Research Article On the Rate of Convergence by Generalized Baskakov Operators

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1 Advaces Mathematcal Physcs Volume 25, Artcle ID , 6 pages Research Artcle O the Rate of Covergece by Geeralzed Basaov Operators Y Gao, Weshua Wag, 2 ad Shgag Yue 3 School of Mathematcs ad Iformato Scece, Befag Uversty of Natoaltes, Ychua, Ngxa 752, Cha 2 School of Mathematcs ad Computer Scece, Ngxa Uversty, Ychua, Ngxa 752, Cha 3 School of Computer Scece, Uversty of Lcol, Lcol LN6 7TS, UK Correspodece should be addressed to Weshua Wag; wws@xu.edu.c Receved 9 December 24; Accepted March 25 Academc Edtor: Hage Nedhardt Copyrght 25 Y Gao et al. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal wor s properly cted. We frstly costruct geeralzed Basaov operators V,α, (f; x) ad ther trucated sum B,α, (f; γ,x).secodly,westudythe potwse covergece ad the uform covergece of the operators V,α, (f; x), respectvely,adestmatethattherateof covergece by the operators V,α, (f; x) s / /2. Fally, we study the covergece by the trucated operators B,α, (f; γ,x)ad state that the fte trucated sum B,α, (f; γ,x)ca replace the operators V,α, (f; x) the computatoal pot of vew provded that lm γ =.. Itroducto Let N={,2,...}, N =N {}, R + =(,+),adr =R + {}. Forafxed N,wetroducetheweghtedfucto w o R by w (x) = {, =, { { ( + x ), N. Assocated wth the above weghted fucto, we also troduce the polyomal weghted space S of all real-valued cotuous fuctos f o R for whch w f s uformly cotuous ad bouded o R,adtheormoS s defed by the formula () f, = sup x R w (x) f (x). (2) Obvously, whe =, the the above orm s the ordary orm f.furthermore,forfxed N,letS be the set of all fuctos f S for whch w (x)f () (x) ( =,, 2,..., ) arecotuousadboudedor ad f () s uformly cotuous o R,wheref () (x) (,,2,...,) deote the th order dervatve of f o R. Let f be a fucto defed o R ;Basaov[] troduced the seuece of lear postve operators V (f; x) as follows: V (f; x) = b, (x) f ( ), (3) where b, (x) s called a Basaov operator s erel, whch s defed by b, (x) = (+) (+ ) x (+x). (4)! Based o the Basaov operators, may Basaov-type operators [2 3] ad ther multvarate Basaov operators [, 4 8] were dscussed. Partcularly, Gupta ad Agarwal studed the Basaov-Katorovch operators, Szász- Basaov operators, ad so forth ther recet boo [6]. Oe of the most famous Basaov-type operators s called geeralzed Basaov operators [9 22]. Oe has V,α (f; x) = f( ), (5)

2 2 Advaces Mathematcal Physcs where = (+α) (+( ) α)! x (+αx) /α, α >. Other modfed Basaov operators are defed as follows []: V, (f; x) = b, (x) f () (/)! (6) (x ), (7) x R, N. By combg the above operators (5) wth (7), we troduce the followg class of operators. Defto. For x R ad N,othergeeralzed Basaov-type operators are defed by V,α, (f; x) = f () (/)! (x ). (8) The actual costructo of Basaov operator ad ts varous modfcatos reures estmatos of fte seres whch a certa sese restrct ther usefuless from the computatoal pot of vew. A uesto aturally arses of whether the Basaov operators ca be replaced by a fte sum. I coecto wth ths uesto we costruct a ew famly of lear postve operators as follows: B,α, (f; γ,x)= [(x+γ )] f () (/)! (x ), where {γ } = saseueceofpostveumberssuchthat lm γ =ad [(x + γ )] deotes the tegral part of (x+γ ). Obvously, whe α=ad =,the operators(8) are (5), whle the operators (9) are degeerated as follows, whch arefrstlyproposedbywalcza[]: [(x+γ )] A (f; γ,x)= b, (x) f( ). () Ad whe α=,theoperators(8) are (7), whle the operators (9) caberepresetedby[2] F, (f; γ,x)= [(x+γ )] b, (x) f () (/)! (9) (x ). () Forthecoveeceofdscussotherestofpaper,we use the otato that K,α, (f; γ,x) deotes the remader term of operators V,α, (f; x) assocated wth the trucated sum B,α, (f; γ,x). Cosder K,α, (f; γ,x) = =[(x+γ )]+ f () (/)! (x ). (2) Ths paper focuses o covergece of the operators V,α, (f; x) ad ther trucated sum B,α, (f; γ,x). The rest of the paper s orgazed as follows. I Secto 2, we gve ma lemmas ad prove that the remader term K,α, (f; γ,x)oftheoperators V,α, (f; x) assocated wth the trucated sum B,α, (f; γ,x)s coverget to provded that lm γ =.ISecto 3, we state the potwse covergece ad the uform covergece of the operators V,α, (f; x) o the polyomal weghted space S,respectvely, whch dcate that the rate of covergece by the operators V,α, (f; x) s / /2.Fally,westudythecovergeceby the trucated operators B,α, (f; γ,x) ad state that the fte trucated sum B,α, (f; γ,x)ca replace the operators V,α, (f; x) the computatoal pot of vew. I ths paper, for better characterzg the degree of approxmato by the geeralzed Basaov operators V,α, (f; x), we troduce the classcal modulus of cotuty of a fucto f S, defed by [23] w (f; t) := sup f( + h) f( ), t R. (3) ht Here, we gve a mportat property of modulus of cotuty,whchwllbeusedtheproofoftheorem 6.Oehas 2. Ma Lemmas w(f;t)(+ t δ )w(f;δ), δ R. (4) I ths secto, we gve some propertes of the above operators, whch wll be used to prove the ma theorems. Lemma 2 (see [22]). If V,α (f; x) s defed by formula (5) the V,α (; x) =; V,α ((t x) 2 ;x)= V,α (t; x) =x; x (+αx). (5) From the frst eualty Lemma 2, forallf(x), x R, we have f(x) = V,α (f(x); x). Lemma 3 (see [9]). If V,α (f; x) s defed by formula (5),for fxed2 N,there exst m-order algebrac polyomals P,,α,, wth coeffcets depedg oly o,,α,such that [/2] V,α ((t x) P,,α (x) ;x)=, (6) where x R ad [/2] deotes the tegral part of /2. Moreover, V,α ((t x) 2m x (+αx) ;x)c( + m 2 ), m N. (7) Here ad the rest of the paper, C deotes a postve absolute costat, whose value may chage from le to le but s depedet of.

3 Advaces Mathematcal Physcs 3 For example, whe =4,wehavethefollowg4-order algebrac polyomal: So we have V,α ((t x) 4 ;x) =3[( α )2 +2( α )2 ]x [α +2(α )2 ]x (3 + α )x2 + 3 x. (8) K,α, (f; γ,x) C =[(x+γ )]+! ( + 2 ( x +x )) x For fxed x R +,obvously,wehave V,α ((t x) 4 ;x)=o x ( ). (9) 2 Furthermore, wth respect to the above weghted fucto w (x), the geeralzed Basaov operators (5) have the followg results, whch demostrate that the weghted fucto w (x) s also mportat to the geeralzed Basaov operators. Lemma 4 (see [5, 2]). If V,α (f; x) ad weghted fucto w (x) are defed by formula (5) ad (), respectvely,forx R +, the there exst postve absolute costats C,suchthat w (x) V,α ( w (t) ;x)c; w (x) V,α ( (t x)2 w (t) (+αx) ;x)cx. Now we wll gve the estmato of K,α, (f; γ,x). (2) Lemma 5. For f S, N, K,α,(f; γ,x)s defed by (2), the K,α, (f; γ,x) 2 x( + αx) C( + /2 2 ) Furthermore, oe has +2 x γ x( + αx) ( + (+)/2 2 ). (2) lm K,α, (f; γ,x)=. (22) C 2 =[(x+γ )]+ ( + 2 x ) C2 V,α ( t x ;x) =[(x+γ )]+ x =[(x+γ )]+ x. x ( + 2 x ) (24) Next, we estmate the sum of the last term, sce >(x+γ ) the last term; for,,...,,weremarthat =[(x+γ )]+ x γ < / x x γ γ < / x x γ V,α ( t x + ;x). + (25) Fally, usg Hölder eualty wth Lemmas 2 ad 3,weget the followg eualty: K,α, (f; γ,x) 2 C(V,α ((t x) 2 ;x)) /2 Proof. By assumpto f S,theresapostveabsolute costat C, suchthat f () (t) C( + t ), =,,...,. Wth the elemetary eualty (a + b) 2 (a +b ) for a, b R +, N,weget f() (t) C(+( t x +x) ) C(+2 ( t x +x )). (23) +2 x (V,α ((t x) 2(+) ;x)) /2 γ 2 x( + αx) C( + /2 2 ) +2 x γ x( + αx) ( + (+)/2 2 ). (26)

4 4 Advaces Mathematcal Physcs Fxg x R, there exst costats C(x) that maybe deped o x ad costats α, but are depedet of,suchthat K,α, (f; γ,x) 2 x( + αx) C( + /2 2 ) + C γ C (x) /2 ( + 2 x x (+αx) )( + (+)/2 2 ) C (x) + /2 γ, (27) ad otcg that lm γ =,thewecaget K,α, (f; γ, x) = o(),. 3. Ma Results I ths secto, we wll study the propertes of the operators V,α, (f; x) ad gve the estmato of degree of approxmato by these operators. Theorem 6. Fx N,foreveryf S ; the there exsts apostveabsolutecostatc,suchthat w 2+ (x) V,α,2+ (f; x) f (x) C (2 + )! [ +αx + +/2 ]ω(f (2+) ; (28) ), where C>s depedet oly o ad α but s depedet of x ad. Proof. By assumpto, usg the modfed Taylor formula [], f (x) = 2+ f (/) (x! ) + (x /)2+ (2)! ( t) 2 (f (2+) ( +t(x )) f(2+) ( )) dt, (29) wth Lemma 2 ad eualty (4),we get V,α,2+ (f; x) f (x) = (x /)2+ (2)! ( t) 2 (f (2+) ( +t(x )) f (2+) ( )) dt x / 2+ (2)! ( t) 2 f(2+) ( +t(x )) f (2+) ( ) dt x / 2+ (2)! ( t) 2 ω(f (2+) ; t(x ) )dt x / 2+ (2)! ( t) 2 ( + t x )ω(f(2+) ; )dt = ω(f(2+) ;/) V (2 + )!,α ( x t 2+ ;x) + ω(f(2+) ;/) B (2, 2 + ) V (2)!,α ((x t) 2+2 ;x), (3) where B(a, b) (a >, b > ) deotes the Beta fucto, B(2, 2+) = /((2+)(2+2)).UsgtheHölder eualty wth Lemmas 2 ad 3,wefurtherhave V,α,2+ (f; x) f (x) ω(f(2+) ;/) (V (2 + )!,α ((x t) 4+2 ;x)) /2 + ω(f(2+) ;/) V (2 + )!,α ((x t) 2+2 ;x) = ω(f(2+) ;/) (2 + )! (( 2+ = ω(f(2+) ;/) (2 + )! /2 P j,4+2,α (x) 4+2 j ) + ( 2+ ( P +/2 j,4+2,α (x)) + + P j,2+2,α (x)). + /2 P j,2+2,α (x) 2+2 j ) (3)

5 Advaces Mathematcal Physcs 5 Thus, we obta w 2+ (x) V,α,2+ (f; x) f (x) = ω(f(2+) ;/) (2 + )! ( 2+ ( P j,4+2,α (x) +/2 ( + x 2+ ) 2 ) + +P j,2+2,α (x) +x 2+ ). /2 (32) O the other had, for f S , smlar to the proof of Theorem 6,weget V,α,2+2 (f; x) f (x) 2 By Lemma 3,weobta f(2+2) (2 + 2)! w 2+2 (x) V,α,2+2 (f; x) f (x) V,α ((x t) 2+2 ;x). (37) Because P j,4+2,α (x) deotes a algebrac polyomal wth order at most 4+2, there exsts a postve absolute costat C, such that P j,4+2,α (x)/(+x 2+ ) 2 C, whle P j,2+2,α (x)/(+ x 2+ ) s a at most -order algebrac polyomal wth respect to x; that s, there exsts a postve absolute costat C depedg o α ad, such that + (P j,2+2,α(x)/( + x 2+ )) C( + αx). 2 f(2+2) + P j,2+2,α (x) (2 + 2)! 2+2 j +x 2+2 2C f(2+2) (2 + 2)! For all x R,wefurtherhave +. (38) Remar 7. The result of V,α,2+2 (f; x) cabeeaslyobtaed by mtatg Theorem 6; here we omt t because t wll be metoed the proof of ext theorem. Theorem 6 s to focus o the potwse approxmato of the operators V,α, (f; x); ow we wll study ther uform approxmato. Theorem 8. Fx N ;foreveryf S,oehas V,α,(f; ) f( ) =O(, ). (33)!/2 Proof. From the proof of Theorem 6,forf S 2+ 2+,wecaget V,α,2+2(f; ) f( ) 2C f(2+2), (2 + 2)! +. (39) Combg the above two eualtes (36) ad (39), for all f S ad fxed N, the desred eualty (33) s obtaed. Remar 9. Theorem 8 dcates that the rate of covergece by the operator V,α, (f; x) s / /2. Corollary. Let f S wth some N,forallx R + ; the lm V,α, (f; x) =f(x). (4) V,α,2+ (f; x) f (x) 2 f(2+) x / 2+ (2 + )! = 2 f(2+) V (2 + )!,α ( x t 2+ ;x). Usg the Hölder eualty wth Lemma 3,weobta w 2+ (x) V,α,2+ (f; x) f (x) 2 f(2+) (2 + )! 2C f(2+) (2 + )! 2+ ( +/2 +/2. P j,4+2,α (x) ( + x 2+ ) 2 ) /2 (34) (35) Fally, we wll dscuss the covergece of the trucated sum B,α, (f; γ,x). Theorem. Let f S wth some N,forfxedx R + ; the lm B,α, (f; γ,x)=f(x). (4) Moreover, the asserto (4) holds uformly o every rectagle x [a,b]wth <a<b. Proof. Notce that B,α, (f; γ,x) f(x) =V,α, (f; x) f (x) K,α, (f; γ,x). (42) Usg Corollary ad Lemma 5, weeaslygettheasserto (4). For all x R,wehave V,α,2+(f; ) f( ) 2C f(2+), (2 + )!. (36) +/2 Remar 2. Theorem demostrates that the geeralzed Basaov operators V,α, (f; x) cabereplacedbythetrucated operators B,α, (f; γ,x) a certa sese from the computatoal pot of vew.

6 6 Advaces Mathematcal Physcs Coflct of Iterests The authors declare that there s o coflct of terests regardg the publcato of ths paper. Acowledgmets Ths wor was supported by the Natoal Natural Scece Foudato of Cha (NSFC) uder Grats os ad 62643, by the EU FP7 Project EYE2E (2698), LIV- CODE (2955), by the Scece Research Project of Ngxa Hgher Educato Isttutos of Cha uder Grat o. NGY2447, ad part by the Scece Research Project of the State Ethc Affars Commsso of Cha uder Grat o. 4BFZ2. Refereces [] V. A. Basaov, A stace of a seuece of lear postve operators the space of cotuous fuctos, Dolady Aadem Nau SSSR, vol. 3, pp , 957. [2] M. Becer, Global approxmato theorems for Szasz- Mraja ad Basaov operators polyomal weght spaces, Idaa Uversty Mathematcs Joural, vol. 27, o., pp.27 42,978. [3] Z. Dtza, O global verse theorems of Szász ad Basaov operators, Caada Mathematcs, vol.3,o.2,pp , 979. [4]N.K.GovladV.Gupta, Covergecerateforgeeralzed Basaov type operators, Nolear Aalyss: Theory, Methods &Applcatos,vol.69,o.,pp ,28. [5] V. Gupta, A estmate o the covergece of Basaov-Bezer operators, Mathematcal Aalyss ad Applcatos, vol. 32, o., pp , 25. [6] V.GuptaadR.P.Agarwal,Covergece Estmates Approxmato Theory, Sprger, New Yor, NY, USA, 24. [7] V.Gupta,P.N.Agrawal,adA.R.Garola, Othetegrated Basaov type operators, Appled Mathematcs ad Computato,vol.23,o.2,pp ,29. [8] N. İspr, O modfed Basaov operators o weghted spaces, Tursh Mathematcs, vol. 25, o. 3, pp , 2. [9] P. Sabloère, Approxmato by Basaov uas-terpolats, [] Z. Walcza, O the rate of covergece for modfed Basaov operators, Lthuaa Mathematcal Joural, vol. 44, o., pp. 2 7, 24. [] Z. Walcza, Basaov type operators, The Rocy Mouta Mathematcs,vol.39,o.3,pp ,29. [2] Z. Walcza ad V. Gupta, A ote o the covergece of Basaov type operators, Appled Mathematcs ad Computato,vol.22,o.,pp ,28. [3] X.-M. Zeg ad V. Gupta, Rate of covergece of Basaov- Bezer type operators for locally bouded fuctos, Computers & Mathematcs wth Applcatos, vol. 44, o. -, pp , 22. [4]M.Gurde,L.Rempulsa,adM.Sorupa, TheBasaov operators for fuctos of two varables, Collectaea Mathematca,vol.5,o.3,pp ,999. [5] Y. Gao, The approxmato by a d of geeralzed Basaov operators of two varables o weghted space of polyomals, Hea Normal Uversty (Natural Scece), vol.39, o.5,pp.6 9,2. [6] Y. Gao, The approxmato propertes by a d of geeralzed Basaov-Katorovch operators, Ier Mogola Uversty (Natural Scece), vol. 44, o. 3, pp , 23. [7] J. Wag, H. Guo, ad J. Jg, Estmato of approxmato wth Jacob weghts by multvarate Basaov operator, Fucto Spaces ad Applcatos, vol.23,artcleid939565, 6pages,23. [8] S. K. Serebay, Ç.Ataut, ad İ. Büyüyazıcı, The geeralzed Basaov type operators, JouralofComputatoaladAppled Mathematcs,vol.259,pp ,24. [9] W. Z. Che, Approxmato Theory of Operators, Xame Uversty Publshg House, Xame, Cha, 989. [2] X. Q. Hou ad Y. C. Xue, Approxmato by geeralzed Basaov operators, Chese Quarterly Mathematcs, vol.5,o.,pp.5 2,99. [2] X. R. Zhag, O uform approxmato characterstcs wth weghts by geeralzed Basaov operators, Ngxa Uversty (Natural Scece),vol.7,o.4,pp.6 2,996. [22] Y. Gao ad Y. C. Xue, O the approxmato wth weghts for the geeralzed Basaov operators, JouralofNgxa Uversty (Natural Scece),vol.25,o.3,pp.96 2,24. [23] Z. Dtza ad V. Tot, Modul of Smoothess, vol. 9ofSprger Seres Computatoal Mathematcs, Sprger, New Yor, NY, USA, 987.

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