ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS

STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVII, Number 4, December 2002 ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS OGÜN DOĞRU Dedicated to Professor D.D. Stacu o his 75 th birthday Abstract. I this paper we itroduced a geeralizatio of Balázs operators [4] which icludes the Bleima, Butzer ad Hah operators [6]. We defie a space of geeral Lipschitz type maximal fuctios ad obtai the approximatio properties of these operators. Also we estimate the rate of covergece of these operators. I the last sectio, we obtai derivative ad bouded variatio properties of these geeralized operators.. Itroductio I [4], K. Balázs itroduced the discrete liear positive operators defied by R fx = + a x f a x, x 0, N b =0 where a ad b are positive umbers, idepedet of x. After simple computatio, we have R e 0 x = R e x = b a x + a x R e 2 x = b 2 2 a x + a x + a x b 2 + a x where e represets the moomial e x = x for = 0,, 2. These equalities show that both of classical Bohma-Korovi theorems i [7], [3] ad weighted Korovi type theorems i [0] ad [9] do ot valid. Received by the editors: 0.0.2002. 2000 Mathematics Subject Classificatio. 4A25, 4A36. Key words ad phrases. Geeralized liear positive operators, Korovi type theorem, Lipschitz type maximal fuctios, differece operator, bouded variatio. 37

OGÜN DOĞRU I [4], Voroosaja type formula was give for operators, uder the some restrictio of sequeces a ad b. I [] ad [2], O. Agratii itroduced a Katorovich type itegral form of operators ad obtaied the degree of approximatio i polyomial weighted fuctio spaces. By choosig a = β, b = β for N ad 0 < β <, the operator was deoted by the symbol R [β]. Also, for some 0 < β < values i [4], [5] ad [7], covergece, derivative ad saturatio properties of R [2/3] K. Balázs, J. Szabados ad V. Toti respectively. were ivestigated by A recet paper is give by O. Agratii i [2] about Voroovsaja type theorem for Katorovich type geeralizatio of the R [β]. O the other had i [6], G. Bleima, P.L. Butzer ad L. Hah itroduced the Berstei type sequece of liear positive operator defied as L fx = + x f + x, x 0, N. 2 =0 I [6], poitwise covergece properties of operators 2 are ivestigated o compact subiterval [0, b] of [0,. I [], T. Herma ivestigated the behavior of operators 2 whe the growth coditio for f is weaer tha polyomial oe. I [2], C. Jayasri ad Y. Sitarama proved direct ad iverse theorems of operators 2 i ν x the some subspaces o positive real axis. I [8], by usig the test fuctios + x for ν = 0,, 2, a Korovi type theorem was give by Ö. Çaar ad A.D. Gadjiev ad they obtaied some approximatio properties of 2 i a subclass of cotiuous ad bouded fuctios o all positive semi-axis. The aim of this paper is to ivestigate the approximatio properties of a geeralizatio of K. Balázs s operators R i Bleima, Butzer ad Hah operators type o the all positive semi-axis. 2. Costructio of the operators 38 We cosider the sequece of liear positive operators A fx = + a x f =0 a x, x 0, N 3

ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION where a ad satisfy the followig coditios for every ad ; a + = c ad c for. 4 Sice replacig b by, these operators differet from the operators R. Clearly, if we choose a =, = + for every ad the c = + the coditios 4 are satisfied. These operators tur out ito Bleima, Butzer ad Hah operators. Therefore, these operators are a Bleima, Butzer ad Hah operators type geeralizatio of Balázs operators. 3. Approximatio properties I this sectio, we will give a Korovi type theorem i order to obtai approximatio properties of operators 3. I [4], B. Leze itroduced a Lipschitz type maximal fuctio as fα ft fx x = sup t>0 x t α. t x Firstly, we defie a space of geeral Lipschitz type maximal fuctios. Let Wα be the space of fuctios defied as { α Wα = f : sup + a t α a f α x, t M, x 0} + a x where f is bouded ad cotiuous o [0,, M is a positive costat, 0 < α ad f α is the followig fuctio f α x, t = ft fx x t α. Example. For ay M >, let the sequece of fuctios f be The for all x, t 0, x t, we have f x = + M a x + a x. ft fx = M a x t + a x + a t. By choosig M M, oe obtais f W. a Also, if + a x is bouded the W α Lip M α where M is a positive costat which satisfies the followig iequality α α a M M. + a x + a t 5 39

OGÜN DOĞRU Really, if f W α the for all x, t 0, x t we ca write ft fx M α α a x t α + a x + a t ad f Lip M α. Clearly that, if a or x the is bouded. + a x Theorem 2. If L is the sequece of positive liear operators actig from Wα to C B [0, ad satisfyig the followig coditios for ν = 0,, 2 ν ν a t a x L x 0 for 6 + a t + a x C B the, for ay fuctio f i W α oe has L f f CB 0 for. a where C B [0, deotes the space of fuctios which is bouded ad cotiuous o [0,. Proof. This proof is similar to the proof of Korovi theorem. Let f Wα,. Sice f is cotiuous o [0,, for ay ɛ > 0 there exists a δ > 0 such that ft fx < ɛ for a t + a t a x + a x < δ ad sice f is bouded o [0,, there is a positive costat M such that ft fx < 2M [ ] 2 a t x δ 2 for a t + a t + a x + a t a x + a x δ. Thus, for all t, x [0, oe has ft fx < ɛ + 2M [ ] 2 a t x δ 2. 7 + a t + a x By usig basic properties of positive liear operators, we have L f f CB L f fx CB + + f CB L CB 8 By usig the iequality 7 ad coditios 6 i 8, the proof is complete. Now, we will give the first mai result about approximatio properties of operators 3 with the help of Theorem 2. 40

ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION Theorem 3. If A is the sequece of positive liear operators defied by 3, the for each f W α A f f CB 0 for. Proof. For the operators i 3, it is easily to verify that A x = a t A x = a x + a t c + a x 2 2 2 a t a x A x = + a x + a t c + a x c c + a x. By usig the coditios 4 ad Theorem 2, the proof is obvious. 4. Approximatio order I this sectio, we give a result about rate of covergece of operators 3. Theorem 4. If f Wα the for all x 0 we have α A fx fx M a 9 c where the costats M ad 0 < α are defied i the defiitio of the space W α ad the operators A are defied i 3. M Proof. If f W α, we ca write a α + a x + a x From the coditios 4, we get A fx fx α α x + a =0 =. c + a If we use this equality i the last iequality, we obtai a x. M α a + a x c α + a x A fx fx xc a α =0 a x. 4

OGÜN DOĞRU By usig the Hölder iequality for p = 2 α, q = 2 2 α A e 0 x 2 α 2 = we have ad cosiderig A fx fx 0 α a α 2 M c + a x + a x xc a a 2 x. =0 =0 O the other had, it is obvious that a x = + a x, a x = a x + a x, = 2 a x = a x 2 + a x 2 + a x + a x. = By usig these equalities, after simplificatios, we obtai + a x x 2 c 2 xc a 2 a x =0 [ ] 2 a c c 2 a 2 x 2 c 2 If we use last iequality i 0, we have a A fx fx M c + a x a x = M + a x α a x Sice, the proof is complete. 2 a. c α x α c α α c a a c α. + a x Sice Theorem 4 is valid for all x 0, this proof gives uiform covergece of the operators A to f without usig Korovi type theorem. α 5. Derivative properties Firstly, explicit formula for derivatives of Berstei polyomials with differece operators is obtaied by G.G. Loretz i [5, p.2]. A lot of studies have icluded derivative properties of positive liear operators. I [6], D.D. Stacu obtaied the 42

ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION mootoicity properties from differet orders of the derivatives of Berstei polyomials with the help of divided differeces. I this part, we will give some derivative properties of operators A defied i 3 with the help of differece operators. We ca easily compute: d dx A fx = a + a x =0 [ f + f ] ad by usig iductio method for derivatives of order, we have a x d dx A fx 2 =... + a + a x f ν b ν=0, ν where f is differece operator defied as Theorem 5. Let f C [0,. The the operators A have the mootoicity properties. ca write f ν = i f i i=0 ν + i. ν a x ν, Proof. If f C [0, the f C [ d dx A fx = a + a x Sice + ad this completes the proof. =0 ], + +. Therefore, from we f ξdξ f ξdξ 0 0 for f x 0 0, we have d dx A fx 0 0 for f x 0 0 a x. I [5, p.23], G.G. Loretz gives a estimate related to the total variatio of Berstei polyomials. Similarly, i the followig theorem, we give a estimate of bouded variatio betwee the operators A ad f. o [0,. Theorem 6. The operators A preserve the fuctios of bouded variatio 43

OGÜN DOĞRU Proof. By usig formula, we get V A f = 0 d dx A fx dx 3 f a =0 Sice > ad + > 0, we ca write 0 a x + a x dx = a x + a x dx. 0 Γ + Γ +. a Γ + If we use properties of Gamma fuctio i this equality, we have 0 a x + a x dx = By usig this equality i 3, we obtai which gives the proof. Refereces V A f V f!!. a! [] Agratii, O., A approximatio process of Katorovich type, Math. Notes, 2, 200, 3-0. [2] Agratii, O., O approximatio properties of Balázs-Szabados operators ad their Katorovich extesio, Korea J. Comput. Appl. Math., 92, 2002, 36-372. [3] Altomare, F. ad Campiti, M., Korovi-Type Approximatio Theory ad its Applicatios, De Gruyter Series Studies i Mathematics, Vol.7, Walter de Gruyter, Berli-New Yor, 994. [4] Balázs, C., Approximatio by Berstei type ratioal fuctios, Acta Math. Acad. Sci. Hugar., 26-2, 975, 23-34. [5] Balázs, C. ad Szabados, J., Approximatio by Berstei type ratioal fuctios II, Acta Math. Acad. Sci. Hugar., 40, 982, 33-337. [6] Bleima, G., Butzer, P.L. ad Hah, L., A Berstei-type operator approximatig cotiuous fuctios o the semi-axis, Math. Proc. A, 833, 980, 255-262. [7] Bohma, H., O approximatio of cotiuous ad aalytic fuctios, Arif für Matemati, 23, 95, 43-56. [8] Çaar, Ö., ad Gadjiev, A.D., O uiform approximatio by Bleima, Butzer ad Hah operators o all positive semi-axis, Tras. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci. 95, 999, 2-26. [9] Doğru, O., O weighted approximatio of cotiuous fuctios by liear positive operators o ifiite itervals, Math. Cluj, 464, 999, 39-46. [0] Gadjiev, A.D., O P.P. Korovi type theorems, Math. Zameti, 20, 976, 78-786 i Russia. [] Herma, T., O the operator of Bleima, Butzer ad Hah, Colloq. Math. Soc. Jáos Bolyai, 58. Approx. Th., Kecsemét Hugary, 990, 355-360. [2] Jayasri, C. ad Sitarama, Y., Direct ad iverse theorems for certai Berstei-type operators, Idia J. Pure Appl. Math. 6, 985, 495-5. 44

ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION [3] Korovi, P.P., Liear Operators ad Approximatio Theory, Delhi, 960. [4] Leze, B., Berstei-Basaov-Katorovich operators ad Lipschitz-type Maximal Fuctios, Colloq. Math. Soc. Jáos Bolyai, 58. Approx. Th., Kecsemét Hugary, 990, 469-496. [5] Loretz, G.G., Berstei Polyomials, Toroto, 953. [6] Stacu, D.D., Applicatio of divided differeces to the study of mootoocity of the derivatives of the sequece of Berstei polyomials, Calcolo, 64, 979, 43-445. [7] Toti, V., Saturatio for Berstei type ratioal fuctios, Acta Math. Hugar., 433-4, 984, 29-250. Aara Uiversity, Faculty of Sciece, Departmet of Mathematics, 0600, Tadoga, Aara, Turey E-mail address: dogru@sciece.aara.edu.tr 45