Bivariate generalization of q-bernstein-kantorovich type operator

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1 Sharma & Mishra Coget Mathematics 06 3: PURE MATHEMATICS RESEARCH ARTICLE Bivariate geeralizatio of -Berstei-Katorovich type operator Preeti Sharma * ad Vishu Naraya Mishra 3 Received: 3 October 05 Accepted: 6 February 06 Published: 5 March 06 *Correspodig author: Preeti Sharma Departmet of Applied Mathematics & Humaities Sardar Vallabhbhai Natioal Istitute of Techology Ichchhaath Mahadev Dumas Road Surat Gujarat Idia; Departmet of Applied Mathematics & Statistics Baasthali Uiversity Tok Rajastha Idia preeti.iita@gmail.com Reviewig editor: Hari M. Srivastava Uiversity of Victoria Caada Additioal iformatio is available at the ed of the article Abstract: I this paper we itroduce a geeralizatio of the Katorovich-type Berstei operators based o -itegers ad get a Bohma Korovki-type approximatio theorem of these operators. We also compute the rate of covergece usig the first modulus of smoothess. Subjects: Advaced Mathematics; Aalysis - Mathematics; Mathematics & Statistics; Operator Theory; Sciece Keywords: -aalysis; -itegral operator; positive liear operators; -Berstei operators; modulus of cotiuity 000 MSC: 4A5; 4A36; 6A5; 6A6. Itroductio Durig the last two decades the applicatios of -calculus emerged as a ew area i the field of approximatio theory. The rapid developmet of -calculus has led to the discovery of various geeralizatios of Berstei polyomials ivolvig -itegers. Lupaş 987 itroduced the first -aalogue of Berstei operators Berstei Berstei ad ivestigated its approximatig ad shape preservig properties. Aother -geeralizatio of the classical Berstei polyomials is due to Phillips 997. Several geeralizatios of well-kow positive liear operators based o -itegers were itroduced ad their approximatio properties have bee studied by several authors. Vishu Naraya Mishra ABOUT THE AUTHORS Preeti Sharma is a assistat professor at Baasthali Uiversity ad pursuig PhD i Mathematics from SVNIT Surat uder the supervisio of Dr Vishu Naraya Mishra. She received the MSc degree i Mathematics & Computig from Idia Istitute of Techology Guwahati i 0. Her research iterests are i the areas of Approximatio Theory ad Operator Theory. Vishu Naraya Mishra received the PhD i Mathematics from Idia Istitute of Techology Roorkee. His research iterests are i the areas of pure ad applied mathematics. He has published more tha 00 research articles i reputed iteratioal jourals of mathematical ad egieerig scieces. He is a referee ad a editor of several iteratioal jourals i frame of Mathematics. He guided may postgraduate ad PhD studets. Citatios of his research cotributios ca be foud i may books ad moographs PhD thesis ad scietific joural articles. PUBLIC INTEREST STATEMENT The approximatio of fuctios by positive liear operators is a sigificat research area i mathematical aalysis with key relevace to studies of computer aided geometric desig umerical aalysis solutio of differetial euatios etc. I this work we itroduce a geeralizatio of the Katorovich-type Berstei operators based o -itegers ad prove the basic covergece of the itroduced operators ad also obtai the rate of covergece i terms of modulus of cotiuity. Further we study the local approximatio property usig the modulus of cotiuity. 06 The Authors. This ope access article is distributed uder a Creative Commos Attributio CC-BY 4.0 licese. Page of 9

2 Sharma & Mishra Coget Mathematics 06 3: For each positive iteger Philips 997 defied -Berstei polyomials as B f ; x = [ [k] f [] k k=0 ] s x x k k s=0. Whe = B f ; x is the classical Berstei polyomial B f x = k f k k=0 x k x k.. Katorovich Loretz 953 modified the Berstei operators ad defied the liear positive operators K : L [0 ] C[0 ] defied for ay f L [0 ] by k K f ; x = p k x f udu.3 k k=0 where p k x = x k x k. These operators are kow as Katorovich operators i literature. Radu 008 has obtaied the statistical covergece of -Berstei Katorovich polyomials. k Also the Katorovich-type geeralizatios of the liear positive operators based o -itegers were studied by some authors see e.g. Mahmudov Mahmudov00; Mishra Khatri & Mishra 0; Mishra Khatri Mishra & Deepmala 03; Mishra Sharma Kiliçma & Jai i press; Mishra Sharma & Mishra 05; Mursalee Kha Srivastava & Nisar 03; Srivastava 0; Srivastava & Choi 0. Gairola Deepmala ad Mishra i press Wafi Rao ad Deepmala 06 studied rate of approximatio ad some approximatio properties of liear positive operators usig uatum calculus approach. Recetly Agrawal Fita ad Kumar 05a itroduced a ew Katorovich-type geeralizatio of the -Berstei Schurer operators I 007 Dalmaoglu 007 defied Katorovich-type -Berstei Operator as follows: B f ; x =[ ] [ k k k=0 ] x k k s=0 s x [k] [] [k] [] f td t..4 Before proceedig further we recall certai otatios of -calculus as follows. Such otatios ca be foud i Erst 000 Kac ad Cheug 00. We cosider as a real umber satisfyig 0 < <. For [] = { = ad { [] [ ] []!= [ ] [] = = 0. The for > 0 ad itegers k k 0 we have [ ] = [] ad [] [k ] =[k]. We observe that x = x; = { x x x x = = 0. Also for ay real umber α we have Page of 9

3 Sharma & Mishra Coget Mathematics 06 3: x α = x α x. I special case whe α is a whole umber this defiitio coicides with the above defiitio. a 0 ad The -Jackso itegral ad -improper itegral i the iterval [0 a] defied as f xd x = a A 0 provided sum coverges absolutely. f a 0 < <. =0 f xd x = a f A A =0 Let 0 a < b ad 0 < <. Followig Mariković Rajković ad Staković 008 we cosider the Riema-type -itegral defied as follows b a f td R t = b a f a b a i i. i=0 This Riema-type -itegral is appropriate to derive the -aalogues of some well-kow itegral ieualities. The the Riema-type -itegral for a bivariate fuctio is give by d b c a f t sd R td R s = b ac d f a b a i c c dj i j. j=0 i=0.5 where 0 a < b0 c < d ad 0 < 0 <. Also f is a R -itegrable fuctio so the series i.5 coverges. The bivariate case for the operators are first itroduced by Stacu 969. He studied the bivariate Berstei polyomials ad estimated the order of approximatio for these operators. The aim of this paper was to costruct bivariate -Berstei Katorovich operators ad ivestigate Korovki-type approximatio properties ad estimate the order of approximatio i terms of a modulus of cotiuity.. The costructio of the bivariate operators of Katorovich type The aim of this part was to costruct the bivariate extesio of the operator.. Let I =[0 p] where p {0 } ad j =[0 ]. For I = I I let CI deote the space of all real-valued cotiuous fuctios o I edowed with the orm f I = sup f x y. Aalogously for J = J J we deote by f xy I J = sup f x y the sup-orm o J. xy J If f CI ad 0 < let us defie the bivariate geeralizatio of operator.4 as follows: f ; = ] [ ] k k k =0 k =0 [k ] [k ] b k k x y. [ ] [ ] [k ] [k ] [ ] ] f t sdr td R s Page 3 of 9

4 Sharma & Mishra Coget Mathematics 06 3: where x y J ad [ ] b x y = k k k [ k ] k x yk x k x k. I 003 Erkuş ad Duma DU proved the statistical Korovki-type approximatio theorem for the bivariate liear positive operators to the fuctios i space H ω. I 009 Ersa ad Doğru 009 obtaied the statistical Korovki-type theorem ad lemma for the bivariate liear positive operators defied i the space H ω as follows Theorem Ersa & Doğru 009 Let be the seuece of liear positive operator actig from H ω R ito C R where R =[0. The for ay f H B ω st lim f f = 0. Lemma The bivariate operators defied i Ersa 007 satisfy the followig : i e 00 ; = ii e 0 ; = iii e 0 ; = [ ] x ] x [ ] y [ ] y iv e 0 ; = ] ] ] x x x ] ] x x v [ ] [ ] e 0 ; = y [ ] y y I order to obtai the covergece properties of bivariate operator. we eed the followig lemma. Lemma Let e ij = x i y j x y I i j {0 } {0 } with i j be the two-dimesioal test fuctios. The we have i e 00 ; = ii e 0 ; = iii e 0 ; = [ ] [ ] y y. ] ] [] x [ ] [ ] [] y [] ] [] [ ] Page 4 of 9

5 Sharma & Mishra Coget Mathematics 06 3: iv 4 [ ] ] e 0 ; = x [] [3] ] v 4 [ ][ ] e 0 ; = y [] [3] [ ] Proof The proof ca be obtaied similar to the proof of bivariate operator i Agrawal Fita ad Kumar 05b. So we shall omit this proof. Lemma 3 For the operator. we have i ] [] [3] ] x 4 [ ] [] [3] [ ] y μ x = t x; ] = x [] ] ] [] ] [3] [ ] [3]. ii iii iv μ y = s y; [ ] = y [] [ ] [ ] [] δ x = t x ; 4 [ ] ] = [ ] 4 [] [3] ] [ ] [] [ ] [] [3] ] ] [] δ y = s y ; 4 [ ][ ] = [ ] 4 [] [3] [ ] [ ] [] [ ] [] [3] [ ] [ ] [] x x ] [3] y y. [ ] [3] 3. Rates of covergece of bivariate operators Let K =[0 [0. The the sup orm o C B K is give by f = sup f x y f C B K. xy K We cosider the modulus of cotiuity ωf ; δ δ where δ δ > 0 for bivariate case give by ωf ; δ δ ={sup f x y fx y : x y x y K ad x x δ y y δ }. 3. Page 5 of 9

6 Sharma & Mishra Coget Mathematics 06 3: It is clear that a ecessary ad sufficiet coditio for a fuctio f C B K is lim ωf ; δ δ =0 δ δ 0 ad ωf ; δ δ satisfy the followig coditio: f x y fx y ωf ; δ δ x x y y δ δ 3. for each f C B K. The observe that ay fuctio i C B K is cotiuous ad bouded o K. The details of the modulus of cotiuity for bivariate case ca be foud i Aastassiou ad Gal 000. Now the rate of statistical covergece of bivariate operator. by meas of modulus of cotiuity i f C B K will be give i the followig theorem. Theorem Let 0 such that as ad as. So we have f ; f x y 4ω f ; δ x δ y 3.3 where δ x ad δ y defied as i Lemma 3. Proof By usig the coditio i 3. for δ δ > 0 ad N we get f ; f x y f x y fx y ; ωf ; δ x δ y ; δ x x ; ; δ y y ; If the Cauchy Schwarz ieuality is applied we have x x ; x x ; ;. So if it is substituted i the above euatio the proof is completed. The ext theorem represets the rate of statistical covergece of bivariate operator. by meas of Lipschitz Lip M α α fuctios for the bivariate case where f C B [0 ad M > 0 ad 0 <α 0 <α the let us defie Lip M α α as f x y f x y M x x α y y α ; x x y y [0. We have the followig theorem. Theorem 3 Let = ad = be seuece satisfyig as ad as ad let Lip M α α x 0 ad 0 <α 0 <α. The f ; f x y M δ α x δ α y 3.4 where δ x ad δ y are defied i Theorem. Page 6 of 9

7 Sharma & Mishra Coget Mathematics 06 3: Proof Sice f ; are liear positive operators ad f Lip M α α x 0 ad 0 <α 0 <α we ca write f ; f x y f x y f x y ; M x x α y y α ; = M x x α ; y y α ;. If we take p = α p = α = α = α applyig Hölder s ieuality we obtai f ; f x y x x α ; x y α α ; y y α ; x y α α ; = Mδ α xδ α y. which is the reuired result. I what follows we shall use the followig otatios: C I ={f CI : f x f y CI } ad C I ={f CI : f xx f xy f yx f yy CI } respectively. We have the ext result. Theorem 4 Let f C I x y J ad 0 such that as ad as. The we have f ; f x y f x I where δ x ad δ y are defied i Lemma 3. δ x f y I δ y 3.5 Proof Let x y J be fixed poit. The we ca write for t s I that t f t s f x y = f u sdu u x s y f v x vdv. Now applyig the operator defied by. o both sides ad Lemma i we obtai t s f ; f x y f u sdu u ; x y f x vdv v ; x y Sice t f u sdu x u f t x ad x I s f x vdv y v f y I s y we have x y f ; f x y f x I t x ; f y I s y ; Now applyig the Cauchy Schwarz ieuality f ; f x y f x I { t x ; } { ; } f { y I s y ; } { ; } f x I δ x f y I δ y. This completes the proof of the theorem. Page 7 of 9

8 Sharma & Mishra Coget Mathematics 06 3: Ackowledgemets The authors would like to express their deep gratitude to the aoymous leared referees ad the editor for their valuable suggestios ad costructive commets which resulted i the subseuet improvemet of this research article. Fudig The first author Preeti Sharma ackowledges the MHRD New Delhi Idia for supportig this research article. The secod author Vishu Naraya Mishra ackowledges that this project was supported by the Cumulative Professioal Developmet AllowaceCPDA SVNIT SuratGujarat Idia. Author details Preeti Sharma preeti.iita@gmail.com ORCID ID: Vishu Naraya Mishra 3 vishuarayamishra@gmail.com ORCID ID: Departmet of Applied Mathematics & Humaities Sardar Vallabhbhai Natioal Istitute of Techology Ichchhaath Mahadev Dumas Road Surat Gujarat Idia. Departmet of Applied Mathematics & Statistics Baasthali Uiversity Tok Rajastha Idia. 3 L. 67 Awadh Puri Coloy Beigaj Phase -III Opposite - Idustrial Traiig Istitute I.T.I. Ayodhya Mai Road Faizabad 4 00 Uttar Pradesh Idia. Citatio iformatio Cite this article as: Bivariate geeralizatio of -Berstei- Katorovich type operator Preeti Sharma & Vishu Naraya Mishra Coget Mathematics 06 3: Refereces Agrawal P. N. Fita Z. & Kumar A. S. 05a. Berstei Schurer Katorovich operators based o -itegers. Applied Mathematics ad Computatio Agrawal P. N. Fita Z. & Kumar A. S. 05b. Bivariate -Berstei Schurer Katorovich pperators. Results i Mathematics Aastassiou G. A. & Gal S. G Approximatio theory moduli of cotiuity ad global smoothess preservatio. Bosto MA: Birkhăuser. Berstei S. N Démostratio du théoréme de Weierstrass fodée sur le calcul de probabilités. Commuicatios of the Kharkov Mathematical Society 3. Dalmaoglu Ŏ 007. Approximatio by Katorovich type -Berstei operators. Proceedigs of the th WSEAS Iteratioal Coferece o Applied Mathematics. Cairo: pp. 3 7.ISSN Erkuş E. & Duma O A-statistical extesio of the Korovki type approximatio theorem. Proceedigs of the Idia Academy of Scieces Erst T The history of -calculus ad a ew method U.U.D.M. Report 000:6. Departmet of Mathematics Uppsala Uiversity. Ersa S Approximatio properties of bivariate geeralizatio of Bleima Butzer ad Hah operators based o the -itegers. Proceedigs of the th WSEAS Iteratioal Coferece o Applied Mathematics pp. 7. Cairo. Ersa S. & Doğru O Statistical approximatio properties of -Bleima Butzer ad Hah operators. Mathematical ad Computer Modellig Gairola A. R. Deepmala & Mishra L. N. i press. Rate of approximatio by fiite iterates of -Durrmeyer operators. Proceedigs of the Natioal Academy of Scieces Idia Sectio A: Physical Scieces. doi:0.007/ s z Kac V. G. & Cheug P. 00. Quatum calculus. New York NY: Uiversitext Sprige-Verlag. Loretz G. G Berstei polyomials. Toroto: Uiversity of Toroto Press. Lupaş A A -aalogue of the Berstei operator. Babeşs-Bolyai Uiversity Semiar o Numerical ad Statistical Calculus Mahmudov N. I. 00. Statistical approximatio properties of Baskakov ad Baskakov Katorovich operators based o the -itegers. Cetral Europea Joural of Mathematics MarikoviĆ S. RajkoviĆ P. & StakoviĆ M The ieualities for some types of -itegrals. Computers & Mathematics with Applicatios Mishra V. N. Kha H. H. Khatri K. & Mishra L. N. 03. Hypergeometric represetatio for Baskakov Durrmeyer- -Stacu type operators. Bulleti of Mathematical Aalysis ad Applicatios Mishra V. N. Khatri K. Mishra L. N. & Deepmala 03. Iverse result i simultaeous approximatio by Baskakov Durrmeyer Stacu operators. Joural of Ieualities ad Applicatios p. Mishra V. N. Khatri K. & Mishra L. N. 0. O simultaeous approximatio for Baskakov Durrmeyer--Stacu type operators. Joural of Ultra Scietist of Physical Scieces Mishra V. N. Sharma P. Kiliçma A. & Jai D. i press. Statistical approximatio properties of Stacu type -Baskakov Katorovich operators. Filomat. Mishra V. N. Sharma P. & Mishra L. N. 05. O statistical approximatio properties of -Baskakov Szász Stacu operators. Joural of Egyptia Mathematical Society 6. doi:0.06/j.joems Mursalee M. Kha A. Srivastava H. M. & Nisar K. S. 03. Operators costructed by meas of -Lagrage polyomials ad A-statistical approximatio. Applied Mathematics ad Computatio Phillips G. M Berstei polyomials based o the -itegers. Aals of Numerical Mathematics Radu C Statistical approximatio properties of Katorovich operators based o -itegers. Creative Mathematics ad Iformatics Srivastava H. M. 0. Some geeralizatios ad basic or - extesios of the Beroulli Euler ad Geocchi polyomials. Applied Mathematics & Iformatio Scieces Srivastava H. M. & Choi J. 0. Zeta ad -Zeta fuctios ad associated series ad itegrals. Amsterdam: Elsevier Sciece. Stacu D. D A ew class of uiform approximatig polyomial operators i two ad several variables. I Proceedigs ad Coferece o Costructive Theory of Fuctios pp Budapest. Wafi A. Rao N. & Deepmala 06. Approximatio properties by geeralized-baskakov Katorovich tacu type operators. Applied Mathematics & Iformatio Scieces Page 8 of 9

9 Sharma & Mishra Coget Mathematics 06 3: The Authors. This ope access article is distributed uder a Creative Commos Attributio CC-BY 4.0 licese. You are free to: Share copy ad redistribute the material i ay medium or format Adapt remix trasform ad build upo the material for ay purpose eve commercially. The licesor caot revoke these freedoms as log as you follow the licese terms. Uder the followig terms: Attributio You must give appropriate credit provide a lik to the licese ad idicate if chages were made. You may do so i ay reasoable maer but ot i ay way that suggests the licesor edorses you or your use. No additioal restrictios You may ot apply legal terms or techological measures that legally restrict others from doig aythig the licese permits. Coget Mathematics ISSN: is published by Coget OA part of Taylor & Fracis Group. Publishig with Coget OA esures: Immediate uiversal access to your article o publicatio High visibility ad discoverability via the Coget OA website as well as Taylor & Fracis Olie Dowload ad citatio statistics for your article Rapid olie publicatio Iput from ad dialog with expert editors ad editorial boards Retetio of full copyright of your article Guarateed legacy preservatio of your article Discouts ad waivers for authors i developig regios Submit your mauscript to a Coget OA joural at Page 9 of 9