Direct Estimates for Lupaş-Durrmeyer Operators

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1 Filomat 3:1 16, DOI 1.98/FIL161191A Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: Direct Estimates for Lupaş-Durrmeyer Operators Ali Aral a, Vijay Gupta b a Departmet of Mathematics, Faculty of Sciece ad Arts, Kirikkale Uiversity, 7145 Yahsiha, Kirikkale, Turkey b Departmet of Mathematics, Netaji Subhas Istitute of Techology, Sector 3 Dwarka, New Delhi 1178, Idia Abstract. The geeralizatio of the Berstei polyomials based o Polya distributio was first cosidered by Stacu [14]. Very recetly Gupta ad Rassias [6] proposed the Durrmeyer type modificatio of the Lupaş operators ad established some results. Now we eted the studies ad here we estimate the covergece estimates, which iclude quatitative asymptotic formula ad rate of approimatio bouded variatio. We also give a ope problem for readers to obtai the momets usig hypergeometric fuctio. 1. Itroductio Stacu [14] itroduced a sequece of positive liear operators P α o-egative parameter α give by P α f, = f k= k p α,k, : C[, 1] C[, 1], depedig o a 1 where p α is the Polya distributio with desity fuctio give by,k k 1 p α,k = k ν= να k 1 µ= 1 µα 1 λ= 1 λα, [, 1]. I case α = these operators reduce to the classical Berstei polyomials. For α = 1/ a special case of the operators 1 was cosidered by Lupaş ad Lupaş [8], which ca be represeted i a alterate form as P 1/ f, =!! f k= k k k, where the risig factorial is give as = Recetly Miclăuş [1] established some approimatio results for the operators 1 ad for the case. 1 Mathematics Subject Classificatio. 41A5; 41A36 Keywords. Polya distributio; Berstei polyomials; Durrmeyer operators; Asymptotic formulae. Received: 4 February 14; Accepted: 6 May 14 Commuicated by Hari M. Srivastava addresses: aliaral73@yahoo.com Ali Aral, vijaygupta1@hotmail.com Vijay Gupta

2 A. Aral, V. Gupta / Filomat 3:1 16, I the last few decades several ew operators were costructed ad their approimatio properties related to covergece behaviour have bee studied by several researchers. We metio here some of them due to Aral ad Acar [1] who cosidered Berstei-Chlodowsky-Gadjiev operators, Mahmudov ad Sabacigil [9] proposed q-berstei Katorovich operators, Srivastava ad Gupta [1], [13] proposed a geeral family of itegral operators ad summatio-itegral type operators established some covergece estimates. Very recetly Gupta ad Rassias [6] proposed the itegral modificatio of the operators, which is based o Polya distributio as follows: D 1/ f, = where K, t = 1 ad p 1/ =!,k! p,k t = t k 1 t k. k K, t f tdt = 1 k= p1/,k k k k k= p,k t with p 1/,k p,k t f t dt, 3 Some approimatio properties related the preset paper ca be foud i [5] ad i the recet book by Gupta ad Agarwal [4]. I this paper, we cosider the operator 3 ad obtai a quatitative Voroovskaja type asymptotic formula ad the rate of covergece for bouded variatio fuctios.. Auiliary Results Lemma.1. Miclăuş [1] For e i = t i, i =, 1, we have ad P 1/ e, = 1, P 1/ e 1, = P 1/ e, = = Lemma.. For e i = t i, i =, 1, we have D 1/ e, = 1, D 1/ e 1, = 1 D 1/ e, = D 1/ e 3, = D 1/ e 4, = 1 { }

3 A. Aral, V. Gupta / Filomat 3:1 16, The proof of the above lemma easily follows by usig Lemma.1 Remark.3. I terms of hypergeometric fuctio, we have the momets i the followig complicated form: Usig 1 k = k k k!, a k = a 1 a k, k, p,ktt r dt =!kr! k!r1! ad k r! = r 1 k.r! we have D 1/ e r, = 1.! k 1 k!k r!!. 1k k. k! 1 k k! r 1! k= = r!. 1!! r 1!! = r!. 1!! r 1!! k= k k r 1 k 1 k 1 k. 1 k! 3F,, r 1; 1, 1 ; 1 = r!. 1! r 1! 3F,, r 1; 1, 1 ; 1 By usig the above form it may be cosidered as a ope problem for the readers to have momets as idicated i Lemma.. This problem was iitially raised by Gupta i [3]. Remark.4. By simple applicatios of Lemma., we have ad D 1/ t, = 1 D 1/ t, = D 1/ t 3, = D 1/ t 4, 7 =

4 A. Aral, V. Gupta / Filomat 3:1 16, Remark.5. For sufficietly large, C > 3 ad, 1, by Remark.4, we have ad D 1/ t, C1, K, t t dt [D 1/ t, C1 ] 1/. Lemma.6. Let, 1 ad C > 3, the for sufficietly large, we have λ, y = y 1 λ, z = K, tdt z K, tdt C1 y, y < C1 z, < z < 1. Proof of the above lemma follows easily by usig Remark.5. For k 1, let us deote by C k = C k [, 1] the subspace of C [, 1] whose elemets f are k-times cotiuosly differetiable ad f k C [, 1].For f C m [, 1], the local Taylor formula at the poit [, 1] is give by f = m k= f k k! k R m f,, [, 1], m 1. I [], remaider term R m f,, was estimated by R m f,, m ω f m,. m! Strictly related to the modulus ω is the well-kow K-fuctioal, itroduced by Peetre ad defied by K f, ε = if { f ε : C 1} f C [, 1] ad ε >. The relatio betwee modulus of cotiuity ad correspodig K-fuctioal is give by K f, ε/ = 1 ω f, ε for f C [, 1],where ω f, ε deotes the least cocave majorat of ω f, ε, see, [11]. Lemma.7. []For m N let f C m ad, [, 1]. The R m f,, m m! 3. Covergece Estimates ω f m,. 4 m 1 I this sectio, we preset some covergece estimates of the operators D 1/ f,.

5 A. Aral, V. Gupta / Filomat 3:1 16, Theorem 3.1. If f L [, 1] the at every poit of cotiuity of f we have lim D1/ f, = f. Moreover if the fuctio f is uiformly cotiuous the we have lim D 1/ f, f =. Proof. Sice D 1/ 1; = 1 we ca write D 1/ f ; f 1/ = D f, = 1 k= k= p 1/ p,k,k t [ f t f ] dt. Let ε > be give. By the cotiuity of f at the poit there eists δ > such that f t f < ε wheever t < δ. For this δ > we ca write D 1/ f ; f = 1 p 1/ p,k,k t ] [ f t f dt t <δ t δ It is obvious that I 1 εd 1/ 1, = ε. := I 1 I. It remais to estimate I. We ca write I f 1 p 1/,k k= f D 1/ t,. δ p,k t dt y δ If we choose δ = 1 3 ad use Remark.4 we have I { } f, 1/3 3 which proves the theorem. The secod part of the theorem is proved similarly. Theorem 3.. Let f C [, 1 ad N. The we have [ D 1/ ] f, f f 1 3 f f f ω f, O 1/ 1 3 = o 1 f f O 1 ω f, O 1/.

6 A. Aral, V. Gupta / Filomat 3:1 16, Proof. By the local Taylor s formula there eists η lyig betwee ad y such that where f y = f f y f y h y, y, h y, := f η f ad h is a cotiuous fuctio which vaishes at. Applyig the operator D 1/ obtai the equality D 1/ f, f = f D 1/ D 1/ f y, h y, y, also we ca write that f 3 f D1/ f, f 1 f D 1/ 1 y, f y D1/, 3 h y, y, D 1/ y D 1/, Usig Remark.4, we ca write [ D 1/ ] f, f f 1 3 f f 4 f h y, y, D 1/ to above equality, we ad usig the fact [, 1] ad ma 1 = 1 4 we have [ D 1/ ] f, f f 1 3 f f f h y, y, D 1/ To estimate the term D 1/ D 1/ h y, y, if we cosider the iequality 4 for m = the we deduce h y, y, 1 D1/ D 1/ y ω f y, 3, y K f y, 6,

7 Let C 3 be fied. The we write D 1/ ad we have D 1/ Sice D 1/ h y, y, D 1/ A. Aral, V. Gupta / Filomat 3:1 16, y f y f D 1/, q 6 D 1/ y 3, y D 1/, f h y, y, y D 1/, K f, y D1/, ω f, 1 3 y 6, D 1/ D 1/ 6 D 1/ D 1/ y 3, y, y 4, D 1/ D 1/ y 4, = O 1/ y ad D 1/, = O 1/ we get D 1/ h y, y, 1 which completes the proof y,. ω f, O 1/, y 3, y, We deote by B D, 1 the class of absolutely cotiuous fuctios f o, 1 havig a derivative f o, 1, which cocide a.e. with a fuctio which is of bouded variatio o every subiterval of, 1. The fuctios f B D, 1 possess the represetatio f = f c φtdt, c >. c Theorem 3.3. Let f B D, 1 ad, 1, the for sufficietly large ad C > 3, we have D 1/ f, f C1 [ C ] 1 [ f f ] k=1 V 1 /k 1 [ f f ] f /k 1 V 1 / f, / where V b a f is the total variatio of f o [a, b] ad the auiliary fuctio is give by f t = f t f, t < ;, t = ; f t f, < t < 1.

8 A. Aral, V. Gupta / Filomat 3:1 16, Proof. By mea value theorem ad usig the methods as give i [4], we have D 1/ f, f f f Usig Remark.4 ad Remark.5, we have t K, tdu dt f f t K, t si u du dt t K, t f udu dt f f D 1/ t, f f [D 1/ t, ] 1/ t K, t f udu dt D 1/ f, f f f 1 f f C1 t K, t f udu dt t K, t f udu dt 5 As b a d tλ, t 1 for each [a, b] [, 1], usig Lemma.6 with y = /, we get t K, t f udu dt = y f t. λ, t dt C1 y y Settig u = / t, we have t f udu d t λ, t Vt f 1 t dt V / f y Vt f 1 t dt = 1 1 V /u f du 1 [ ] V /k f. k=1 Thus t K, t f udu dt C1 [ ] V /k f V / f 6 k=1

9 Fially = z t K, t f udu dt t f udu K, tdt C 1 1 [ C ] 1 k=1 A. Aral, V. Gupta / Filomat 3:1 16, z V 1 / f du V 1 / f du t f udu d t 1 λ, tdt 1 V 1 / f 1 V 1 / f 7 Combiig the estimates 5, 6 ad 7, ad usig 1 1, we are let to the coclusio of the theorem. Remark 3.4. Very recetly Gupta ad Tachev [7] cosidered the combiatios of certai Durrmeyer type operators ad Verma et al. [15] cosidered the Stacu variat of some Durrmeyer type operators. O ca cosider the methods here to apply combiatios ad Stacu variats of these operators. At this momet it is ot possible to establish aalogous results. We may cosider such results i future. Refereces [1] A. Aral, T. Acar, Weighted approimatio by ew Berstei-Chlodowsky-Gadjiev operators, Filomat [] H. Goska, P. Pitul, I. Rasa, O Peao s form of the Taylor remaider,voroovskaja s theorem ad the commutator of positive liear operators, Proc.It. Cof. o Numerical Aalysis ad Approimatio Theory, Cluj-Napoca,Romaia, July [3] V. Gupta, Europea Mathematical Society Newsletter March 14, p. 64 Problem 13 [4] V. Gupta, R. P. Agarwal, Covergece Estimates i Approimatio Theory, Spriger 14. [5] V. Gupta, A. J. Lopez-Moreo ad Jose-Mauel Latorre-Palacios, O Simultaeous Approimatio of the Berstei Durrmeyer Operators, Applied Math. Comput [6] V. Gupta, Th. M. Rassias, Lupaş-Durrmeyer operators based o Polya distributio, Baach J. Math. Aal [7] V. Gupta, G. Tachev, Approimatio by Szász-Mirakya Baskakov operators, J. Appl. Fuct. Aal [8] L. Lupaş, A. Lupaş, Polyomials of biomial type ad approimatio operators, Studia Uiv. Babes-Bolyai, Mathematica [9] N. I. Mahmudov, P. Sabacigil, Approimatio theorems for q-berstei-katorovich operators, Filomat [1] D. Miclăuş, The revisio of some results for Berstei Stacu type operators, Carpathia J. Math [11] R. T. Rockafellar, Cove Aalysis, Priceto Uiversity Press, Priceto, 197. [1] H. M. Srivastava, V. Gupta, A certai family of summatio-itegral type operators, Math. Comput. Modellig [13] H. M. Srivastava, Z. Fita, V. Gupta, Covergece of a certai family of summatio itegral type operators, Applied Math. Comput [14] D. D. Stacu, Approimatio of fuctios by a ew class of liear polyomial operators, Rew. Roum. Math. Pure. Appl [15] D. K. Verma, V. Gupta, P. N. Agrawal, Some approimatio properties of Baskakov-Durrmeyer-Stacu operators, Applied Math. Comput