Appl Math If Sci, No 4, 55-556 6 55 Applied Mathematics & Iformatio Scieces A Iteratioal Joural http://ddoiorg/8576/amis/433 p, q-basaov-katorovich Operators Vijay Gupta Departmet of Mathematics, Netaji Subhas Istitute of Techology, Sector 3 Dwara, New Delhi-78, Idia Received: 4 Apr 6, Revised: 3 May 6, Accepted: Ju 6 Published olie: Jul 6 Abstract: I the last decade the applicatios of quatum calculus i the field of approimatio theory is a active area of research The -calculus is further etesio of q-calculus, which provides a ew directio for researchers I the preset article, we propose the p, q-variat of the Basaov-Katorovich operators, usig p, q-itegrals We estimate momets ad establish direct results, usig liear approimatig methods viz Stelov mea ad K-fuctioals i terms of modulus of cotiuity Also, i a weighted space, we obtai a direct estimate Keywords: p, q-basaov operators, Katorovich variat, direct estimates, Stelov mea, K-fuctioal, modulus of cotiuity, weighted approimatio Itroductio The quatum calculus q-calculus i the field of approimatio theory was discussed widely i the last two decades Several geeralizatios to the q variats were recetly preseted i the boos 5 ad related to covergece behaviours of differet operators Quatum calculus has may applicatios i special fuctios ad may other areas see, 6 Also, Araci et al studied o the fermioic p-adic q-itegral represetatio associated with weighted q-berstei ad q-geocchi polyomials Further there is possibility of etesio of the q-calculus to post-quatum calculus, amely the p, q-calculus Actually such etesio of quatum calculus ca ot be obtaied directly by substitutio of q by q/p i q-calculus The q-calculus may be obtaied by substitutig p i p, q-calculus Sahai ad Yadav 5 established some basic properties of p, q-calculus based o two parameters Recetly Mursalee et al 4 discussed some approimatio properties of p, q-berstei-stacu operators Very recetly the author defied p, q-szász-miraya-basaov operators ad established some approimatio results Some basic otatios ofp, q-calculus are metioed below: Thep, q-umbers are defied as : p p q p 3 q pq q p q p q Obviously, it may be see that p q/p, where q/p is the q-iteger i quatum-calculus give by q/p q/p q/p The-factorial is defied by!,,! Thep, q-biomial coefficiet is give by!!!, The-power basis is defied by a apqap q a p q a Defiitio The -derivative of the fuctio f is defied as D f f p f q, p q Correspodig author e-mail: vijaygupta@hotmailcom c 6 NSP Natural Scieces Publishig Cor
55 V Gupta: -Basaov-atorovich operators As a special case whe p, the p, q-derivative reduces to the q derivative The p, q-derivative fulfils the followig product rules: D fg fpd ggqd f D fg gpd f fqd g Defiitio Let f be a arbitrary fuctio ad a be a real umber The-itegral of f o,a is defied as a ad a f d q pa f d p qa p q f p q a q q p f p a if if p q < q p < I the year, Aral ad Gupta 3 proposed q-basaov operators, which was further eteded to Durrmeyer variat i by usig q-itegral The -aalogue of Basaov operators for, ad <q< p may be defied as where B, f, b, b p, f q, p / q / I case p, we get the q-basaov operators 3, 8 If p q, we get at oce the well ow Basaov operators Defiitio 3For,, < q < p the p, q-variat of Basaov-Katorovich operators are defied as K f, b, p q /q /q ftd t where b, is as defied i For the special case of the operators, oe may see 3 I the preset paper, we estimate the recurrece formula for momets of the p, q-basaov operators For p, q-basaov-katorovich operators we estimate direct results usig liear approimatig methods viz Stelov mea, K-fuctioals ad also obtai approimatio estimate i weighted space Momets First we estimate the followig Loretz type lemma for -Basaov basis, which will be used i the sequel Lemma For,, we have pd b, p q q b qp, q ProofBy simple computatio usig the defiitio of p, q-derivative, we have D p p,d Applyig product rule D fg fpd ggqd f, forp, q-derivative, we ca write D q p q p pp q Thus usig p q, we get pd pp q p q p q q q qp q p q Therefore, we have pd b, p q q b qp, q RemarWe may ote doe here that for the special case pq of the above lemma, we may capture at oce the Loretz type relatio of the Basaov operators, viz d d b, b,, where the Basaov basis is give by b, c 6 NSP Natural Scieces Publishig Cor
Appl Math If Sci, No 4, 55-556 6 / wwwaturalspublishigcom/jouralsasp 553 The momets of p, q-basaov operators, satisfy the followig: Lemma If we defie T,m : B,e m, b p m, q, where e i t i,i,,,, the for m, we have the followig recurrece relatio: T,m q qp pd T,m qt,m q I particular, we have ad B, e,,b, e, B, e, p pq ProofUsig Lemma, we have q pd T,m q pd b p, q p q q T p,m q qt p m p b p, q q,mq m This completes the proof of the recurrece relatio Obviously p, q-calculus may be related with the q-calculus ad we may write ad p q/p a p / a q/p Usig the defiitio of q-basaov operators see 3, 5, we get B, e, The other cosequeces follow from recurrece relatio Lemma 3For,, <q< p, we have K e, K e, qp 3K e, q pqp p q 3 3 q 3 p ProofBy, usig p q ad Lemma, we have K e, b, p q /q /q d t b, p q q q B, e, By, usig the idetity p q q p ad applyig Lemma, we have K e, b, p q q q b, p q q q b p q p q, p q b, q q B, e, qp B,e, qp Agai, usig the idetity p q q p ad by Lemma, we get K e, b, p q 3 b, 3 b, 3 b, 3 3 q 3 3 q 3 3 q q q p q pq q pqp q q 3 q pqp q q q p B pq,e, qp B, e, 3 3 B, e, q p p pq pq qp 3 3 q 3 p p q q pqp 3 3 c 6 NSP Natural Scieces Publishig Cor
554 V Gupta: -Basaov-atorovich operators 3 Direct Estimates By C B, we deote the class of all real valued cotiuous ad bouded fuctios f o, The orm CB is defied as f CB sup f, For f C B, the Stelov mea is defied as f h t 4 h h h f t uv f t uvdudv 3 By simple computatio, it is observed that i f h f CB ω f,h ii If f is cotiuous ad f h, f C B the f h CB 5 h ω f,h, f CB 9 h ω f,h, where the first ad secod order modulus of cotiuity for δ are respectively defied as ad ω f,δ sup f u f v,u,v u v δ ω f,δ sup f u fuv f v,u,v u v δ Theorem Let q, ad p q, The operator K maps space C B ito C B ad K f CB f CB ProofLet q, ad p q, From Lemma 3, we have K f, b, p q /q /q b h ft d t, p q /q d t sup f, sup f K, f CB, /q Theorem Let q, ad p q, If f C B, the K f, f 5 ω f, qp 9 ω f, q 3 p qp p q q pqp 3 3 ProofFor ad N ad usig the Stelov fuctio f h defied by 3, we ca write K f, f K f f h, K f h f h, f h f First by Theorem ad property i of Stelov mea, we have K f f h, K f f h CB f f h CB ω f,h Also, by Taylor s epasio, we have K f h f h, f h K t, By Lemma 3, we have K f h f h, 5 ω f,h h where K t, q 3 p f CB K 9 h ω f,hk p q 3 q 3 p qp p q q pqp 3 3 t, qp t, q pqp 3 qp, for, h > Settig h, we get the desired result A differet form to obtai the direct result is the applicatios of K-fuctioal The Peetre s K fuctioal is defied by K f,δ if g W { f g δ g }, where W {g C B, : g,g C B,} By 7, p 77, Theorem 4, there eists a positive costat C > such that K f,δ Cω f, δ,δ >, where ω f, δ sup <h< δ,, fh fh f is the secod order modulus of cotiuity of fuctio f C B, Also, for f C B, the first order modulus of cotiuity is give by ω f, δ sup <h< fh f δ,, c 6 NSP Natural Scieces Publishig Cor
Appl Math If Sci, No 4, 55-556 6 / wwwaturalspublishigcom/jouralsasp 555 Theorem 3Let f C B, The for all N, there eists a absolute costat C > such that where ad K f, f Cω f,δ ω f,α, δ { K t,k t, } / α qp ProofFor,, we cosider the auiliary operators K f, defied by K f,k f, f f qp It is observed that K f, preserve liear fuctios Let, ad g W Applyig the Taylor s formula we have K K K t gtgg t t ug udu, g, g t t ug udu, t t ug udu, qp K t t ug udu, qp O the other had, t ad t ug udu g qp qp u g udu qp u g udu qp t t u du t g, qp u g udu g Therefore, we have K g, g t K t ug udu, qp g K t, δ g Also, we have K Therefore, K f, f K qp u qp f, K f, f 3 f f g, f g f qp f K g, g K f g, f g f qp f K g, g 4 f g ω f, qp δ g g udu g Fially taig the ifimum o the right-had side over all g W, we get K f, f 4K f,δω f,α By the property of K fuctioal, we have K f, f Cω f,δ ω f,α This completes the proof of the theorem 4 Weighted Approimatio We cosider the followig class of fuctios: Let H, be the set of all fuctios f defied o, satisfyig f M f, where M f is certai costat depedig oly o f By C,, we deote the subspace of all cotiuous fuctios belogig to H, Also, let C, be the subspace of all fuctios f C,, for which f is fiite The orm o C, is f sup f, Fially, we discuss the weighted approimatio theorem, where the approimatio formula holds true o the iterval, Theorem 4Let p p ad qq satisfies <q < p ad for sufficietly large p, q ad q ad p For each f C,, we have c 6 NSP Natural Scieces Publishig Cor
556 V Gupta: -Basaov-atorovich operators Kp,q f f ProofUsig the methods of 9, i order to complete the proof of theorem, it is sufficiet to verify the followig three coditios K p,q e ν, ν, ν,, 4 Sice K p,q e, the first coditio of 4 is fulfilled for ν We ca write K p,q e, q p p,q p,q q p ad K p,q e, p,q p,q q 3 p 3 p,q p,q which implies that p sup, p q p q p,q q 3 p,q sup, K p,q e ν, ν,ν, Thus the proof is completed RemarFor q, ad p q, it is see that /p q I order to obtai covergece estimates of p, q-basaov-katorovich operators, we assume pp, qq such that <q < p ad for sufficietly large p, q ad p ad q ad p,q 5 Coclusio By cosiderig the p, q-variat of the Basaov-Katorovich operators, we may have better results for suitable choices of p ad q Also, for special case p q of our operators, we capture the approimatio properties of the usual Basaov-Katorovich operators Oe may cosider the other form of p, q-basaov-katorovich operators by etedig the results of 4 to p, q settig, as the aalysis is differet, we may discuss it elsewhere S Araci, M Acigoz ad H Jolay, O the families of q- Euler polyomials ad their applicatios, J Egyptia Math Soc 3, -5 5 3 A Aral ad V Gupta, Geeralized q Basaov operators, Math Slovaca 64, 69-634 4 A Aral ad V Gupta, O q-basaov type operators, Demostratio Mathematica 4, 9-9 5 A Aral, V Gupta ad R P Agarwal, Applicatios of q Calculus i Operator Theory, Spriger 3 6 G Bagerezao, Variatioal q-calculus, J Math Aal Appl 89, 65-665 4 7 R A DeVore ad G G Loretz, Costrutive Approimatio, Spriger, Berli, 993 8 Z Fita ad V Gupta, Approimatio properties of q- Basaov operators, Cet Eur J Math 8, 99-9 A D Gadzhiev, Theorems of the type of P P Korovi type theorems, Math Zameti, 5,78-786 976 ; Eglish Traslatio, Math Notes 5-6, 996-998 976 V Gupta, -Szász-Miraya-Basaov operators, Comple Aalysis ad Operator Theory, to appear V Gupta ad R P Agarwal, Covergece Estimates i Approimatio Theory, Spriger 4 V Gupta ad A Aral, Some approimatio properties of q-basaov Durrmeyer operators, Appl Math Comput 8 3, 783-788 3 V Gupta ad C Radu, Statistical approimatio properties of q-basaov-katorovich operators, Cet Eur J Math 7 4, 89-88 9 4 M Mursalee, K J Asari ad A, Kha, Some approimatio results by p, q-aalogue of BersteiStacu operators, Appl Math Comput 64, 39-4 5 5 V Sahai ad S Yadav, Represetatios of two parameter quatum algebras ad p, q-special fuctios, J Math Aal Appl 335, 68-797 Vijay Gupta is professor at the Departmet of Mathematics, Netaji Subhas Istitute of Techology, New Delhi, Idia He obtaied his PhD degree from Uiversity of Rooree ow IIT Rooree, i 99 His area of research is Approimatio theory, especially o liear positive operators ad he is the author of two boos, boo chapters ad over 5 research papers to his credit Curretly, he is actively associated editorially with several iteratioal scietific research jourals Refereces S Araci, D Erdal ad J Seo, A study o the fermioic p-adic q-itegral represetatio o Z p associated with weighted q- Berstei ad q-geocchi polyomials, Abstract ad Applied Aalysis 78, pages c 6 NSP Natural Scieces Publishig Cor