A Study on the Rate of Convergence of Chlodovsky-Durrmeyer Operator and Their Bézier Variant

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1 IOSR Joural of Matheatics (IOSR-JM) e-issn: , p-issn: X. Volue 3, Issue Ver. III (Mar. - Apr. 7), PP -8 A Study o the Rate of Covergece of Chlodovsky-Durreyer Operator ad Their Bézier Variat S. K. Tiwari, V.K. Gupta, Yogita Parar 3 : Departet of Matheatics, School of Studies i atheatics, Vikra Uiversity, Ujjai (M.P.) : Departet of Matheatics, Govt. Madhav Sciece College Ujjai (M.P.),3: Preseach Scholar, School of Studies i atheatics, Vikra Uiversity, Ujjai (M.P.) Abstract: I this paper, we have studied the Bézier variat of Chlodovsky-Durreyer operators D, for fuctio f easurable ad locally bouded o the iterval [, ). I this we iproved the result give by Ibikl E. Ad Karsli H. [4]. We estiate the rate of poitwise covergece of D, f () at those > at which the oe-sided liits f +, f( ) eist by usig the Chaturiodulus of variatio. I the special case = the recet result of Ibikl E. Ad Karsli H. [4] cocerig the Chlodowsky-Durreyer operators D is essetially iproved ad eteded to ore geeral classes of fuctios. Keywords: Rate of covergece, Chlodovsky-Durreyer operator, Bézier basis, Chaturiodulus of variatio, p-th power variatio. I. Itroductio For a fuctio the classical Berstei-Durreyer operators (see [7]) M applied to f are defie as M f = + p,i f t p,i t, [,] () i= where p,i = i i ( ) i. Several researchers have studied approiatio properties of the operators M ([8], []) for fuctio of bouded variatio defied o the iterval [, ]. After that Zeg ad Che [] defied the Bézier variat of Durreyer operators as where Q,i = J,i J,i+ M, f = + i= DOI:.979/ Page Q,i f t p,i t, () ad J,i = j =i p,j for i =,,,,, J, +i = are the Bézier basis fuctio which is itroduced by P. Bézier [4] ad estiated the rate of covergece of M, f for fuctios of bouded variatio o the iterval [,]. Let X loc [, ) be the class of all cople valued fuctio easurable ad locally bouded o the iterval [, ). For f X loc [, ) the Chlodowsky Durreyer operator D are defied as D f = + P,i f t P t,i i= where ( ) is a positive icreasig seuece with the properties,. (3) li a = ad li = (4) For f X loc [, ) ad, we itroduce the Bézier variat of Chlodowsky Durreyer operators D, as follows D, f = + i= ( ) Q,i f t P,i t,. (5) Obviously, D, is a positive liear operator ad D, =. I particular, whe = the operators (5) reduce to operators (3). Recetly Agratii [], Aiol ad Pych Taberska [3], Pych Taberska [], ad Gupta [, ] have ivestigated the rate of poitwise covergece for Katorovich ad Durreyer Type Baskakov Bézier ad Bézier operators usig a differet approach. They have proved their theores i ters of the Chaturiodulus of variatio, which is a geeralizatio of the classical Jorda variatio. It is useful to poit out that a deeper aalysis of the Chaturiodulus of variatio ca be foud i [6], but actually the odulus of variatio was itroduced for the first tie by Lagrage [8]. Although the Chaturiodulus of variatio was defied as a

2 A Study o the Rate of Covergece of Chlodovsky-Durreyer Operator ad Their Bézier Variat geeralizatio of the classical variatio early four decades years ago, it was ot used to a sufficiet etet to solve the proble etioed above. The paper is cocered with the rate of poitwise covergece of the operators (5) whe f belog to X loc [, ). Usig the Chaturiodulus of variatio defied i [6], we eaie the rate of poitwise covergece of D, f at the poits of cotiuity ad at the first kid discotiuity poits of f. For soe iportat papers o differet operators related to the preset study we refer the readers to Gupta et. Al. [9, ] ad Zeg ad Piriou [3]. It is ecessary to poit out that i the preset paper we eted ad iprove the result of Ibikli E. ad Karsli H.[4] for Chlodowsky-Durreyer operators. We beig by givig Defiitio. Let f be a bouded fuctio o a copact iterval I = [a, b]. The odulus of variatio μ (f; a, b ) of a fuctio f is defied for oegative itegers as μ f; a, b = ad for as μ f; a, b = sup f i+ f( i ), π i= where π is a arbitrary syste of disjoit itervals i, i+, i =,,,, i.e., a < < 3 < b. The odulus of variatio of ay fuctio is a o-decreasig fuctio of. Soe other properties of this odulus ca be foud i [6]. If f BV p I, p, i.e., if f of p-th bouded power variatio o I, the for every i N, μ i f; I i p V p f, I, (6) where V p f, I deotes the total p-th bouded power variatio of f o I, defied as the upper boud of the set of ubers f i j f(l j ) p p j over all fiite systes of o-overlappig itervals i j, l j I. We also cosider the class BV p loc,, p, cosistig of all fuctio of bouded p-th power variatio o every copact iterval I [, ). I the seuel it will be always assued that the seuece satisfies the fudaetal coditios (4). The sybol [a] will be deote the greatest iteger ot greater tha a. Reark. Now, let us cosider the special case =, p =, ad let us suppose that fuctio f is of bouded variatio i the Jorde sese o the whole iterval [, ) (f BV, ). The, for all itegers such that > ad 4, we have the followig estiatio for the rate of covergece of the Chlodowsky-Durreyer operators (3): f + + f( ) D, f V g ; H ( + + V g ; H i i= + 4M + f + f +, where M = Sup < f() ad V(g ; H) deotes the Jorda variatio of g o the iterval H. The above estiatio is essetially better tha the estiatio preseted i [4]. Naely, it is easy to see that the right-had side of the ai ieuality give i Theore. i [4] is ot coverget to zero for all fuctio f BV[, ) ad for all seueces satisfyig (4). II. Auilary Result I this sectio we give certai results, which are ecessary to prove the ai result. For this, let us itroduce the followig otatio. Give ay [, ] ad ay o-egative iteger, we write ψ t (t ) for t,, W, D ψ Lea. If N, [, ], the + W, =, i= P,i t P,i t. (7) DOI:.979/ Page W, = +,

3 A Study o the Rate of Covergece of Chlodovsky-Durreyer Operator ad Their Bézier Variat ad, for >, W, = ( 3)( ) + ( + 3) W, = j = β j, + ( + )( + 3) DOI:.979/ Page j j, (8) where β j, are real ubers idepedet of ad bouded uiforly i. Moreover, for W, + ad, for >, W, c where c is a positive costat depedig oly o. + (9), () Proof. Forulas for W,, W,, W, ad ieuality (9) follow by siple calculatio. Suppose > ad put y. The y [,] ad = + W, = + i= P,i y i= P,i y t y P,i t s y P,i s ds = a M ψ y where M is the classical Berstei-Durreyer operator (). The represetatio forula (8) follows at oce fro the kow idetity M ψ y y = j = β j,, y( y) j, y, () where β j,, are real ubers idepedet of y ad bouded uiforly i (see [3], Lea 4.8 with c = ). Now, let us observe that for y [, ] y,,, oe has y( y) ad M ψ y y = β j,, 3 j η j,, j = where η j, are o-egative ubers depedig oly o j ad. If y, the ad y ( y) Coseuetly, M ψ y M ψ y y = y( y) y( y) j = j = y C y y + β j,, η j, j. j = (y y ) j, wit C = j = η j, j. Takig i () ad i the above ieuality y = we easily obtai estiatio (). Lea. Let (, ) ad let The ad t K t,, + t K u,, K u,, du du i= () Q,i P,i t t W, if < t () t W, if < t <, (3).

4 A Study o the Rate of Covergece of Chlodovsky-Durreyer Operator ad Their Bézier Variat where W, is give by (7) (with = ). Proof. As is kow Q,i P,i for. Hece, if < t, the t K u,, du t t (u ) K u,, t D,ψ t D ψ = t W,. The proof of (3) is siilar. I order to forulate the et lea we itroduce the followig itervals. If >, we write I u + u,, if u < I u, + u if u > Lea.3 Let f X loc [, ) ad let the oe-sided liits f +, f( ) eist at a fied poit (, ). Cosider the fuctio g defied by (6) ad write d. If = or =, the for all itegers such that d we have g (t)k t,, v a g ; I (d ) I () + 8W,() d v j g ; I (jd ) j 3 + v g ; I (), where = d ad W, () is estiated i (9). Proof. Restrictig the proof to = we defie the poit t j = + jd for j =,,3,, + ad we deote t + =. Put T j = [t j, ] for j =,,3,, + ad we have Clearly, I () g (t)k t,, t j g (t)k t,, + g (t j ) K t t,, + g a t g (t j ) K,, t j + t j + = I, + I, + I 3,, say I, t t t j g t g () K t,, v g ; T By the Abel lea o suatio by parts ad by (3) we have I, g t t K t,, K t,, + g t j + g (t j ) du = v g ; T. t j + W,() g d t g + g t j + g (t j ) (j + ) j = W,() g d t g + g t i+ g (t i ) + g t i+ g (t i ) i= i= W,() v j + g ; T j + v d g ; T + (j + ) 3 + v g ; T K t,, (j + ) (j + ) DOI:.979/ Page

5 A Study o the Rate of Covergece of Chlodovsky-Durreyer Operator ad Their Bézier Variat W,() v j g ; T j d j 3 Net, i view of (3) ad the Abel trasforatio, = W,() d I 3, v g ; [t j +, t j ] i= W,() d v g ; [t i+, t i ] + v g ; T +. K t,, DOI:.979/ Page t t j + v g ; [t j +, t j ] j j + v g ; t i+, t i i= j j + W,() v g ; T + v j g ; T j + d + 6 (j + ) 3 W,() d v g ; T j = v j g ; T j j 3 Cobiig the result ad observig that T j = I (jd ) we get the desired estiatio for h = -. I the case of h = the proof rus aalogously; we use ieuality () istead of (3). III. Mai Results I this sectio we prove our ai theores. Theore 3. Let f X loc [, ) ad let the oe-sided liits f +, f( ) eist at a fied poit (, ). The, for all itegers such that > ad 4 oe has f + + f( ) D, f μ + g ; H ( where = μ j g ; H (j j 3 + μ g ; H () + C φ ; f a + a f + f +,, H u = u, + u for u, φ a; f = sup t b f(t) g t = f t f + f t f if t >, if t =, if t <, is a arbitrary positive iteger ad c is a positive costat depedig oly o. Proof. We decopose f(t) ito four parts as f + + f( ) f + f( ) f t = + sg + t g t f + f + δ t f + where g (t) is defied as (4) ad sg t sg t, δ t =, = t,, t, Fro (5) we have f + + f( ) f + f( ) D, f = + D +, g + D, sg + f + f( ) + f D, δ. + For operators D, usig (6) we ca observe that the last ter o the right had side vaishes. I additio it is obvious that D, =. Hece we have (4) (5) (6)

6 A Study o the Rate of Covergece of Chlodovsky-Durreyer Operator ad Their Bézier Variat f + + f D, f + D, g + f + f D, sg + +, (7) I order to prove our theore we eed the estiates for D, g ad D, sg +. + To estiate D, g with the help of the fied poits ad, we decopose it ito three parts as follows: g (t)k t,, g (t)k t,, + g (t)k t,, + g a (t)k t,, = E,, + E,, + E 3,,, (8) where K t,, is defied i Lea.. The estiatios for E,, ad E,, are give i Lea.3 i which we put = ad =, respectively. Usig the obvious ieuality µ j g ; I ( u) + µ j g ; I (u) µ j g ; H (u), where H u = u, + u, < u, we obtai E,, + E,, µ j g ; H + 6W,() µ j g ; H j d j 3 Now, we estiate E 3,, Clearly, give ay N, we have E 3,, φ ; f + φ ; f + i= φ ; f + i= i= ( ) Q,i ( ) Q,i + µ g ; H. (9) P,i P,i t (t ) P,i t (t ) P,i t = φ ; f W,. () Fially, replacig by i the result of X. M. Zeg ad W. Che [] (sect. 3, pp. 9-) we iediately get D, sg Puttig (8), (9), () ad () ito (7), we get the reuired result. Thus the proof of Theore is coplete. Fro Theore 3. ad ieuality (6) we get Theore 3. Let f BV p loc [, ), p ad let,. The, for all itegers such that > ad 4 we have f + + f( ) D, f V + p g ; H ( + 7+ p + p + + i= V p g ; H i i p DOI:.979/ Page

7 A Study o the Rate of Covergece of Chlodovsky-Durreyer Operator ad Their Bézier Variat + C φ ; f + f + f +. I order to show this it is ecessary to prove that the right-had sides of the ieualities etioed i the hypotheses of the theores ted to zero as. I view of (4) we have = as. So, i Theore it is eough to cosider oly the ter Clearly, = = 4d µ j g ; H (jd ) j 3 µ g ; H (jd ) j + µ g ; H s Sice the fuctio g is cotiuous at ad µ g ; H i we have ad coseuetly,, were d =. 4d ds 4 li i µ g ; H i d d i= µ g ; H (t) t µ g ; H i li = As regards Theore 3., it is easy to verify that i view of the cotiuity of g at,. deotes the oscillatio of g o the iterval H i li V + p i p p g ; H =. i i= Thus we get the followig approiatio theore. Proof. Let f BV p loc,, p. I view of (6) ad the otatio d =, =, we have d µ j g ; H jd V p g ; H jd V + p p g ; H t j 3 d j+ p t+ p ad oreover, (+) + p V p g ; H s s + p ds µ g ; H = V p g ; H + p d (+) + p V p g ; H i i= µ g ; H V p g ; H The estiatio give i Theore 3. ow iediately follows fro Theore 3.. Corollary. Suppose that f X loc, (i particular, f BV p loc,, p ) ad that there eists a positive iteger such that a li φ a ; f =. The at every poit [, ) at which the liits f +, f( ) eist we have li D f + + f( ),f = + Obviously, the above relatios hold true for every easurable fuctio f bouded o [, ), i particular for every fuctio f of bouded p-th power variatio (p ) o the whole iterval [, ). i + p, DOI:.979/ Page

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