DEMONSTRATIO MATHEMATICA Vol. XLI No 4 8 Deepak Kumar Dubey, V. K. Jai RATE OF APPROXIMATION FOR INTEGRATED SZASZ-MIRAKYAN OPERATORS Absrac. Recely Jai e al. [3] proposed a iegral modificaio of Szasz-Mirakya operaors S,α (f,, α > ad sudied some direc approimaio heorems i simulaeous approimaio. The prese paper deals wih he rae of approimaio of such operaors, for fucios which have derivaives of bouded variaio. 1. Iroducio To approimae iegrable fucios o he ierval [,, ad for α >, we proposed i [] ad [3] he iegral modificaio of he Szasz-Mirakya operaors as (1 S,α (f, = = v= W,α (, f(d s,v ( b,v,α (f(d, where he kerel W,α (, is defied as: W,α (, = s,v (b,v,α (, v= [,, ad he Szasz ad Bea basis fucios are give by s,v ( = ep( (v, b,v,α ( = α Γ( α + v + 1 (α v v! Γ(v + 1Γ( α (1 + α ( +v+1. α I case α = 1, he above operaors (1 reduce o he Szasz-Bea operaors sudied i [6]. Some direc resuls i simulaeous approimaio o S,α (f, for ieraive combiaios ad wihou combiaios were discussed i [] ad [3] respecively. Very recely Gupa ad Siha [5] iroduced similar ype of operaors, bu hey have cosidered he value of fucio a zero eplicily, he operaors discussed i [5] for α >, are
88 D. K. Dubey, V. K. Jai defied as V,α (f, = ( α v=1 where s,v ( is as defied above ad s,v ( p,v 1,α (f(d + e f(, [, p,v,α ( = Γ( α + v (α v Γ(v + 1Γ( α (1 + α α +v. The above wo iegral modificaios S,α (f, ad V,α (f, of Szasz- Mirakya operaors are very similar. The mai differece bewee hese wo are ha V,α (f, defied i [5] are discreely defied a f( o preserve he cosa fucios, while he operaors S,α (f, are he usual iegral modificaio of he Szasz-Mirakya operaors havig he weigh fucio of geeralized Baskakov operaors. As he operaors (1 are he geeralizaio of he operaors discussed i [6], his moivaed us o sudy furher o such operaors. We defie Ì Ì β,α (, = W,α(, sds, he as a special case we have β,α (, = W,α(, sds = 1. Le DB γ (,, γ be he class of absoluely coiuous fucios f defied o (, saisfyig he growh codiio f( = O( γ, ad havig a derivaive f o he ierval (, coicidig a.e. wih a fucio which is of bouded variaio o every fiie subierval of (,. I ca be observed ha all fucios f BD γ (, posses for each c > a represeaio f( = f(c + ψ(d, c. c Aoher opic of ieres is he rae of covergece for fucios havig derivaives of bouded variaio. Such ype of problems were discussed i [1] ad [4], where he rae of covergece have bee discussed for Bersei ad some oher iegral operaors. I he prese paper, we eed he sudy ad obai he rae of approimaio for differeial fucios of bouded variaio.. Auiliary resuls We shall use he followig Lemmas o prove our mai heorem. Lemma 1 ([]. Le he fucio µ,m,α (, m ℵ ad α > be defied as µ,m,α ( = s,v ( b,v,α (( m d. v= The by easy compuaio, we have
Rae of approimaio 881 µ,,α ( = 1, µ,1,α ( = (1 + α α, µ,,α ( = (4α + α + + (α +. ( α( α Also for > α(m + 1, we have he recurrece relaio: [ α(m + 1]µ,m+1,α ( = µ (1,m,α( + [(m + 1(1 + α α]µ,m,α ( + m(α + µ,m 1,α (. Cosequely for each [,, i follows from he recurrece relaio ha µ,m,α ( = O( [(m+1/]. Remark 1. Paricularly for ay umber λ > 1 ad [,, usig Lemma 1, for sufficiely large, we have ( S,α (( λ( + α, µ,,α (. Remark. I view of Remark 1, i ca be easily verified by Holder s iequaliy ha (3 S,α (, [µ,,α (] 1/ λ( + α. Lemma. Le [,, λ > 1, he for sufficiely large, we have y λ( + α (i β,α (, y = W,α (, d ( y, y <, λ( + α (ii 1 β,α (, z = W,α (, d (z, < z <. Proof. Firs we prove (i, usig (, we have z y y ( W,α (, d ( y W,α(, d ( y µ,,α ( The proof of (ii is similar, so we omi he deails. 3. Mai resul This secio deals wih he followig mai heorem. λ( + α ( y. Theorem 1. Le f DB γ (,,γ > ad (,. The for λ > ad sufficiely large, we have
88 D. K. Dubey, V. K. Jai S,α (f, f( λ( + α ([ ] +/v v=1 /v ((f + +/ / ((f λ( + α + ( f( f( f ( + + f( λ( + α + (C γ O( γ + f ( + + 1 λ( + α f ( + f ( + 1 + α ( α f ( + + f (, where b a f( is he oal variaio of f o he ierval [a, b] ad he auiliary fucio f is defied as f( f(, < ; f ( =, = ; f( f( +, < < ; f( ad f( + represes he lef ad righ had limis a. Ì Proof. I is easily observed from Lemma 1 ha W,α(, d = 1, so we ca wrie Also, we ca wrie S,α (f, f( = = W,α (, (f( f(d ( W,α (, (f (udud. f (u = [f ( + + f ( ] + (f (u + [f ( + f ( ] sg(u [ + f ( [f ( + + f ( ] ] χ (u, where Ne, we have ( χ ( = { 1, = u, u. f ( [f ( + + f ( ] χ (udu W,α (, d =,
hus S,α (f, f( = Also ( ad ( ( + Rae of approimaio 883 W (, ( [f ( + + f ( ] ( W (, [f ( + f ( ] [f ( + f ( ] sg(u du W,α (, d + (f (udud sg(u dud. = [f ( + f ( ] S,α (, 1 [f ( + +f ( ]du W,α (, d = 1 [f ( + +f ( ]S,α ((,. We ca wrie S,α (f, f( ( (f (udu W,α (, d ( (f (udu W,α (, d + 1 f ( + f ( S,α (, + 1 f ( + + f ( S,α ((, A,α (f, + B,α (f, + 1 f ( + f ( S,α (, + 1 f ( + + f ( S,α ((,. By applyig Lemma 1 ad Remark, we have (4 S,α (f, f( = A,α (f, + B,α (f, + 1 f ( + f ( S,α (, + 1 f ( + + f ( S,α ((, A,α (f, + B,α (f, + 1 f ( + f ( + 1 f ( + + f ( (1 + α α. λ( + α
884 D. K. Dubey, V. K. Jai I order o complee he proof of he heorem i is sufficie o esimae he erms A (f, ad B (f,. Usig iegraio by pars, Lemma ad seig y = /, we have ( B,α (f, = (f (udu = d (β,α (, β,α (, (f (d y λ( + α y λ( + α y λ( + α (y + y ((f β,α (, d ((f 1 ( d + ((f d y ((f 1 ( d + ((f ((f 1 ( d + Le u =. The we obai y λ( + α ((f 1 λ( + α ( d = Therefore (5 B,α (f, Ne, we have (6 A,α (f, = = ( + [ ] λ( + α v=1 v ( (f (udu λ( + α / 1 [ ] ((f + W,α (, d ( (f (udu W,α (, d + (f (udu (f( f(w,α (, d + f ( + (f (udu (1 β,α (, + ((f. / ((f du u v=1 v ((f. ((f. d d(1 β,α (, d ( W,α (, d (f ( 1 β (, d
C W,α (, γ d + f( + f ( + + λ( + α Rae of approimaio 885 W,α (, ( d + [ ] v=1 + v ((f + + W,α (, ( d λ( + α ( f( f( f ( + ((f. Usig Holder s iequaliy, ad Lemma 1, we esimae he firs wo erms i he righ had side of (6 as follows: (7 C W,α (, γ d + f( C ( + f( W,α (, γ d C γ O( γ/ W,α (, ( d 1 ( W,α (, ( d λ( + α W,α (, ( d + f( λ( + α. Fially he hird erm of he righ side of (6 is esimaed as follows: (8 f ( + W,α (, d f ( + W,α (, d ( 1 ( f ( + W,α (, ( d W,α (, d = f ( + λ( + α. Combiig he esimaes (6 (8, we ge (9 A,α (f, + λ( + α [ ] v=1 + C γ O( γ/ λ( + α + v ( f( f( f ( + ((f + + λ( + α + f( ((f + f ( + λ( + α. 1 1 λ( + α
886 D. K. Dubey, V. K. Jai Fially combiig (4, (5 ad (9, we ge he desired resul. This complees he proof of Theorem 1. Remark 3. I may be oed ha uder he assumpio of Theorem 1, he covergece rae of he operaors S,α (f, o f is O ( 1, which is he covergece rae of he classical Szasz-Mirakya operaors. Ackowledgme. The auhors are hakful o he referee, for his/her criical suggesios, for he overall improveme of he paper. We are also hakful o Professor Maciej Maczyski for his effors i sedig he repors imely. Refereces [1] R. Bojaic ad F. Cheg, Rae of covergece of Bersei polyomials for fucios wih derivaives of bouded variaio, J. Mah. Aal. Appl. 141 (1989, 136 151. [] D. K. Dubey ad V. K. Jai, O he ieraive combiaios of Szasz-Mirakya- Durrmeyer operaors, Aal. Uiv. Oradea Fasc. Mah., i press. [3] S. Jai, D. K. Dubey ad R. K. Gagwar, Rae of approimaio for cerai Szasz- Mirakya-Durrmeyer operaors, Georgia Mah. J., o appear. [4] V. G u p a ad P. N A g r a w a l, Rae of covergece for cerai Baskakov Durrmeyer ype operaors, Aal. Uiv. Oradea Fasc. Mah. 14 (7, 33 39. [5] V. Gupa ad J. Siha, Direc resuls o cerai Szasz-Mirakya operaors, Appl. Mah Compu. 195 (8, 3 39. [6] V. Gupa, G. S. Srivasava ad A. Sahai, O simulaeous approimaio by Szasz Bea operaors, Soochow J. Mah. 1 (1 (1995, 1 11. DEPARTMENT OF MATHEMATICS BAREILLY COLLEGE BAREILLY 431, INDIA e-mail: deepak486@gmail.com Received November 3, 7; revised versio March 19, 8.