hah operators for f C0, )by L τ,q (f; x) l (f τ k] k+] q ) k where q k(k )/2 k ]τ(x)k, (3) l τ (x) ( + q s ), (4) s0 ad τ is a cotiuously differetiable fuctio o 0, ) such that τ (0) 0, if x 0, ) τ (x). (5) A example of such a fuctio τ is give i 26]. Note that, i the settig of the operators (3), we have L τ,q f fl L,q (f τ ) τ, (6) where the operators L,q are defied by (7). If τe,the L τ,q L,q. Obviously, we have L τ,q (; x), L τ,q ( τ +τ ;x) ] +] +τ(x), L τ,q (( τ +τ )2 ;x) ] ] +] 2 q2 τ 2 (x) (+τ(x)) ( + q) + ] +] 2 +τ(x). (7)

3 Abstract ad Applied Aalysis 3 I this study, we cosider a geeralizatio of q-bleima, Butzer, ad Hah operators i the sese of 26], we ivestigate uiform covergece of {L τ,q (f; x)} N to f(x) o 0, ) for f H τ ω, ad we obtai the degree of approximatio. Moreover, we study shape preservig properties uder τcovexity of the fuctio. Our results show that the ew operators are sesitive to the rate of covergece to f, depedig o the selectio of τ. Fortheparticularcase τ(x) x, the previous results for q-bleima Butzer ad Hah operators are obtaied. I order to esure that the covergece properties holds, the author will assume qq is a sequece such that q as for 0<q <,asi22]. Defiitio. Let x 0,x,...,x be distict poits i the domai of f.deote f(x f x 0,x,...,x ] r ) r0 jr (x r x j ), (8) where r remais fixed ad j takes all values from 0 to, excludig r. Defiitio 2. A cotiuous, real valued fuctio f is said to be covex i D 0, ),if m f( i α i x i ) m i α i f(x i ) (9) for every x,x 2,...,x m D ad for every oegative umber of α,α 2,...,α m such that α +α 2 ++α m. I 25] Cárdeas-Morales et al. itroduced the followig defiitio of τ-covexity of a cotiuous fuctio. Defiitio 3. A cotiuous, real valued fuctio f is said to be τ-covex i D, iff τ is covex i the sese of Defiitio Approximatio Properties I this sectio we deal with the promised approximatio properties of the sequece of q-bleima, Butzer, ad Hah operators. I 27], the followig Korovki-type theorem was give. Theorem 4. Let {T τ (f)} N be a sequece of liear positive operators from H τ ω to C B0, ). If V τ (t) lim Tτ (( +τ(t) ) ;x) ( +τ(x) V C ) 0 (20) B is satisfied for V 0,,2,theforf H τ ω oe has lim Tτ f f C B 0. (2) Now we are ready to give the followig theorem. Theorem 5. Suppose that qq, 0<q <,adletq as.if L τ,q is the operator defied by (3), the for ay f H τ ω oe has lim Lτ,q f f C B 0. (22) Proof. Accordig to Theorem 4 we will show that (20) holds for L τ,q.obviously,from(7)weeasilyobtaithat L τ,q (; x). Lτ,q ( τ (t) +τ(t) ;x) ] +τ(x) C B +] q q +], L τ 2,q (( τ (t) +τ(t) ) ;x) ( +τ(x) 2 C ) B sup x 0 2 ( +τ(x) ) ( ] ] +] 2 q2 +τ(x) q 2 (+τ(x)) ( + q ) )+ ] +] 2 ( q 2 2 +] + +] 2 ). (23) Therefore, the coditios (20) are satisfied. By Theorem 4, the proof is completed. Theorem 6. Let q q, 0 < q <,adletq as.the oe has for all f H τ ω. Here, μ τ Lτ,q (f; x) f (x) 2ω( μτ (x)), (24) 2 (x) ( +τ(x) ) ( 2 ] +] + ] ] +] 2 + ] +] 2 +τ(x). q2 (+τ(x)) ( + q ) ) Proof. Firstly, from (2) ad (0) we ca write f (t) f(x) ω( τ (t) +τ(t) +τ(x) ) (+ τ (t) / (+τ(t)) τ(x) / (+τ(x)) )ω(δ) δ ad cosiderig (7), we get (25) (26) Lτ,q (f; x) f (x) Lτ,q ( f (t) f(x) ;x). (27)

4 4 Abstract ad Applied Aalysis Equatios (27) ad (26) together imply that Lτ,q (f; x) f (x) ω(δ) ( + τ (t) δ Lτ,q +τ(t) +τ(x) ;x). Usig the Cauchy-Schwarz iequality, we obtai Lτ,q (f; x) f (x) ω(δ) ( + 2 /2 τ (t) δ Lτ,q (( +τ(t) +τ(x) ) ;x)] ). (28) (29) ()+ l τ + (x) q(+)/2 (f τ ] q ) (f τ +] q + )] + l+ k (f τ ) k] ( k+] q k )qk(k )/2 k ]τ(x)k + l+ (f τ k] k+] q ) k q k(k )/2 q k ]τ(x)k+ l+ k k] ( k+2] q k )qk(k )/2 + k ]τ(x)k (f τ ) By choosig δ μ τ(x) Lτ,q ((τ(t)/( + τ(t)) τ(x)/( + τ(x))) 2 ;x)] /2,weget where Lτ,q (f; x) f (x) 2ω( μτ (x)), (30) which cocludes the proof. sup μ τ (x) x 0 +] 2 (3) 3. Shape Preservig Properties Theorem 7. Let f be a τ-covex fuctio that is oicreasig o 0, ); the oe has for N. L τ,q (f; x) Lτ +,q (f; x) (32) ()+ l τ + (x) q(+)/2 (f τ ] q ) (f τ +] q + )] + l+ (f τ ) ( k+] k] q k+ )qk(k )/2 q k k+ ]τ(x)k+ + l+ (f τ k] k+] q ) k q k(k )/2 q k ]τ(x)k+ l+ (f τ ) k+] ( k+] q k+ )qk(k )/2 q k + k+ ]τ(x)k+. (33) Moreover, we have the followig equalities that are proved i Lemma 3. of 24]: + k+ ] ]+] k]k+] k ], Proof. From (3), oe ca write L τ,q (f; x) Lτ +,q (f; x) s0 ( + qs ) (f τ k] k+] q k )qk(k )/2 k ]τ(x)k + (+q ) s0 ( + (f τ ) qs ) k] ( k+2] q k )qk(k )/2 + k ]τ(x)k which imply ] ] k k] k ], k+ ] ] k+] k ], L τ,q (f; x) (τ L)+ +,q (f; x) l+ q (+)/2 (f τ ] q ) (34)

5 Abstract ad Applied Aalysis 5 (f τ +] q + )] + l+ q k q k(k )/2 k ]τ(x)k+ {(f τ k+] k] q ) ] k+ k+] +(f τ k] ] )q k k+] qk k] (f τ k+] k+] q ) ]+] k+ k]k+] } l τ + (x) q k q k(k )/2 ]+] k]k+] k ]τ(x)k+ {(f τ k+] k] ) k] qk+ +] +(f τ k] k+] k+] q k )q k +] (f τ k+] k+] q )}. k+ (35) Proof. From (7), we have L τ,q (f; x) f (x q ) s0 ( + qs ) (f τ k] k+] q ) f(x k q )] q k(k )/2 k ]τ(x)k l τ (x) ( q k] k+] q k ) k+] q k ;(f τ )] q k(k )/2 k ]τ(x)k. (39) Usig the equality (k]/ k+]) k ] k ],weowget L τ,q (f; x) f (x q ) l τ (x) k+] q k ;(f τ )] From + ]/q + ]/q /q + >0ad by hypothesis, we have (f τ ] q ) (f τ +] q + )>0. (36) Sice f is τ-covex, by usig Lemma 3.2 i 24] we obtai L τ,q (f; x) Lτ +,q (f; x) 0 (37) for x 0, )ad N. This proves the theorem. Theorem 8. Suppose that x 0, ) \ {k]/ k + ]q k :k 0,,...,},adτ is liear. The oe has L τ,q (f; x) f (x q τ )] q ( )/2 + ) τ (x)+ l τ l τ (x) k+] q, k qk(k )/2 k 2 k] + k ]τ(x)k. (x) τ (x) q, ] ;(f q k+] k] q k+ ;(f τ )] (38) q k(k )/2 k ]τ(x)k+ + k l τ (x) k+] q k ;(f τ )] q k(k )/2 k τ k ]τ(x)k (x)+ (x) l τ τ (x) q, ] q ;(f τ )] q ( )/2 + l { q, k+] k] q k+ ;(f τ )] k+] q k ;(f τ )]} q k(k )/2 k ]τ(x)k+. O the other had we ca write k+] k] q k] k+ k+] q k +] k] k+] q k+, (40) (4)

6 6 Abstract ad Applied Aalysis which implies q, k+] k] q k+ ;(f τ )] k+] q k ;(f τ )] k+] q, k+] k k] q k+ ;(f τ )] +] k] k+] q k+. Now takig (42) i (40), we have L τ,q (f; x) f (x q τ )] q ( )/2 + ) τ (x)+ l τ l τ (x) k+] q, k (x) τ (x) q, ] ;(f q k+] k] q k+ ;(f τ )] +] k] k+] q k+ qk(k )/2 k ]τ(x)k+. Thustheproofiscompleted. (42) (43) Now, from Theorem 8, we have the followig corollary immediately. Corollary 9. Let f be a τ-covex, oicreasig fuctio o 0, ) ad τ is liear. The oe has f( x q ) Lτ,q (f; x) (44) for ay x 0, )\{k]/ k+]q k :,,...,}, N. Coflict of Iterests The author declares that there is o coflict of iterests regardig the publicatio of this paper. Refereces ] A. Lupaş, A q-aalogue of the Berstei operators, i Semiar o Numerical ad Statistical Calculus, vol. 9, Techical Uiversity of Cluj-Napoca, Cluj-Napoca, Romaia, ] G. M. Phillips, Berstei polyomials based o the q-itegers, Aals of Numerical Mathematics,vol.4,pp.5 58,997. 3] A. Il iskii ad S. Ostrovska, Covergece of geeralized Berstei polyomials, JouralofApproximatioTheory, vol. 6, o., pp. 00 2, ] H. Oruç ad N. Tucer, O the covergece ad iterates of q- Berstei polyomials, JouralofApproximatioTheory, vol. 7, o. 2, pp , ] S. Ostrovska, q-berstei polyomials ad their iterates, Joural of Approximatio Theory, vol.23,o.2,pp , ] A. Aral ad V. Gupta, O q-baskakov type operators, Demostratio Mathematica,vol.42,o., ] A. Aral ad V. Gupta, q-derivative ad applicatios to the q- Szász Mirakya operators, Calcolo,vol.43,pp.5 70, ]M.MursaleeadK.J.Asari, Approximatioofq-Stacu- Beta operators which preserve x 2, Bulleti of the Malaysia Mathematical Scieces Society,205. 9]M.Mursalee,F.Kha,A.Kha,adA.Kiliçma, Some approximatio results for geeralized Katorovich-type operators, Joural of Iequalities ad Applicatios,vol.203,article 585, ] M. Mursalee, F. Kha, ad A. Kha, Approximatio properties for Kig s type modified q-berstei-katorovich operators, Mathematical Methods i the Applied Scieces,205. ] M. Mursalee, F. Kha, A. Kha, ad A. Kiliçma, Some approximatio properties of Bivariate q-stacu-β operators, Joural of Fuctio Spaces, vol. 204, Article ID , 7 pages, ] M. Mursalee ad A. Kha, Geeralized q-berstei-schurer operators ad some approximatio theorems, Joural of Fuctio Spaces ad Applicatios, vol.203,articleid79834,7 pages, ] M. Mursalee ad A. Kha, Statistical approximatio properties of modified q-stacu-β operators, Bulleti of the Malaysia Mathematical Scieces Society,vol.36,o.3,pp ,203. 4] G. Bleima, P. L. Butzer, ad L. Hah, A Berstei-type operator approximatig cotiuous fuctios o the semiaxis, Idagatioes Mathematicae (Proceedigs), vol. 83, o. 3, pp , ] A. D. Gadjiev ad Ö. Çakar, O uiform approximatio by Bleima, Butzer ad Hah operators o all positive semi-axis, Trasactios of AS Azerbaija. Series of Physical Techical ad Mathematical Scieces,vol.9,o.5,pp.2 26,999. 6] U. Abel ad M. Iva, Some idetities for the operator of Bleima, Butzer ad Hah ivolvig divided differeces, Calcolo,vol.36,o.3,pp.43 60,999. 7] R. A. Kha, A ote o a Berstei-type operator of Bleima, Butzer, ad Hah, Joural of Approximatio Theory,vol.53, o. 3, pp , ] J. A. Adell, J. De la Cal, ad M. Sa Miguel, Iverse beta ad geeralized Bleima-Butzer-Hah operators, Joural of Approximatio Theory,vol.76,o.,pp.54 64,994. 9] B. Della Vecciha, Some Properties of a Ratioal Operator of Berstei-Type, Istituto per Applicazioi della Matematica- C.N.R.ViaP.Castellio,Napoli,Italy,99. 20] T. Erst, The history of q-calculus ad a ew method, U.U.U.D.M Report 6, Departmet of Mathematics, Upsala Uiversity, ] G. M. Phillips, Iterpolatio ad Approximatio by Polyomials, CMSBooksiMathematics/Ouvrages demathématiques de la SMC4,Spriger,NewYork,NY,USA, ] A. Aral ad O. Doğru, Bleima, butzer ad hah operators o the q-itegers, Joural of Iequalities ad Applicatios, vol. 2007, Article ID 7940, 2 pages, ] O. Agratii ad G. Nowak, O the geeralizatios of Bleima, ButzeradHahoperatorsbasedoq-itegers, Mathematical ad Computer Modellig,vol.53,pp ,20.

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