ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS

Transcription

1 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: 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

2 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

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

4 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 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

5 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 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

6 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

7 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

8 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, [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, [5] Balázs, C. ad Szabados, J., Approximatio by Berstei type ratioal fuctios II, Acta Math. Acad. Sci. Hugar., 40, 982, [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, [7] Bohma, H., O approximatio of cotiuous ad aalytic fuctios, Arif für Matemati, 23, 95, [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, [9] Doğru, O., O weighted approximatio of cotiuous fuctios by liear positive operators o ifiite itervals, Math. Cluj, 464, 999, [0] Gadjiev, A.D., O P.P. Korovi type theorems, Math. Zameti, 20, 976, i Russia. [] Herma, T., O the operator of Bleima, Butzer ad Hah, Colloq. Math. Soc. Jáos Bolyai, 58. Approx. Th., Kecsemét Hugary, 990, [2] Jayasri, C. ad Sitarama, Y., Direct ad iverse theorems for certai Berstei-type operators, Idia J. Pure Appl. Math. 6, 985,

9 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, [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, [7] Toti, V., Saturatio for Berstei type ratioal fuctios, Acta Math. Hugar., 433-4, 984, Aara Uiversity, Faculty of Sciece, Departmet of Mathematics, 0600, Tadoga, Aara, Turey address: 45