QUANTITATIVE ESTIMATES FOR GENERALIZED TWO DIMENSIONAL BASKAKOV OPERATORS. Neha Bhardwaj and Naokant Deo

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1 Korea J Math , No 3, pp QUANTITATIVE ESTIMATES FOR GENERALIZED TWO DIMENSIONAL BASKAKOV OPERATORS Neha Bhardwaj ad Naoat Deo Abstract I this paper, we obtai quatitative estimates for geeraized doube Basaov operators We cacuate goba resuts for these operators usig Lipschitz-type spaces ad estimate the error usig moduus of cotiuity 1 Itroductio The Basaov operator V f; x was itroduced by VA Basaov [2] give by: 11 V f; x p, x f 0, 1 where p, x x 1 x, f C B [0,, C B [0, is the set of fuctios which are bouded o the set By ow a umber of resuts about the operator have bee obtaied [1, 3 7] I this paper, we address the ivestigatio for the mutivariate Basaov operator defied as foows Received May 3, 2016 Revised Juy 11, 2016 Accepted August 1, Mathematics Subject Cassificatio: 41A25, 41A30 Key words ad phrases: Basaov operators, Lipschitz-type space, moduus of cotiuity c The Kagwo-Kyugi Mathematica Society, 2016 This is a Ope Access artice distributed uder the terms of the Creative commos Attributio No-Commercia Licese -c/30/ which permits urestricted o-commercia use, distributio ad reproductio i ay medium, provided the origia wor is propery cited

2 336 Neha Bhardwaj ad Naoat Deo Let P f; x, y are the foowig two variate Basaov operators: 12 P f; x, y p, x p, yf,,,0 where 0, 0, f x, y C [0, [0, Ozarsa ad Duma [9] have itroduced a differet approach i order to get a faster approximatio without preservig the test fuctios I [8], Özarsa ad Atuǵu have cacuated quatitative goba estimates for doube Szasz-Miraja operators I this paper, same method is used for geeraized doube Basaov operators Cosider the cassica Basaov operators defied by 11 Sice for f i t i, i 0, 1, 2 V f 0 ; x 1, V f 1 ; x x, V f 2 ; x 1 1 x 2 x Foowig the simiar argumets as used i [9], the best error estimatio amog a the geera doube Basaov operators ca be obtaied from the case by taig a 1, b e 0, c 1 1, d 1 for a N where a, b, c, d ad e are sequeces of oegative rea umbers satisfyig the coditios give i [9] Now observe that u x 2a x d 2c 2x [0,, where u is a fuctioa sequece, u : I A where A deote R ad assume that I be subiterva of A So, u x R if ad oy if x 1 ad 1 Hece, choosig 2 [ 1 I 2, R The best error estimatio amog a the geera doube Basaov operators ca be obtaied from the case u x 2x 1 2 1, v y 2y ; N

3 Estimates for geeraized two dimesioa Basaov operators 337 for a f C B [0, C B [0, ad x, y [ 1 2, Hece, 12 becomes 13 Q f; x, y : P f; u x, v y h 1 1,0 f, u x 1 u x v y 1 v y, f C B [0, [0, For the operators Q f; x, y, we have foowig Lemma: Lemma 11 Let x x, y, t t, s; e i,j x x i y j, i, j 0, 1, 2 ad ψ 2 x t x 2 The, for each x, y 0 ad > 1, we have i Q e 0,0 ; x, y 1 ii Q e 1,0 ; x, y u x iii Q e 0,1 ; x, y v y iv Q e 2,0 e 0,2 ; x, y 1 1 u 2 x v 2 y uxvy v Q ψ 2 x t ; x, y u x x 2 v y y 2 1 u2 xv 2 y u x v y 2 Goba Resuts We have used foowig defiitios i this paper for goba resuts of the operators Q f; x, y Otto Szasz [10] earier cosidered this space of bivariate extesio of Lipschitz-type space, give as: { Lip x α M α : f C [0, [0, : f f x M t x y α /2 } ; t, s; x, y 0, where t t, s, x x, y ad M is ay positive costat ad 0 < α 1,

4 338 Neha Bhardwaj ad Naoat Deo For a f C [0, [0,, the moduus of f deoted by ω f; δ is defied as { ω f; δ : sup f t, s f x, y : x 2 s y 2 < δ, t, s, } x, y [0, [0, Now, for the space Lip M α with 0 < α 1, we have the foowig approximatio resut Theorem 21 For ay f Lip M α, α 0, 1], ad for each x, y 0,, N, we have 21 Q f; x, y f x, y 1 M x y α /2 [ u x x 2 v y y 2 u 2 x v 2 y u x v y ]α /2 Proof Let α 1 For each x, y 0, ad for f Lip M 1, we have Q f; x, y f x, y Q f t, s f x, y ; x, y x MQ t x y 1 ; x, y /2 Appyig Cauchy-Schwarz iequaity, we get M x y 1 /2 Q x ; x, y M Q f; x, y f x, y x y 1 Q ψx 2 t ; x, y /2 M x y 1 u x x 2 v y y 2 1 /2 u2 x v 2 y u x v y

5 Estimates for geeraized two dimesioa Basaov operators 339 Now, et 0 < α < 1 The for each x, y 0, ad for f Lip M α, we obtai Q f; x, y f x, y Q f t, s f x, y ; x, y x α MQ t x y α ; x, y /2 M x y α /2 Q x α ; x, y For Hoder iequaity with p 2 α ad q 2 2 α, for ay f Lip M α, we have M [ Q f; x, y f x, y x y α Q /2 ψ 2 x t ; x, y ]α /2 [ M x y α u x x 2 v y y 2 /2 1 u 2 x v 2 y u x v y ]α /2 which is the required resut Lemma 22 For each x, y > 0, 22 Q x 2 s y 2; x, y 1 u x x 2 u2 x u x x 1 y v y y 2 v2 y v y

6 340 Neha Bhardwaj ad Naoat Deo Proof We have c d c d c, d 0, therefore Q x 2 s y 2; x, y, x 2 y u x 1 u x v y 1 v y 1 x u x 1 u x 0 1 y v y 1 v y 0 1 x 0 u x 1 u x x 1 y 0 v y 1 v y y 1 1 x x u x 1 u x y y v y 1 v y 0 Usig the Cauchy-Schwarz iequaity, Q x 2 s y 2; x, y 1 x 0 1 y x u x 1 u x 2 y v y 1 v y

7 Estimates for geeraized two dimesioa Basaov operators 341 Usig Lemma 11, Q x 2 s y 2; x, y 1 u x x 2 u2 x u x 1 v y y 2 v2 y v y x y which is the desired resut 0, Theorem 23 Let g x, y f x 2, y 2 The we have for each x, y > Q f; x, y f x, y 2ω g; δ x, y, where δ x, y 1 x u x x 2 u2 xu x 1 y v y y 2 v2 yv y Proof We have Q f; x, y f x, y Q f t, s f x, y ; x, y g Q t, s g x, y ; x, y Q ω g; x 2 s y 2 ; x, y,0 1 1 ω g; 2 x u x 1 u x v y 1 v y x 2 y ω g; t 2 2; Q x s y x, y,0 y 2 2; Q x s y x, y ; x, y 2 ; x, y Now, we have ω f; λδ 1 λ ω f; δ

8 342 Neha Bhardwaj ad Naoat Deo Therefore, Q f; x, y f x, y 2 2; g; Q x s y x, y ω 2ω 1 1,0 2 x 1 2 y t 2 2; Q x s y x, y u x 1 u x v y 1 v y 2 2; g; Q x s y x, y Now, usig Lemma 22, competes the proof Theorem 24 Let g x, y f x 2, y 2 Let g Lip M α : { g C B [0, [0, : g g x M x α ; t, s; x, y 0, }, where t t, s, x x, y ad M is ay positive costat ad 0 < α 1 The, 23 Q f; x, y f x, y Mδ α x, y, where δ x, y 1 x u x x 2 u2 xu x 1 y v y y 2 v2 yv y

9 Estimates for geeraized two dimesioa Basaov operators 343 Proof We have Q f; x, y f x, y Q f t, s f x, y ; x, y g Q t, s g x, y ; x, y MQ x 2 s y 2 α / 2; x, y M,0 1 1 u x 1 u x v y 1 v y 2 2 x y For Hoder iequaity with p 2 ad q 2, we have α 2 α [ Q f; x, y f x, y M By usig Lemma 22, competes the proof Q x 2 s y 2; x, y ] α α / 2 Refereces [1] Ade JA, Bada FG ad Ca J de a, O the iterates of some Berstei type operators, J Math Aa App , [2] Basaov VA, A exampe of a sequece of iear positive operators i the spaces of cotiuous fuctios, Do Aad Nau SSSR , [3] Becer M, Goba approximatio theorems for SzszMiraja ad Basaov operators i poyomia weight spaces, Idiaa Uiv Math J , [4] Deo N, O the iterative combiatios of Basaov operators, Geera Mathematics, , [5] Deo N ad Bhardwaj N, O the degree of approximatio by modified Basaov operators, Lobachevsii J Math , [6] Ditzia Z ad Toti V, Modui of Smoothess, Spriger-Verag, Beri, 1987 [7] Guo F, O covergece rate of approximatio for two-dimesioa Basaov operators, Aaysis i Theory ad Appicatios, , [8] Özarsa MA ad Atuǵu H, Quatitative Goba Estimates for Geeraized Doube Szasz-Miraja Operators, Joura of Appied Mathematics, [9] Özarsa MA ad Duma O, A ew approach i obtaiig a better estimatio i approximatio by positive iear operators, Commu Fac Sci Uiv A Ser A1 Math Stat, , [10] Szasz O, Geeraizatio of S Berstei s poyomias to the ifiite iterva, J Research Nat Bur Stadards, ,

10 344 Neha Bhardwaj ad Naoat Deo Neha Bhardwaj Departmet of Mathematics Amity Istitute of Appied Scieces Amity Uiversity Noida , Idia E-mai: eha Naoat Deo Departmet of Appied Mathematics Dehi Techoogica Uiversity Formery Dehi Coege of Egieerig Bawaa Road, Dehi , Idia E-mai: dr aoat