Research Article The Rate of Convergence of Lupas q-analogue of the Bernstein Operators

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1 Hidawi Publishig Corporatio Abstract ad Applied Aalysis, Article ID 52709, 6 pages Research Article The Rate of Covergece of Lupas q-aalogue of the Berstei Operators Hepig Wag ad Yabo Zhag 2 School of Mathematical Scieces, BCMIIS, Capital Normal Uiversity, Beijig 00048, Chia 2 Departmet of Basic Courses, Shadog Moder Vocatioal College, Jia, Shadog 25004, Chia Correspodece should be addressed to Hepig Wag; waghp@cu.edu.c Received 24 December 203; Accepted 2 Jauary 204; Published 9 April 204 Academic Editor: Sofiya Ostrovska Copyright 204 H. Wag ad Y. Zhag. 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. We discuss the rate of covergece of the Lupas q-aalogues of the Berstei operators R,q (f; x) which were give by Lupas i 987. We obtai the estimates for the rate of covergece of R,q (f) by the modulus of cotiuity of f, ad show that the estimates are sharp i the sese of order for Lipschitz cotiuous fuctios.. Itroductio I 92,Berstei (see []) defied the Berstei polyomials. Later, it was foud that the Berstei polyomials possess may remarkable properties, which made them a area of itesive research. Due to the developmet of q-calculus, geeralizatios of Berstei polyomials coected with qcalculus have emerged. The first perso to make progress i this directio was Lupas, who itroduced a q-aalogue of the Berstei operator R,q (f; x) ad ivestigated its approximatig ad shape-preservig properties i 987 (see [2]). If q,ther, (f; x)} are the classical Berstei polyomials. For q,theoperatorsr,q (f; x) are ratioal fuctios rather tha polyomials. Other geeralizatios of the Berstei polyomials, for example, the q-berstei polyomials (see [3]), the two-parametric geeralizatio of q-berstei polyomials (see [4]), ad the q-berstei- Durrmeyer operator (see [5]), had also bee cosidered i recet years. Amog these geeralizatios, q-berstei polyomials proposed by Phillips attracted the most attetio ad were studied widely by a umber of authors (see [3, 6 5]). The Lupas q-aalogues of the Berstei operators R,q (f; x)} are less kow; see [2, 6 2]. However, they have a advatage of geeratig positive liear operators for all q > 0,whereasq-Berstei polyomials geerate positive liear operators oly if q (0, ). I this paper, we will study the rate of covergece of the Lupas q-aalogues of the Berstei operators R,q (f; x)}.we will obtai the estimates for the rate of covergece of R,q (f) by the modulus of cotiuity of f, ad show that the estimates are sharp i the sese of order for Lipschitz cotiuous fuctios. Our results demostrate that the estimates for the rate of covergece of R,q (f; x)} are essetially differet from those for the classical Berstei polyomials; however, theyare very similarto thosefortheq-berstei polyomials i the case q (0, ). Throughout the paper, we always assume that f is a cotiuous real fuctio o [0, ], q > 0, q.deoteby C[0, ] (or C [0, ], )thespaceofallcotiuous (correspodigly, times cotiuously differetiable) realvalued fuctios o [0, ] equipped with the uiform orm. The expressio A() B() meas that A() B() ad A() B(),adA() B() meas that there exists a positive costat c idepedet of such that A() cb(). To formulate our results, we eed the followig defiitios. Let q>0. For each oegative iteger k, theq-iteger [k] ad the q-factorial [k]! are defied by ( q k ) [k] : [k] q : ( q), q k, q,

2 2 Abstract ad Applied Aalysis [k][k ] [], [k]! : k,. For itegers 0k,theq-biomial coefficiet is defied by [ k ] : []! [k]! [ k]!. (2) I [2], Lupas proposed the q-aalogue of the Berstei operator R,q (f; x): foreachpositiveiteger, adf C[0, ], where f( [k] R,q (f, x) : [] )r,k (q,x), f (), r,k (q; x) : [ k ] 0x< x, q k(k )/2 x k ( x) k ( xqx) ( xq x) [ k ] q k(k )/2 (x/ ( x)) k j0 ( qj (x/ ( x))). I [9], Ostrovska proved that, for each f C[0, ] ad q (0, ), thesequecer,q (f, x)} coverges to the limit operator R,q (f, x) uiformly o [0, ] as,where f( q )r k (q;x), R,q (f, x) 0x< f (), x, q k(k )/2 (x/ ( x)) k r,k (q; x) : ( q) k [k]! j0 ( qj (x/ ( x))). (6) Whe q >, the followig relatios (see [9])allow usto reduce to the case q (0, ): R,q (f; x) R,/q (g; x), (7) R,q (f; x) R,/q (g; x), where g(x) f( x) C[0, ]. The problem to fid the rate of covergece occurs aturally ad this paper deals with the problem of fidig estimates for the rate of covergece for a sequece of the qaalogue of the Berstei operator R,q (f; x) for 0<q<. For f C[0, ], t > 0, themodulusofcotiuityω(f, t) ad the secod modulus of smoothess ω 2 (f, t) are defied as follows: ω(f;t): sup f (x) f(y) ; x y t x,y [0,] ω 2 (f, t) : sup sup 0<ht x [0, 2h] () (3) (4) (5) f (x2h) 2f(xh) f(x). The mai results of the paper are as follows. (8) Theorem. Let q (0, ) ad let f C[0, ].The R,q (f) R,q (f) C qω(f;q ), (9) where C q 2 6/( q). This estimate is sharp i the followig sese of order: for each α, 0 < α, thereexists afuctiof α (x) which belogs to the Lipschitz class Lip α: f C[0,] ω(f;t) t α } such that R,q (f α ) R,q (f α ) qα. (0) Theorem 2. Let 0<q<.The Furthermore, R,q (f) R,q (f) cω 2 (f; q ). () sup R,q (f) R,q (f) 0<q< cω 2 (f; /2 ), (2) where c is a absolute costat. Remark 3. From (2), it follows that, for each f C[0, ], lim R,q (f; x) R,q (f; x) (3) uiformly ot oly i x [0,],adbutalsoiq (0, ], which geeralizes the Ostrovska s result i [9]. Remark 4. It should be emphasized that Theorem caot be obtaied i a way similar to the proof of the Popoviciu Theorem for the classical Berstei polyomials (see [22]). It requires differet estimatio techiques due to the ifiite product ivolved. Also, the proof i the paper is more difficult tha the oe used for q-berstei polyomials (see [4]), sice the Lupas q-aalogue of Berstei operators has the sigular ature at the poit xadeedsaewmethod (whe x, x/( x) ). Remark 5. Results similar to Theorems ad 2 for q- Berstei polyomials were obtaied i [4]ad[2], respectively. Note that whe f(x) x 2,forq (0, ),wehave(see (46)) R,q (f; x) R,q (f; x) q x ( x) q ω 2 (f; q ). ( xqx)[] (4) Hece,theestimate() is sharp i the followig sese: the sequece q i () caotbereplacedbyayother sequece decreasig to zero more rapidly as. However, () is ot sharp for the Lipschitz class Lip α(α (0, ]) itheseseoforder.this,combiigwiththeorem, shows that i the case 0<q<the modulus of cotiuity is more appropriate to describe the rate of covergece for the Lupas q-aalogue Berstei operators tha the secod modulus of smoothess. This is differet from that i the case q.

3 Abstract ad Applied Aalysis 3 Remark 6. The umbers c i () adc q i (9) are both the costats idepedet of f ad. However,while c i () does ot deped o q, thecostatc q i (9) depeds o q ad teds to as q.hece,() doesotfollow from (9). Let f C[0, ] ad g(x) f( x). Usig(7) adthe relatios ω (f, t) ω(g, t) ; ω 2 (f, t) ω 2 (g, t), (5) we have the followig corollaries. Corollary 7. Let f C[0, ].Theforayq (, ), R,q (f) R,q (f) C qω (f; where C q is a costat idepedet of f ad. ), q (6) Corollary 8. Let f C[0, ].Theforayq (, ), Furthermore, R,q (f) R,q (f) cω 2 (f; ). (7) q sup R,q (f) R,q (f) q>0 cω 2 (f; /2 ), (8) where c is a absolute costat. 2. Proofs of Theorems ad 2 For the proofs of Theorems ad 2, we eed the followig lemmas. Proof. Usig (9) ad(3), we get q k r,k (q; x) (q q k ) q r,k (q; x) r,k (q; x) (q [k] ) [] r,k (q; x) (q )R,q (t; x) xq x. Similarly, usig (9) ad(5), we have q k r,k (q; x) ( (q k )r,k (q; x) ( q k )r,k (q; x)) R,q (t; x) x. The proof of Lemma 0 is complete. r,k (q; x) r,k (q; x) For itegers, k,adq (0, ), x [0,],wehave r,k (q; x) r k (q; x) [ k ] q k(k )/2 (x/ ( x)) k s0 ( qs (x/ ( x))) q k(k )/2 (x/ ( x)) k ( q) k [k]! s0 ( qs (x/ ( x))) (22) (23) Lemma 9 (see [2]). The followig equalities are true: R,q (; x) R,q (; x), R,q (t; x) R,q (t; x) x, (9) [ k ] q k(k )/2 (x/ ( x)) k s0 ( qs (x/ ( x))) ( s ( qs (x/ ( x))) ) R,q (t 2 ;x) x 2 x ( x) [] x2 ( x) ( q) xxq ( [] ). (20) Lemma 0. With the defiitios of r,k (q; x) ad r,k (q; x), we have q k r,k (q; x) xq x, q k r,k (q; x) x. (2) q k(k )/2 (x/ ( x)) k s0 ( qs (x/ ( x))) ([ k ] ( q) k [k]! ) r,k (q; x) ( r,k (q; x) ( s ( qs (x/ ( x))) ) s k r,k (q; x) J r,k (q; x) J 2, ( q s )) (24)

4 4 Abstract ad Applied Aalysis where J : s ( qs (x/ ( x))), J 2 : s k We will prove the followig lemma. ( q s ). (25) Lemma. Let 0<q<.Theforitegers, k ad for 0< x</(q ), q k r,k (q; x) r,k (q; x) 3q q. (26) Proof. It is easy to prove by iductio that 0J 2 : s k ( q s ) q s q s q k q. s k s k (27) Sice exp( x) x ad l( x) x for all x [0,), we obtai 0J exp ( l ( q s x x )) Hece, s q s s s l ( q s x x ) x x r,k (q; x) r k (q; x) ad therefore, by (2) ad(9) we get q k r,k (q; x) r,k (q; x) q x ( q) ( x) q x ( q) ( x). q x ( q) ( x) r,k (q; x) q k q r,k (q; x), q (28) (29) q k r,k (q; x) r q,k (q; x) q x ( q) ( x) ( xq x) q q. (30) Sice 0<x</(q )<,itfollowsthat0<x/( x)< /q ad thece q k r,k (q; x) r,k (q; x) 3q q. (3) This completes the proof of Lemma. Proof of Theorem. It follows from the defiitio of R,q (f; x) ad R,q (f; x) that both of them possess the ed poit iterpolatio property; i other words, R,q (f; 0) R,q (f; 0) f (0), R,q (f; ) R,q (f; ) f (). (32) It follows from the defiitio of r,k (q; x) ad r,k (q; x) that r,k (q; x) 0 ad r,k (q; x) 0 for 0x<.Ifx, the x/( x).so,thelupasq-aalogue of Berstei operators has the sigular ature at the poit xad the rate of covergece ear the poit eeds to be cosidered idepedetly. First we suppose x (/( q ), ); thatis, x<q /( q )<q.the Sice we get I I R,q (f; x) R,q (f; x) (f ( [k] [] ) f())r,k (q; x) (f ( q k ) f())r,k (q; x) f([k] [] ) f() r,k (q; x) f( qk ) f() r,k (q; x). [k] [] q k q ω(f;λt)(λ) ω(f;t), ω(f;q k )r,k (q; x) qk ( q k ) q q k, ω(f,q )( qk q )r,k (q; x) ω(f;q )( qk q )r,k (q; x) 2ω(f;q ) ω(f,q ) q ω(f,q ) q q k r,k (q; x). (0 k), λ>0, ω(f;q k )r,k (q; x) q k r,k (q; x) (33) (34) (35)

5 Abstract ad Applied Aalysis 5 By Lemma 0 ad x<q, x<,wehave I2ω(f;q ) ω(f,q ) q ( xq x) ω(f,q ) q ( x) 5ω(f;q ). (36) Next, we assume that 0<x</(q ).The0 x/( x) /q.wehave I R,q (f; x) R,q (f; x) (f ( [k] [] ) f( qk )) r,k (q; x) (f ( q k ) f())(r,k (q; x) r,k (q; x)) (f ( q k ) f())r,k (q; x) k f([k] [] ) f( qk ) r,k (q; x) k : δ δ 2 δ 3. f( qk ) f() r,k (q; x) r,k (q; x) f( qk ) f() r,k (q; x) First we estimate δ ad δ 3.Sice [k] [] ( qk ) q k q ( qk ) we get ( qk ) qk q, (k ), δ ω(f,q ) δ 3 ω(f,q ) k (37) q ( q k ) q q, (0 k) (38) r,k (q; x) ω (f, q ), (39) r,k (q; x) ω (f, q ). (40) Now we estimate δ 2.Siceω(f, λt) ( λ)ω(f, t), by Lemma we get δ 2 ω(f,q k ) r,k (q; x) r,k (q; x) ω(f,q )( qk q ) r,k (q; x) r,k (q; x) 2ω (f; q ) q q k r,k (q; x) r,k (q; x) 6ω (f; q ). q (4) From (39) (4), we have for 0x/(q ), I(2 6 q )ω(f;q ). (42) Hece from (36)ad(42), we coclude that, for q (0, ), R,q (f; x) R,q (f; x) C q ω(f;q ), (43) where C q 26/( q). At last we show that the estimate (9) issharp.foreach α, 0<α,supposethatf α (x) is a cotiuous fuctio, which is equal to zero i [0, q] ad [ q 2,],equalto (x ( q)) α i [ q, qq( q)/2], ad liear i the rest of [0, ].Itisobviousthatω(f α ;t)ct α,ad R,q (f α ) R,q (f α ) qα r, (q; ) qα. (44) The proof of Theorem is complete. I order to prove Theorem 2,weeedthefollowigresult. Theorem A (see [2]). Let the sequece L } of positive liear operators o C[0, ] satisfy the followig coditios. (A) The sequece L (e 2 )} coverges to a fuctio L (e 2 ) i C[0, ],wheree i (x) x i, i 0,, 2. (B) The sequece L (f, x)} is oicreasig for ay covex fuctio f ad for ay x [0,]. The there exists a operator L o C[0, ] such that L (f) L (f) 0 for ay f C[0, ].Furthermore, L (f, x) L (f, x) cω 2 (f; λ (x)), (45) where λ (x) L (e 2,x) L (e 2,x)ad c is a costat which depeds oly o L (e 0 ). Proof of Theorem 2. From [2], we kow that the Lupas qaalogues of the Berstei operators satisfy Coditio (B).

6 6 Abstract ad Applied Aalysis It follows from [9]that,forq (0, ), R,q (f; x)} coverges to R,q (f; x) uiformly i x [0,]as ;ad 0λ (x) R,q (t 2,x) R,q (t 2,x) R,q (t 2,x) lim R,q (t 2 ;x) x ( x) [] x2 ( x) ( q) xxq ( [] ) x( x) ( q) x2 ( x)( q) q xxq x( x) ( [] ( q)) x2 ( x) ( q) xxq x ( x) ( q) q xxq q q. ( [] ( q)) Theorem 2 follows from (46) ad Theorem A. Coflict of Iterests (46) The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmets Hepig Wag is supported by the Natioal Natural Sciece Foudatio of Chia (Project o ), the Beijig Natural Sciece Foudatio (3200), ad BCMIIS. Yabo Zhag is supported by the Sciece ad Techology Pla Projects of Higher Istitutios i Shadog Provice (J3LI55). Refereces [] S. N. Berstei, Démostratio du théorème de Weierstrass fodée sur la calcul des probabilités, Commuicatios of the Mathematical Society of Charkow Séries 2,vol.3,pp. 2,92. [2] A.Lupas, Aq-aalogue of the Berstei operator, Uiversity of Cluj-Napoca semiar o umerical ad statistical calculus, 9, 987. [3] G. M. Phillips, Berstei polyomials based o the q-itegers, Aals of Numerical Mathematics, vol.4,o. 4,pp.5 58, 997. [4] S. Lewaowicz ad P. Woźy, Geeralized Berstei polyomials, BIT Numerical Mathematics, vol.44,o.,pp.63 78, [5] M.-M. Derrieic, Modified Berstei polyomials ad Jacobi polyomials i q-calculus, Redicoti del Circolo Matematico di Palermo Serie 2,supplemet76,pp ,2005. [6] A. Il iskii ad S. Ostrovska, Covergece of geeralized Berstei polyomials, JouralofApproximatioTheory, vol. 6, o., pp. 00 2, [7] S. Ostrovska, O the q-berstei polyomials, Advaced Studies i Cotemporary Mathematics,vol.,o.2,pp , [8] S. Ostrovska, The approximatio by q-berstei polyomials i the case q, Archiv der Mathematik, vol.86,o.3,pp , [9] S. Ostrovska, The first decade of the q-berstei polyomials: results ad perspectives, Joural of Mathematical Aalysis ad Approximatio Theory,vol.2,o.,pp.35 5,2007. [0] G. M. Phillips, Iterpolatio ad Approximatio by Polyomials, Spriger,NewYork,NY,USA,2003. [] G. M. Phillips, A survey of results o the q-berstei polyomials, IMA Joural of Numerical Aalysis,vol.30,o.,pp , 200. [2] H. Wag, Korovki-type theorem ad applicatio, Joural of Approximatio Theory,vol.32,o.2,pp ,2005. [3] H. Wag, Voroovskaya-type formulas ad saturatio of covergece for q-berstei polyomials for 0<q<, Joural of Approximatio Theory,vol.45,o.2,pp.82 95,2007. [4] H. Wag ad F. Meg, The rate of covergece of q-berstei polyomials for 0<q<, Joural of Approximatio Theory, vol. 36, o. 2, pp. 5 58, [5] H. Wag ad X. Wu, Saturatio of covergece for q-berstei polyomials i the case q, Joural of Mathematical Aalysis ad Applicatios,vol.337,o.,pp ,2008. [6] Z. Fita, Quatitative estimates for the Lupaş q-aalogue of the Berstei operator, Demostratio Mathematica, vol.44,o., pp.23 30,20. [7] L.-W.Ha,Y.Chu,ad Z.-Y.Qiu, Geeralized Bézier curves ad surfaces based o Lupaş q-aalogue of Berstei operator, Joural of Computatioal ad Applied Mathematics,vol.26,pp , 204. [8] N. I. Mahmudov ad P. Sabacigil, Voroovskaja type theorem for the Lupaş q-aalogue of the Berstei operators, Mathematical Commuicatios,vol.7,o.,pp.83 9,202. [9] S. Ostrovska, O the Lupas q-aalogue of the Berstei operator, The Rocky Moutai Joural of Mathematics,vol.36, o.5,pp ,2006. [20] S. Ostrovska, O the Lupaş q-trasform, Computers & Mathematics with Applicatios,vol.6,o.3,pp ,20. [2] S. Ostrovska, Aalytical properties of the Lupaş q-trasform, Joural of Mathematical Aalysis ad Applicatios,vol.394,o.,pp.77 85,202. [22] G. G. Loretz, Berstei Polyomials, Chelsea,NewYork,NY, USA, 986.

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