(p, q)-baskakov-kantorovich Operators

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1 Appl Math If Sci, No 4, Applied Mathematics & Iformatio Scieces A Iteratioal Joural 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 vijaygupta@hotmailcom c 6 NSP Natural Scieces Publishig Cor

2 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

3 Appl Math If Sci, No 4, / 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

4 554 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

5 Appl Math If Sci, No 4, / 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

6 556 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, A Aral ad V Gupta, Geeralized q Basaov operators, Math Slovaca 64, A Aral ad V Gupta, O q-basaov type operators, Demostratio Mathematica 4, 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, R A DeVore ad G G Loretz, Costrutive Approimatio, Spriger, Berli, 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, ; Eglish Traslatio, Math Notes 5-6, 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, V Gupta ad C Radu, Statistical approimatio properties of q-basaov-katorovich operators, Cet Eur J Math 7 4, M Mursalee, K J Asari ad A, Kha, Some approimatio results by p, q-aalogue of BersteiStacu operators, Appl Math Comput 64, V Sahai ad S Yadav, Represetatios of two parameter quatum algebras ad p, q-special fuctios, J Math Aal Appl 335, 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