Research Article Approximation by Lupas-Type Operators and Szász-Mirakyan-Type Operators

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1 Hidawi Publishig Corporatio Joural of Applied Mathematics Volume, Article ID , 8 pages doi:.55// Research Article Approximatio by Lupas-Type Operators ad Szász-Mirakya-Type Operators Hee Su Jug ad Ryozi Sakai Departmet of Mathematics Educatio, Sugkyukwa Uiversity, Seoul -745, Republic of Korea Departmet of Mathematics, Meijo Uiversity, Nagoya , Japa Correspodece should be addressed to Hee Su Jug, hsu9@skku.edu Received 8 July ; Accepted 5 Jauary Academic Editor: Yuatog Gu Copyright q H. S. Jug ad R. Sakai. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Lupas-type operators ad Szász-Mirakya-type operators are the modificatios of Berstei polyomials to ifiite itervals. I this paper, we ivestigate the covergece of Lupas-type operators ad Szász-Mirakya-type operators o,.. Itroductio ad Mai Results For f C,, Berstei operator B f is defied as follows: Let p,k x k x k, k,. k ad the we defie B f k p,k f.. Derrieic gave a modified operator of B f such as M f p,k p,k tftdt,.3

2 Joural of Applied Mathematics ad obtaied the result that for f C,, lim M f f xf x xf..4 Lupas ivestigated a family of liear positive operators which mapped the class of all bouded ad cotiuous fuctios o, ito C, such that L f k x k k k x f, x,..5 k Moreover, Sahai ad Prasad modified Lupas operators as follows: Let f be itegrable o, ad let be a positive iteger. The we defie where [ ] M f P,k P,k f dy, x,,.6 k x k P,k k x. k.7 I this paper, we assume that is a positive iteger. The they obtaied the followig; Theorem. see, Theorem. If f is itegrable o, ad admits its r th ad r th derivatives, which are bouded at a poit x,, ad f r Ox α α is a positive iteger asx,the lim M [ ] r f f r r xf r x xf r..8 Theorem. holds oly for bouded x K, so it does ot mea the orm covergece o,. I this paper, we improve Theorem. with respect to the orm covergece o,. Let <p ad let w be a positive weight, that is, w forx R. For a fuctio g o,, we defie the orm by g Lp, : /p gt dt p, <p<,.9 gt, p. sup, For coveiece, for oegative itegers, r, ad r, we let A,r :!! r! r!..

3 Joural of Applied Mathematics 3 The we have the followig results: Theorem.. Let <p. Letα ad r be oegative itegers ad r. Letf C r, satisfy f r x α, f r x α.. The we have uiformly for f ad, [ ] r A,r M f f r /3 x α.. I particular, if x α w Lp, <, the we have uiformly for, [ ] r A,r M f f r w Lp, Remark.3. a We see that for oegative itegers, r, ad r, /3..3!! as..4 r! r! b The followig weight is useful. λ> α, <p<, w λ p x λ λ α, p..5 Let ψ : x..6 Theorem.4. Let r ad β be oegative itegers ad r. Letf C r, satisfy f r ψ β <, L, f r ψ β <. L,.7 The we have uiformly for f ad, [ ] r A,r M f f r ψ β f r ψ β f r ψ β L, L,..8

4 4 Joural of Applied Mathematics Let us defie the weighted modulus of smoothess by ω k f; η; t : sup Δ k h f η L, t, k,,,.9 ht where Δ h f fx h f, Δ hf f fx h fx h.. Theorem.5. Let β ad r be oegative itegers ad r. Letf C r,. Thewe have uiformly for f ad, [ ] r A,r M f f r ψ β C ω f r ; ψ β ; L, ω f r ; ψ β ;.. TheSzász-Mirakya operators are also geeralizatios of Berstei polyomials o ifiite itervals. They are defied by: k S f S,k f,. where S,k e x k..3 k! I 3, the class of Szász-Mirakya operators S ;r,q f; x was defied as follows: rk rk S ;r,q f; x : A r rk! f, x,,.4 q where q>ad A r t t rk, t,..5 rk! Theorem.6 see 3. Let q> ad r N be fixed umbers. The there exists M q,r cost. > depedig oly o q ad r such that, for every uiformly cotiuous ad bouded fuctio fe qx o,, the followig iequalities hold;

5 Joural of Applied Mathematics 5 a S ;q,r f; x f ϕe qx L, M q,r f e qx q L, f e qx L ;,.6 b S ;q,r f; x f ϕe qx L, M q,r δ,q ω f; e qx ; δ,q ω f; e qx.7 ; δ,q, q where δ,q : q /. c for every fixed x,, we have for every cotiuous f with f j e qx, j,,, bouded o,, lim S ;q,r f; x f qxf x f..8 Now, we modify the Szász-Mirakya operators as follows: let f be itegrable o,, the we defie [ ] Q,β f β S,k where β is a oegative iteger. The we have the followig results: S β,k f dy, x,,.9 Theorem.7. Let α, β ad r be oegative itegers. Let f C r, satisfies f r e βx x α, f r e βx x α..3 The oe has uiformly for f ad, β r [ ] r Q,β f f r /3 e βx x α..3 I particular, let <p. If oe supposes e βx x α w Lp, <, the oe has uiformly for f ad, β r [ ] r Q,β f f r w Lp,. /3.3

6 6 Joural of Applied Mathematics Remark.8. a We ote that for oegative itegers β ad r, β r as..33 b The followig weight is useful. w λ,β e βx w λ,.34 where w λ is defied i Remark.3. Theorem.9. Let β, γ, ad r be oegative itegers. Let f C r, satisfies f r e βx ψ γ <, L, f r e βx ψ γ <. L,.35 The oe has uiformly for f ad, [ β r ] [ ] r Q,β f f r f r e βx ψ γ e βx ψ γ f r e βx ψ γ L, L,..36 Theorem.. Let β, γ, ad r be oegative itegers. The oe has for f C r,, [ β r ] [ ] r Q,β f f r C ω f; e βx ψ γ ; e βx ψ γ L, ω f; e βx ψ γ ;..37. Proofs of Results First, we will prove results for Lupas-type operators such as Theorems.,.4, ad.5. To prove theorems, we eed some lemmas. Lemma.. Let m ad r be oegative itegers ad >m r. Let T,m,r : r P r,k P r,kr x mdy.. The i T,,r,

7 Joural of Applied Mathematics 7 ii T,,r T,,r r x r x r r x r r 3 r r 3 ;,. iii for m, m r T,m,r x x T,m,r mt,m,r m r xt,m,r,.3 where >mr ; iv for m, T,m,r m/ q,m,r,.4 where q,m,r is a polyomial of degree m such that the coefficiets are bouded idepedetly of ad they are positive for >m r. Proof. i, ii, adiii have bee proved i, Lemma. So we may show oly the part of.4. For m,,.4 holds. Let us assume.4 for m.weote T,m,r m/ q m,r, q m,r P m..5 So, we have by the assumptio of iductio, m r T,m,r x x T,m,r mt,m,r m r xt,m,r x x C m/ q m,r m q m,r m/ m r x q m,r. m/.6 Here, if m is eve, the [ m ] m m [ m ], [ m ] m m [ m ],.7

8 8 Joural of Applied Mathematics ad if m is odd, the [ m ] m [ m ], [ m ] m m [ m ]..8 Hece, we have T,m,r m/ q,m,r,.9 ad here we see that q,m,r is a polyomial of degree m such that the coefficiets of q,m,r are bouded idepedetly of. Moreover, we see from.6 that the coefficiets of q,m,r are positive for >m r. Lemma. see, Lemma. Let r be a oegative iteger ad r. The oe has for f C r, : M [ f ] r r! r!!! P r,k P r,kr f r dy.. Let [ ] r M f r P r,k P r,kr f r dy.. The we have M [ f ] r A,r M [ f ] r,. where A,r is defied by.. Proof of Theorem.. Let y x. By the secod iequality i., f r f r y x f r ξ y x ξ α y x x α..3

9 Joural of Applied Mathematics 9 Let ε : γ, <γ<, [ ] r M f f r r P r,k P r,kr y f r f r dy y x <ε y x ε P r,kr y f r f r dy : A B..4 First, we see by.3 ad Lemma., A r P r,k y x <ε ε T,,r x α εx α. P r,kr y y x x α dy.5 Next, we estimate B. By the first iequality i., B C r P r,k C r P r,k P r,kr y f r f r dy α P r,kr y y x α dy..6 Here, usig α α α α y ix y y x x x α i i i.7 ad the otatio:, i :odd i, i :eve,.8 we have α α y ix B C r P r,k P r,kr y x α i x α dy i i

10 Joural of Applied Mathematics α α y i y x C r P r,k P r,kr y x i i ε y x C r P r,k P r,kr y x α dy ε i x α i dy : B B..9 The, we obtai α α B C T,i i,r i x α i i i ε α α γ i C O q,i i,r x α i i i i i / x α i x α. γ i i / γ i. Here, we used the followig that for i, [ i i ] γ i γ,. because [ i i ] γ i i γ, i :odd,. i, i :eve. Ad we kow that B C T,,r ε γ 3/ x α q,,r 3/ q,,r x α γ ε x α. x α.3 Thus, we obtai B x α. γ.4

11 Joural of Applied Mathematics Therefore, we have uiformly o, [ ] r M f f r x γ α Here, if we let γ /3, the we have [ ] r M f f r /3 γ x α, x α..5.6 that is,. is proved. So, we also have a orm covergece.3. Proof of Theorem.4. We kow that for f C r,, t x f r t f r f r t x t uf r f udu C r ψ β L, t x t uf r udu, x β t β t x,.7.8 where ψt / x. The we obtai from. ad.7, M [ f ] r f r f r T,,r r P r,k x y P r,kr y y u f r udu dy.9 ad from.8, r P r,k f r ψ β y P r,kr y y u f r udu dy L, x β T,,r r P r,k x P r,kr y β x dy..3 Usig y β Cy x β x β, we have r P r,k P r,kr y β x dy C T,β,r β T,,r..3

12 Joural of Applied Mathematics Therefore, we have [ ] r M f f r ψβ f r ψ β T,,r ψ Cf r ψ β L, xβ T,,r ψ β Cf r ψ β T,β,r ψ β β L, f r ψ β g,,r ψ f r ψ β L, xβ g,,r ψ β f r ψ β L, g,β,r ψ β..3 Sice we kow that for x,, g,,r ψ C, x β g,,r ψ β C, g,β,r ψ β C,.33 we have [ ] r M f f r ψβ f r ψ β f r ψ β L, L,..34 Lemma.3. Let r ad β be oegative itegers ad r. Letf C r, satisfies f r ψ β L, <..35 The oe has uiformly for, f ad x,, [ ] r M f ψ β Cf r ψ β. L,.36

13 Joural of Applied Mathematics 3 Proof. Usig y β Cy x β x β, we have r P r,k βdy P r,kr y y C ψ β T,β,r..37 The assumptio.35 meas f r C y β..38 The we ca obtai by., [ ] r M f C r P r,k P r,kr y βdy f r ψ β L, C ψ β T,β,r f r ψ β L, C ψ β q,β,r f r ψ β L,. β.39 Cosequetly, sice q,β,r ψ β is uiformly bouded o,, we have the result. The Steklov fuctio f h for f C, is defied as follows: [ f ]h : 4 h/ [ ] fx s t fx s t ds dt, x,h>..4 h The for the Steklov fuctio f h with respect to f C,,wehavethefollowig properties. Lemma.4 cf.4. Let f C, ad η be a positive ad oicreasig fuctio o,. Theif h C, ; ii [ f ] h f η ω L, f; η; h ;.4 iii [ f ] h η L 4, h ω f; η; h η ηx h/ h ω f; η; h η ηx h ;.4

14 4 Joural of Applied Mathematics iv [ f ] h η L 4 [ ω, h f; η; h 4 ω ] f; η; h..43 Proof. i For f C,, we have the Steklov fuctios f h ad f h as follows. We ote [ f ]h 4 h h/ xh/ x xh fu tdu fu tdu dt, x, h>..44 x The, we ca see from.44, [ ] f h 4 h/ [ f x h h t fx t fx h t fx t ] dt 4 h/ [Δ h/ h fx t ] Δh fx t dt..45 Similarly to.44,wekow [ ] f h 4 [ h xh/ x f u h fu du 4 xh x ] fu h fu du..46 Therefore, we have from.46, [ ] f h 4 [ fx h f x h f ] fx h fx h f h 4 4 h [ Δ h/ f 4 Δ h f ]..47 Therefore, i is proved. ii We easily see from.44 that h/ f [ f ] h η 4 Δ h st fηds dt ω f; η; h..48

15 Joural of Applied Mathematics 5 iii From.46, we have [ f ] h η 4 h 4 h 4 h ω h/ h/ η Δ h/ fx tηx t ηx t dt η Δ h fx tηx t ηx t dt f; η; h η ηx h/ h ω η f; η; h ηx h..49 iv From.47, we have [ f ] h η 4 [ω h f; η; h 4 ω ] f; η; h..5 Proof of Theorem.5. We kow that for f C r,, [ f ] r h [ f r] h, [ ] r f h [f r], h [ ] r f h [f r]. h.5 The, we have [ ] r M f f r ψ β [ [ ] M f f h] r ψ β L, L, [ ] r M fh f r h ψ β [ ] f r h f r ψ β. L, L,.5 From.5 ad.4 of Lemma.4, [ [ ] M f f h] r [ ψ β f r [ f ] r ]ψ h β L, [ f r [f r] ] ψ β h ω f r ; ψ β ; h. L, L,.53

16 6 Joural of Applied Mathematics Here, we suppose <h ad the we kow that ψ ψ, ψx h. ψx h/.54 From Theorem.4,.5,.4, ad.43 of Lemma.4, we have [[ ] ] r r M f h f ψ h β L, [ ] r f h ψ β [ ] r f L, h ψ β h ω f r ; ψ β ; h h ω f r ; ψ β ; h. L,.55 Therefore, we have M f r f r ψ β L, h ω f r ; ψ β ; h h ω f r ; ψ β ; h ω f r ; ψ β ; h..56 If we let h /, the [ ] r M f f r ψ β C ω f r ; ψ β ; L, ω f r ; ψ β ;,.57 because ω f r ; ψ β ;/ ω f r ; ψ β ;/. From ow o, we will prove Theorems.7,.9, ad., which are the results for the Szász-Mirakya operators, aalogously to the case of Lupas-type operators. Lemma.5. Let r be a oegative iteger. The oe has for f C r,, [ ] r r Q,β f β S,k β S β,kr f r dy, x,..58

17 Joural of Applied Mathematics 7 Proof. We kow that r r e y r i k i y S r,k i k! i r r e y i k r i y S r,k i k! i r r r i r S,k i, i i r r i r S,k ri. i i.59 Therefore, we have Q,β [ f ] r β S r,k S β,k y f y dy β r r r i r S,k i S β,k y f y dy i ki i β r r r S,k r i β r Sβ,ki y f y dy β i β r S,k β β r S,k β i r S r β,kr y f dy S β,kr f r dy..6 Lemma.6. Let a, b, ad m be oegative itegers. R,m,r a, b; x : b S a,k S b,kr x mdy..6 The oe has i R,,r a, b; x ad R,,r a, b; x a bx r / b; ii For m br,m,r a, b; x xr,m,ra, b; x a bx m r R,m,r a, b; x xmr,m,r a, b; x;.6

18 8 Joural of Applied Mathematics iii R,m,r a, b; x m/ g,m,r a, b; x,.63 where g,m,r a, b; x is a polyomial of degree m such that the coefficiets of g,m,r a, b; x are bouded idepedetly of. Proof. Let R,m,r : R,m,r a, b; x. The i R,,r b S a,k R,,r b S a,k k r S a,k b k S a,k b ax b r b S b,kr dy, S b,kr x dy x r b x a bx r x. b.64 ii Usig xs,k k xs,k,weobtai x R,m,r mr,m,r b xs b S,k a,k S b,kr x mdy k as b,kr x mdy..65 Here, we see k as b,kr k r by r a b b x S b,kr y.66 ys b,kr y r a bsb,kr y bsb,kr y y x.

19 Joural of Applied Mathematics 9 The substitutig.66 for.65, we cosider the followig; x R,m,r mr,m,r b S,k mdy bs b,kr y y x y x :. 3 ys b,kr y r a bsb,kr y.67 The, we have b S,k b S a,k x b S a,k ys b,kr y x mdy S b,kr y x mdy S b,kr y x mdy.68 m R,m,r xmr,m,r. Here the last equatio follows by parts of itegratio. Furthermore, we have r a br,m,r br,m,r..69 Therefore, we have br,m,r xr,m,r a bx m r R,m,r xmr,m,r..7 iii It is proved by the same method as the proof of Lemma. iv. Proof of Theorem.7. Let y x. By the secod iequality i.3, f r f r y x f r ξ C y x e βξ ξ α C y x e βx x α..7

20 Joural of Applied Mathematics Let ε γ,<γ<, β r [ ] r Q,β f f r β S,k S β,kr y f r f r dy y x <ε y x ε S β,kr y f r f r dy.7 : A B. First, we see that by.7 ad Lemma.6i, A C β S,k y x <ε Cεe βx x α γ S β,kr y x dy e βx x α e βx x α..73 Next, to estimate B, we split it ito two parts: B β S,k β S,k β S,k y x ε y x ε y x ε S β,kr y f r f r dy S β,kr y f r f r dy S β,kr e βy α e βx x α dy.74 : B B. First, we estimate B β S,k S β,kr y e βy α dy..75 The, usig the followig facts: S β,kr y e βy β kr S,kr y,.76

21 Joural of Applied Mathematics β kr β r S,k S β,ke βx,.77 α α α α y ix y y x x x α i,.78 i i we have B Ce βx β β r S β,k β r Ce βx S β,k S,kr αdy α α y ix S,kr y x α i dy i i β r Ce βx S β,k S,kr y x α dy.79 : Ce βx B B. The, usig.8 ad Lemma.6, we have α α y ix B S β,k S,kr y x α i dy i i α α y i y x i S β,k S,kr y x i i ε x α i dy α α y i i i S β,k S,kr y x x α i dy i i ε α α i R,i i r β, ; x x α i i i ε α α i q i i i /,i i,r β, ; x x α i. i.8 The by. we have B x α. γ.8

22 Joural of Applied Mathematics For B, we have B S β,k S β,k S,kr x α dy y x S,kr y dyx α ε R,,r β, ; x x α γ ε x α. γ 3/ q,,r β, ; x x α.8 From.8,.8 ad.79, we have B e βx x α. γ.83 We estimate B β S,k S β,kr y e βx x α dy..84 The we ca estimate B by the same method as B, B β S,k β S,k S β,kr e βx x α dy y x S β,kr y dy e βx x α ε.85 R,,r,β; x e βx x α, ε so we have B γ 3/ q,,r,β; x e βx x α γ e βx x α..86 Cosequetly, we obtai from.83 ad.86, B O e βx x α e βx x α. γ γ γ.87

23 Joural of Applied Mathematics 3 Therefore, from.73 ad.87, we have uiformly o, β r [ ] r Q,β f f r e βx γ x α γ e βx x α..88 Here, if we let γ /3, the we have β r [ ] r Q,β f f r /3 e βx x α,.89 that is,.3 is proved. So, we also have a orm covergece.3. Lemma.7. Let m ad b be oegative itegers. Let U,m,r b; x : b S,k S b,kr x me by dy..9 The oe has b r U,m,r b; x e bx R,m,rb, ; x..9 Proof. From.76 ad.77 we have b kr S b,kr e bx S,kr,.9 b kr b r S,k S b,ke bx..93 We have from.9,.93, ad otig.6, U,m,r b; x b S,k S b,kr x me by dy b kr mdy b S,k S,kr y y x b r b e bx S b,k S,kr x mdy.94 b r e bx R,m,rb, ; x.

24 4 Joural of Applied Mathematics Proof of Theorem.9. We prove this theorem, similarly to the proof of Theorem.4. Usig.93 ad.7, we have for f C r,, β r [ ] r Q,β f β S,k S β,kr f r dy f r f r β S,k β S,k f r f r R,,r,β; x β S,k S β,kr y x S β,kr y x S β,kr x dy y u f r udu dy y u f r udu dy..95 We estimate the last term. We ote the give coditio: f r e βx ψ γ <. L,.96 Usig the iequality t γ C x γ t x γ ad Lemma.7, we have : β S,k C β S,k S β,kr e βy ψ γ e βx ψ γ y x dy S β,kr e βy ψ γ e βy γ y y x e βx ψ γ dy x C ψ γ U,,r β; x U,γ,r β; x e βx ψ γ R,,r,β; x C e βx ψ γ β R,,r β, ; x r.97 e βx β r R,γ,r β, ; x e βx ψ γ R,,r,β; x. The, we ca estimate as follows: y β S,k S β,kr y y u f r udu dy x

25 Joural of Applied Mathematics 5 f r e βx ψ γ Cf r e βx ψ γ L, L, eβx ψ γ β r β r R,,r β, ; x R,γ,r β, ; x ψ γ R,,r,β; x..98 The, we have by iv of Lemma.6, β r [ ] r Q,β f f r e βx ψ γ f r e βx ψ γ L, ψ g,,r β, ; x O γ g,γ,r β, ; x ψ γ O ψ g,,r,β; x f r e βx ψ γ. g,,r,β; x ψ.99 Cosequetly, we have [ β r ] [ ] r Q,β f f r f r e βx ψ γ e βx ψ γ f r e βx ψ γ L, L,,. sice we kow that g,,r β, ; xψ, g,,r,β; xψ, g,γ,r β, ; xψ γ, ad g,,r,β; xψ are uiformly bouded o,. Theorem.8. Let β ad γ be oegative itegers ad r> be a positive iteger. The oe has uiformly for, f ad x,, [ Q,β [ ] ] r f e βx ψ γ C β f r e βx ψ γ. L,.

26 6 Joural of Applied Mathematics Proof. Usig y γ C x γ y x γ, we have β S,k S β,kr e βy y γ dy β S,k S β,kr y e βy x γ x γ dy C U,,r β; x ψ γ U,γ,r β; x.. By Lemma.7 ad i of Lemma.6,wekow β r U,,r β; x e βx β r R,,r β, ; x e βx, β r U,γ,r β; x e βx R,γ,r β, γ; x.3 γ β r e βx g,γ,r β, γ; x. Therefore, we have β S,k S β,kr e βy y γ dy β r C e βx ψ γ γ g,γ,r β, γ; x..4 Sice g,γ,r β, γ; xψ γ is uiformly bouded o,, we have β r [ ] re Q,β f βx ψ γ β S,k S β,kr y e βy y γ dy e βx ψ γ f r e βx ψ γ L,.5 β C r γ f r e βx ψ γ L,. Therefore, we have the result.

27 Joural of Applied Mathematics 7 Proof of Theorem.. We will prove this theorem by the same method as the proof of Theorem.5. First, we split it as follows: [ β r ] [ ] r Q,β f f r e βx ψ γ L, β r [ [ ] re Q,β f f h] βx ψ γ L, [ β r [[ ] ] r [ ] r Q,β f h f ]e h βx ψ γ [ [f ] ] r f r e βx ψ γ. L, h L,.6 The for the first term, we have, usig Theorem.8 ad.4, Q,β f f h r e βx ψ γ C β L, [ ] f r f r h e βx ψ γ Cω L, f r ; e βx ψ γ ; h..7 For the secod term, we have from Theorem.9, [ β r [ ] r r Q,β fh f ]e h βx ψ γ L, f r h e βx ψ γ f r L, h e βx ψ γ L,..8 Here, we suppose < h ad the we kow that e βx ψ/e βh ψx h ad e βx /e βh are uiformly bouded o,. Therefore, we have from.4 ad.43 of Lemma.4, f r h f r h e βx ψ γ C L, h ω f; e βx ψ γ ; h, e βx ψ γ C L, h ω f; e βx ψ γ ; h..9 Therefore, we have [ β r [ ] r r Q,β fh f ]e h βx ψ L, h ω f; e βx ψ ; h h ω f; e βx ψ γ ; h..

28 8 Joural of Applied Mathematics Cosequetly, we have [ β r ] [ ] r Q,β f f r e βx ψ γ h ω f; e βx ψ γ ; h h ω f; e βx ψ γ ; h Cω f r ; e βx ψ γ ; h.. If we let h /, the [ β r ] [ ] r Q,β f f r C ω f; e βx ψ γ ; e βx ψ γ ω f; e βx ψ γ ;,. sice ω f r ; e βx ψ γ ;/ ω f; e βx ψ γ ;/. 3. Coclusio I this paper, Lupas-type operators ad Szász-Mirakya-type operators are treated ad the various weighted orm covergece o, of these operators are ivestigated. Moreover, this paper proves theorems o degree of approximatio of f C r, by these operators usig the modulus of smoothess of f. Ackowledgmets The authors thak the referees for may valuable suggestios ad commets. H. S. Jug was supported by SEOK CHUN Research Fud, Sugkyukwa Uiversity,. Refereces M. M. Derrieic, Sur l approximatio de foctios itégrables sur, par des poliômes de Berstei modifies, Joural of Approximatio Theory, vol. 3, pp , 98. A. Sahai ad G. Prasad, O simultaeous approximatio by modified Lupas operators, Joural of Approximatio Theory, vol. 45, o., pp. 8, L. Rempulska ad S. Graczyk, O certai class of Szász-Mirakya operators i expoetial weight spaces, Iteratioal Joural of Pure ad Applied Mathematics, vol. 6, o. 3, pp ,. 4 M. Becker, D. Kucharski, ad R. J. Nessel, Global approximatio theorems for the Szász-Mirakja operators i expoetial weight spaces, i Liear Spaces ad Approximatio, pp , Birkhäuser, Basle, Switzerlad, 978.

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