q-durrmeyer operators based on Pólya distribution

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1 Available olie at wwwtjsacom J Noliear Sci Appl Research Article -Durrmeyer operators based o Pólya distributio Vijay Gupta a Themistocles M Rassias b Hoey Sharma c a Departmet of Mathematics Netaji Subhas Istitute of Techology Sector 3 Dwarka New Delhi-0078 Idia b Departmet of Mathematics Natioal Techical Uiversity of Athes Zografou Campus Athes Greece c Departmet of Applied Scieces Gulzar Group of Istitutes Khaa Ludhiaa Pujab Idia Commuicated by Y J Cho Abstract We itroduce a aalogue of Durrmeyer type modificatio of Berstei operators based o Pólya distributios We study the ordiary approimatio properties of operators usig modulus of cotiuity ad Peetre K-fuctioal of secod order Further we establish the weighted approimatio properties for these operators c 206 All rights reserved Keywords: Pólya distributio -itegers -Berstei operators modulus of cotiuity Peetre K-fuctioal weighted modulus of cotiuity 200 MSC: 4A25 4A35 Itroductio The uatum calculus -Calculus is a importat brach of mathematics ad physics I the theory of approimatio for liear positive operators -calculus was applied first by Lupaş [2] who itroduced the -aalogue of the classical Berstei polyomials I 997 Phillips [5] proposed a ew aalogue of the Berstei polyomials which became popular amogst researchers Recetly Aral Gupta ad Agarwal [] preseted some results o covergece of several differet operators We use the otatios of calculus as metioed i [] Also Nowak [3] itroduced the -variat of the Lupaş operators which is based o Pólya distributio For f C[0 ] 0 < < ad α 0 the operators cosidered i [3] are defied by f; = f[k] /[] p α k [0 ] Correspodig author addresses: vijaygupta200@hotmailcom Vijay Gupta trassias@mathtuagr Themistocles M Rassias prosharmah@gmailcom Hoey Sharma Received

2 V Gupta T M Rassias H Sharma J Noliear Sci Appl where p α k = [ K ] k i=0 [i] α k j=0 j [j] α l=0 [l] α Note that empty product i the above basis fuctio p α k is deoted by Nowak i [3] ad Nowak ad Gupta i [4] estimated the momets ad established some direct estimates for the operators I the year 2008 Gupta [6] itroduced -Durrmeyer operators Some other results ad forms of - Durrmeyer type operators were discussed i [ ] ad [8] etc We ow itroduce the -aalogue of Lupaş Durrmeyer operators for f C[0 ] ad 0 < < by where ad f; = [ ] [ p /[] k = k ] p k t = [ k k p /[] k k i=0 ] 0 p k tftd t [0 ] 2 [i] [] k i=0 [i] [i] i=0 [] k k t k j t As a special case whe we get the operators due to Gupta ad Rassias [9] j=0 [] 2 Lemmas Lemma 2 For e i = t i i = the momets of -Lupaş operators based o Pólya distributio are where e 0 ; = e ; = e 2 ; = α e 3 ; = e 4 ; = W 0 α; = α [2] α W α; = α2 W 2 α; = [2] α 2 i=0 [i] α 3 i=0 [i] α W 0 α; = α [2] α [3] α 2 3 W α; = α [2] α [] W k α; [] k W k α; [] k W 2 α; = α α [2] 2 α [2] [3] α W 3 α; = α[2] [3] 2 α

3 V Gupta T M Rassias H Sharma J Noliear Sci Appl Lemma 22 For all [0 ] N ad 0 we have We have e 0 ; = e ; = [] e 2 ; = e 3 ; = e 4 ; = [] 3 3 [] [ 3] 2 [] [ 3] /[] [ 3] a [] 3 [ 3] [ 4] 2 i=0 [i] /[] a 2 [] 3 [] [ 3] [ 4] a 3 [] [ 3] [ 4] [] 2 W k /[] ; [] k [] /[] a 4 [ 3] [ 4] b [] 4 [ 3] [ 4] [ 5] 3 i=0 [i] /[] b 2 [] 3 [ 3] [ 4] [ 5] 2 i=0 [i] /[] b 3 [] 3 [] [ 3] [ 4] [ 5] b 4 [] [ 3] [ 4] [ 5] 0 p k tts d t = k [s k]![]! [s ]![k]! 3 /[] W k /[] ; [] k 2 W k /[] ; [] k [] a 4 [ 3] [ 4] [ 5] usig the methods described i [6] ad Lemma 2 The proof of the lemma follows immediately ad thus we omit the details Corollary 23 For cetral momets deoted by φ m = D /[] t m m = 2 we have φ = [] φ 2 = [] [] 2 3 [] [ 3] 2[] 2[] 2 3 [] [ 3] 2 [] [ 3] [ 3] Lemma 24 For 0 ad > 3 we have where δ 2 = 2 2 φ 2 = 3 δ [ 2] 2 2 ϕ 2 [2] ad ϕ 2 =

4 V Gupta T M Rassias H Sharma J Noliear Sci Appl Proof By usig Corollary 23 ad simple computatio we obtai φ 2 = [] 2 2 [] [] [ 3] 2 [] [] [] [] [ 3] [ 3] Clearly for > 3 we have ad Also as 0 we obtai [] 2 2 [] > 0 22 [] 2 2 [] [] [ 3] [] [] [] [ 3] 23 = [] 3 2 [] [] = [] 3 [] [] Fially by usig the above ieuality 22 ad 23 we get φ 2 3 [ 3] [ 3] 3 = 3 Local Approimatio Let us cosider: The Peetre s K-fuctioal for δ > 0 is defied by K 2 f δ = W 2 = {g C[0 ] : g g C[0 ]} if g C 2 B [0 { f g δ g : g W 2} where is the uiform orm o C[0 ] There eists a costat C > 0 due to [3] such that for δ > 0 we have K 2 f δ Cω 2 f δ 3 where the secod order modulus of cotiuity for f C[0 ] is defied by ω 2 f δ = 0<h δ We prove below the followig local direct result: f 2h 2f h f 2h [0]

5 V Gupta T M Rassias H Sharma J Noliear Sci Appl Theorem 3 Let 0 < < The D /[] f; f Kω 2 f φ 2 φ2 ω f φ for every [0 ] ad f C[0 ] where K is a positive costat Here φ ad φ 2 are the first ad secod cetral momets of the operator Proof We cosider modified operators where [0 ] The operators defied by f; = D /[] f; f preserve liear fuctios: Let g C 2 [0 ] ad t [0 ] Usig Taylor s epasio we have [] f 32 t ; = 0 33 gt = g g t t t ug udu The usig 33 we get t g; = g t ug udu; Therefore from 32 we have g; g t D/[] t ug udu; D /[] [ t t u g u du; t 2 ; [] [2] [] [2] ] [] 2 g [] u g udu [] u g u du O usig Euatio 32 we see that f; D /[] f; 2 f f D /[] ; 2 f = 3 f 35 Now usig Euatios ad 35 we obtai D /[] f; f f g; f g g; g f [] f 4 f g φ 2 φ2 g f [] f Thus takig ifimum o the right-had side over all g W 2 we get f; f 4K 2 f φ 2 φ2 Coseuetly we get D /[] f; f Kω 2 f This completes the proof of the theorem ω f φ φ 2 φ2 ω f φ 34

6 V Gupta T M Rassias H Sharma J Noliear Sci Appl Global Approimatio The Ditzia Totik moduli of the first order is give by ω ψ f η = 0 h η hϕ [0] f hϕ f where ψ is a admissible step-weight fuctio o [0] The secod order Ditzia Totik modulus of smoothess for f C[0 ] ϕ = ad [0 ] is defied by ω ϕ 2 f η = 0 h η f hϕ 2f f hϕ ±hϕ [0] For W 2 ϕ = {g C[0 ] : g AC loc [0 ] ϕ 2 g C[0 ]} ad g AC loc [0 ] meas that g is differetiable ad g is absolutely cotiuous o every closed iterval [a b] [0 ] K-fuctioal correspodig to the secod order Ditzia Totik modulus of smoothess is defied by It is well kow that see [4] K 2ϕ f η = if { f g g W 2 ϕ η ϕ2 g η 2 g } K 2ϕ f η Cω ϕ 2 f η 4 Theorem 4 For f C[0 ] 0 ad ψ = 2 [0 ] there eists a absolute costat C > 0 such that D /[] f; f Cω ϕ 2 f ω ψ f Proof Agai cosider modified operators f; = D /[] f; f defied by [] where [0 ] Usig Taylor s epasio for g W 2 ϕ we have The usig 33 we get Therefore from 42 we have g; g gt = g g t g; = g D/[] t t t t ug udu; t t ug udu t ug udu; [] [2] t u g u [] [2] du; f 42 [] u g udu [] u g u du As the fuctio δ 2 is a cocave fuctio o [0 ] we have for u = t τ t τ [0 ] that the followig estimate holds true t u δu 2 = τ t δt 2 τ t τ t t δt 2 τδ 2 δt 2 δ 2

7 V Gupta T M Rassias H Sharma J Noliear Sci Appl Hece Also g; g D /[] where [0 ]; thus we obtai δ 2 t t u δu 2 du; δ 2 g = ϕ 2 g t 2 ; g; g 0 [] δg 2 [2] [] [] [2] δu 2 2 u du δg 2 δ 2 g g ϕ 2 g g ϕ 2 g Usig 32 ad 43 we get D /[] f; f f g; f g f [] f 4 f g 0 ϕ 2 g g 43 g g; g f [] f Takig ifimum o the right-had side over W 2 ϕ we have D /[] f; f 0K 2ϕ f f [] f 44 Cosider f [] f f ψ [] ψ f ttψt [] [2] ψ[2] [0] ω ψ f [] ψ f t ψt [] ft ψ = ω ψ f Fially usig 4 44 ad the above ieuality we get D /[] f; f Cω ϕ 2 f ω ψ f Refereces [] A Aral V Gupta R P Agarwal Applicatios of Calculus i Operator Theory Spriger New York 203 [2] I Büyükyazıcı H Sharma Approimatio properties of two-dimesioal -Berstei-Chlodowsky-Durrmeyer operators Numer Fuct Aal Optim [3] R A De Vore G G Loretz Costructive Approimatio Spriger-Verlag Berli [4] Z Ditzia V Totik Moduli of Smoothess Spriger-Verlag New York 987 4

8 V Gupta T M Rassias H Sharma J Noliear Sci Appl [5] Z Fita V Gupta Approimatio by -Durrmeyer operators J Appl Math Comput [6] V Gupta Some approimatio properties of -Durrmeyer operators Appl Math Comput [7] V Gupta Z Fita O certai -Durrmeyer type operators Appl Math Comput [8] V Gupta W Hepig The rate of covergece of -Durrmeyer operators for 0 < < Math Methods Appl Sci [9] V Gupta T M Rassias Lupaş-Durrmeyer operators based o Pólya distributio Baach J Math Aal [0] V Gupta H Sharma Recurrece formula ad better approimatio for -Durrmeyer Operators Lobachevskii J Math [] V Gupta H Sharma T Kim S Lee Properties of -aalogue of Beta operator Adv Differece Eu pages [2] A Lupaş A -aalogue of the Berstei operator Semiar o Numerical ad Statistical Calculus Cluj-Napoca [3] G Nowak Approimatio properties for geeralized -Berstei polyomials J Math Aal Appl [4] G Nowak V Gupta The rate of poitwise approimatio of possitive liear operators based o -iteger Ukraiia Math J [5] G M Phillips Berstei polyomials based o the -itegers A Numer Math [6] D D Stacu Approimatio of fuctios by a ew class of liear polyomial operators Rev Roumaie Math Pures Appl

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