Approximation theorems for localized szász Mirakjan operators

Transcription

1 Joural of Approximatio Theory 152 (2008) Approximatio theorems for localized szász Miraja operators Lise Xie a,,1, Tigfa Xie b a Departmet of Mathematics, Lishui Uiversity, Lishui , Zhejiag Provice, People s Republic of Chia b Departmet of Mathematics, Chia Jiliag Uiversity, Hagzhou , Zhejiag Provice, People s Republic of Chia Received 12 March 2007; accepted 28 November 2007 Commuicated by Kirill Kopotu Available olie 14 December 2007 Abstract I the preset paper, we ivestigate the covergece ad the approximatio order of the localized Szász Miraja operators, ad obtai some ew results to improve the results due to Omey [Note o operators of Szász Miraja type, J. Approx. Theory 47 (1986) ] Elsevier Ic. All rights reserved. MSC: 41A36 Keywords: Localizatio; Szász Miraja operators; Covergece; Approximatio order 1. Itroductio Let C[0, ) deote the set of all cotiuous fuctios o [0, ). Some subsets of C[0, ) are defied as follows: C α ={f : f C[0, ) ad there exist A, α (0, ) such that f(t) Ae αt } ad C m ={f : f C[0, ) ad there exist A, B (0, ) ad m N such that f(t) B + At 2m }. Correspodig author. addresses: lisexie@tom.com (L. Xie), xietf@cjlu.edu.c (T. Xie). 1 Supported by Natioal Natural Sciece Foudatio of Chia ( ) ad Zhejiag Provicial Natural Sciece Foudatio of Chia (102005) /$ - see frot matter 2007 Elsevier Ic. All rights reserved. doi: /j.jat

2 126 L. Xie, T. Xie / Joural of Approximatio Theory 152 (2008) For f C α, the well-ow Szász Miraja operator is give by S (f, x) = e x =0 (x) f, x 0. For applicatio ad calculatio, several authors cosidered the followig localized Szász Miraja operator S,N (f, x) = e x N =0 (x) f, x 0, for various choices of N. Grof [1] obtaied that if f C α ad N() lim =, the for N = N() ad x [0, ) lim S,N (f, x) = f(x). Lehhoff [2] proved that if f C m, the for N =[(x + δ )] ad x [0, ), lim δ = implies (1). Omey [3] further proved the followig results. Let f C α. (i) If N = N(, x) such that N x lim =, (1) (2) (3) the (1) holds. (ii) If (3) holds uiformly i [x 1,x 2 ], 0 x 1 <x 2 <, the (1) holds also uiformly i this iterval. (iii) If N x lim = C(C is a fiite costat), the lim S,N (f, x) = f(x) C x e 1 2 u2 du. Later, Su [6] cosidered a ecessary coditio for (1) ad he poited out that (1) holds for every f C α ad every x [0, ) if ad oly if (2) holds. Zhou ad Zhou [7] also proved that (1) holds for every x [0, ) if ad oly if f 0 whe (2) does ot hold. From the above results, we ca see the coditio f C α is very importat. However, whe costructig the operator S,N (f, x) these authors oly used the value of f(x) o [0,x + δ ] or o [ 0, N ]. Therefore, there arises a atural problem: whether the coditio f Cα could be deleted. I the preset paper we cosider this problem ad get a positive aswer uder the hypothesis that {δ } =1 is bouded. I Sectio 2, we obtai a ew estimate about the erel of

3 L. Xie, T. Xie / Joural of Approximatio Theory 152 (2008) the Szász Miraja operator by usig oe of the cetral limit theorems i probability theory. I Sectio 3, we study the covergece of S,N (f, x). I Sectio 4, we cosider the approximatio properties of S,N (f, x). 2. Lemmas I studyig the approximatio of positive operators, may mathematicias always apply probabilistic methods. The essece of these methods is the applicatio of the so called Berry Essee uiform estimate. The result [5] is idicated below: For S = η 1 + η 2 + +η, oe has sup <t< P S σ t 1 t e 2 1 u2 du < 0.8E η 1 3 σ 3, where η 1, η 2,...,η,... is a sequece of idepedet idetically distributed radom variables with E(η 1 ) = 0, variace Var(η 1 ) = σ 2 ad E η 1 3 <. From the above, we ca easily obtai for x>0 sup <t< e x [x+t x] =0 (x) 1 t e 2 1 u2 du 0.8 3x + 1. (4) x The deficiecy of the iequality (4) lies i the fact that the left is a bouded variable whereas the right is ot whe x 0. To overcome the deficiecy, it is ecessary to establish a ew iequality. We have the followig result. Lemma 1. For x>0ad y (, ), there holds [x+y ] (x) 1 x y e 1 2 u2 du Ax 3x + 1 ( x + y ) 3, (5) e x =0 where A is a absolute costat. For x = 0 ad y>0, the iequality (5) also holds if we regard y x as. Proof. We eed the followig ouiform estimated cetral limit theorem established by Petrov [4]: Let η 1, η 2,...,η,...be a sequece of idepedet idetically distributed radom variables with Eη 1 = 0, Eη 2 1 = σ2 ad E η 1 3 <, the for all t (, ) P S σ t 1 t e 1 2 u2 du AE η 1 3 σ 3 (1 + t ) 3, (6) where A is a absolute costat. For x>0, we cosider the radom variable ξ 1 which has the Poisso distributio with x as parameter, that is, x x P(ξ 1 = ) = e, = 0, 1,... (7) It is easy to calculate that E(ξ 1 ) = x ad E(ξ 2 1 ) = x(x + 1).

4 128 L. Xie, T. Xie / Joural of Approximatio Theory 152 (2008) Therefore E(ξ 1 x) = 0 ad E(ξ 1 x) 2 = x. (8) Notig that E(ξ 1 x) 4 = x(3x + 1), the by the Cauchy iequality we get E ξ 1 x 3 x 3x + 1. (9) We deote η 1 = ξ 1 x, the (8) ad (9) imply E(η 1 ) = 0, σ 2 = E(η 2 1 ) = x ad E η 1 3 x 3x + 1. Let η1, η 2,...,η,... be a sequece of idepedet idetically distributed radom variables ad S = η 1 +η 2 + +η. From (7), it is easy to obtai by iductio that for a arbitrary costat M [M+x] P(S M) = e x (x). Therefore, for M = t x, P =0 S σ t = P(S t [x+t x] x) = e x =0 As such, by (6), we have [x+t x] (x) e x =0 1 t (x). e 2 1 u2 du A 3x + 1 x(1 + t ) 3. Let t x = y, the deductio of (5) is complete. Whe x = 0 ad y>0, we regard y x as, so the two sides of (5) equal to zero. This fiishes the proof of Lemma Covergece of S,N I this sectio we give the theorems o the covergece of the operators S,N (f, x). Theorem 1. If f C[0, ), ad if N = N(, x) such that (i) (N x)/ is uiformly bouded i [x 1,x 2 ],0<x 1 <x 2 <, ad (ii) N x lim = C(x) holds uiformly i the same iterval, the lim S,N (f, x) = f(x) C(x) x e 2 1 u2 du (10) holds uiformly i [x 1,x 2 ].

5 L. Xie, T. Xie / Joural of Approximatio Theory 152 (2008) Proof. Deote δ = (N x)/, G = sup 1 { δ }, ad for η > 0 ω(f, η) = max f(x) f(y). x,y [0,x 2 +G], x y <η Obviously, for λ > 0, we have ω(f, λη) (1 + λ)ω(f, η). (11) For x [x 1,x 2 ] we ca write S,N (f, x) = e x N =0 ( f ) (x) N f(x) + f(x)e x (x) =0 I 1 + I 2. Usig (11) we ca fid that I 1 (1 + 1 x)ω f,. (12) By Lemma 1, we have for x [x 1,x 2 ] x+ δ I 2 = f(x)e x (x) =0 1 = f(x) δ x ( ) e 2 1 x u2 2 3x2 + 1 du + O ( x1 + δ ) 3. (13) Combiig (12) ad (13), we ca deduce that (10) holds uiformly i [x 1,x 2 ]. The proof of Theorem 1 is ow complete. Usig the same method as i the proof of Theorem 1 we also have Theorem 2. If f C[0, ), ad if N = N(, x) such that (i) (N x)/ is uiformly bouded i [x 1,x 2 ],0 x 1 <x 2 <, ad (ii) N x lim = C(x), C(x) ρ > 0, holds uiformly i the same iterval, the lim S,N (f, x) = f(x) C(x) x e 2 1 u2 du holds uiformly i [x 1,x 2 ], where we regard C(x) x as for x = 0. Remar 1. From the costructio of S,N (f, x), we may see that S,N (f, x) will be meaigful oly whe C(0) >0. Corollary 1. If f C[0, ), ad if N = N(, x) such that (i) (N x)/ is uiformly bouded i [x 1,x 2 ],0 x 1 <x 2 <, ad (ii) N x lim =

6 130 L. Xie, T. Xie / Joural of Approximatio Theory 152 (2008) holds uiformly i the same iterval, the lim S,N (f, x) = f(x) holds uiformly i [x 1,x 2 ]. Corollary 2. Let f C[0, ) ad x 0 (0, ). If (N x 0 )/ is bouded, ad whe, (N x 0 )/ does ot coverge to, the lim S,N (f, x 0 ) = f(x 0 ) implies f(x 0 ) = 0. Corollary 3. Let x 0 (0, ). If (N x 0 )/ is bouded, the lim S,N (f, x 0 ) = f(x 0 ) holds for every f C[0, ) if ad oly if N x 0 lim =. 4. Approximatio order of S,N Cocerig the approximatio order of the operators S,N (f, x) approximatig to f(x), Omey [3] obtaied the followig result by methods of probability theory. Assumig f C α, ad for fixed x>0ad δ > 0 assumig f(t) f(x) t x C(x,δ) for t x δ ad t 0, (14) the for N = N(, x), N x lim if > 0 (15) l() implies sup S,N (f, x) f(x) <. (16) 1 Furthermore, if f (x) exists, the for N = N(, x),(15) ad N x = o( 1/6 ) (17) implies ( lim S,N (f, x) f(x) ) = 0. (18) Note that i (14) C(x,δ) is a costat depedig oly o x ad δ.

7 L. Xie, T. Xie / Joural of Approximatio Theory 152 (2008) I this sectio we will prove by a ew method that the coditios i the above result ca be weaeed. Our result is Theorem 3. If f C α, ad if for a fixed x>0, there exists δ > 0 such that f satisfies (14), the for N = N(, x), N x lim if > 1 (19) x l() implies (16). Furthermore, if f (x) exists, the for N = N(, x), (19) implies (18). Remar 2. Obviously, the coditio (19) is weaer tha the coditio (15). I Theorem 3 we o loger eed the coditio (17). Proof. To prove (16), ote that N ( ) (x) S,N (f, x) f(x)= e x f f(x) f(x)e x (x) =0 =N+1 I 1 + I 2. (20) We first discuss the case N x > δ. I this case, I 1 e x f f(x) (x) + e x x δ +e x f f(x) (x) By (14), we have x+δ< N I 11 + I 12 + I 13. I 11 C(x,δ)e x By f C α, we obtai =0 x (x) I 12 2 max 0 t x f(t) e x x >δ 0 <x δ f f(x) (x) (21) x C(x,δ). (22) (x) 2Ae αx x δ 2 (23) ad I 13 A e x>δ A e x>δ x (xe α ) + (α )x (x) e x >δ x(e α 1) e xe α (xe α ) + e αx x δ 2.

8 132 L. Xie, T. Xie / Joural of Approximatio Theory 152 (2008) Because e α α 1 = + 1 ( α ) 2 +, 2! we ca tae 1 such that, for > 1, e α 1 2α ad xe α x< δ 2. Thus, we have for > 1 Therefore e x>δ I 13 = O x ( 1 x(e α 1) e xe α (xe α ) ). e 2αx e 2αx e x>δ xe α > δ 2 xe α (xe α ) e xe α (xe α ) 4e 2αx xe α δ 2. (24) Combiig (21) (24), we have 1 I 1 =O x. (25) Next we estimate I 2. Usig Lemma 1, we get e x =N+1 Notig that for z 1 z (x) e 1 2 u2 du 1 2z = 1 z 2 N x x e 1 e 1 2 v dv = 1 z e 2 1 z2, we immediately obtai that for N x x 1 ad f C α I N x x ( e 1 N x 2 2 u2 x 3x + 1 du + O ( ) x N x 3. (26) + ) 2 x Ae αx Ae αx x 3x O ( ) x N x 3. + As such, by (19), there exists 2 such that for > 2 N x x l(), so I 2 =O x ( 1 ). (27)

9 L. Xie, T. Xie / Joural of Approximatio Theory 152 (2008) Combiig (20), (25) ad (27), we obtai that there exists 0 such that for > 0 we have S,N (f, x) f(x) C (x, δ) 1, (28) where C (x, δ) is a costat depedig oly o x ad δ. Ad for 1 0, sice f C α we also have (28). I fact, sice f C α we have N S,N (f, x) f(x) Ae x (xe α ) =0 1 A 0 (e 0xe α + e αx). + Ae αx A (e 0xe α + e αx) So we complete the proof of (28) for satisfyig the case N x > δ. For the case N x δ the proof of (28) follows similarly. The proof of (16) is complete. Now we prove (18). Sice f (x) exists, for a arbitrary ε>0, there exists δ (0 < δ x) such that f(t) f(x) f (x)(t x) <ε t x for t x < δ. (29) We first discuss the case N x δ. I this case, we write ( ) (x) S,N (f, x) f(x)= e x f f(x) x δ ( ) (x) + e x f f(x) Usig (29), we have J 1 f (x)e x = f (x)e x f (x) x x + ε δ. Sice f C α, we obtai J 2 2Ae αx e x 0 <x δ [x+δ] ( ) (x) e x f f(x) f(x)e x (x) =N+1 =N+1 J 1 + J 2 + J 3 + J 4. (30) x δ x >δ 0 <x δ ( x ( x ) (x) + εe x x (x) x δ ) (x) + εe x x (x) x δ (31) (x) 2Ae αx x δ 2. (32)

10 134 L. Xie, T. Xie / Joural of Approximatio Theory 152 (2008) By f C α,(19) ad (26), we have for large ( J 3 2Ae α(x+δ) e x (x) 2Ae2αx 2Ae 2αx ) 3x O l() x( l()) 3 ad =N+1 ( f(x) f(x) ) 3x + 1 J 4 + O. (34) l() x( l()) 3 Combiig (30) (34), we ca arrive at the coclusio that for large satisfyig the case N x δ S,N (f, x) f(x) C (x) ε, (35) where C (x) is a costat depedig oly o x. For large satisfyig the case N x > δ, the proof of (35) follows similarly. So we complete the proofs of (18) ad Theorem 3. From the proof of Theorem 3, we ca see f C α is a very importat coditio. The problem is what will happe if we oly suppose f C[0, ). The followig theorem gives a aswer. Theorem 4. If f C[0, ), ad if for a fix x>0, there exists δ > 0 such that f satisfies (14), the the boudedess of (N x)/ ad (19) imply (16). Furthermore, if f (x) exists, the the boudedess of (N x)/ ad (19) imply (18). Proof. This proof is similar to that of Theorem 3. However, we eed to ote that there exists a costat B>0such that N x B. I other words, whe 0 N we have 0 x + B. Thus, i this proof we ca use D = max 0 t x+b f(t) to estimate f ad f(x), i which D depeds o the property of f(x)o [0,x+ B]. So we omit the details. This fiishes the proof of Theorem 4. Acowledgmets The authors express their sicere gratitude to both the editor ad referees for their id suggestios which improved the expositio ad the shape of this paper. Refereces [1] J. Grof, Über approximatio durch polyome mit Belegfutioe, Acta Math. Acad. Sci. Hugar. 35 (1980) [2] H.G. Lehhoff, O a modified Szász Miraja operator, J. Approx. Theory 42 (1984) [3] E. Omey, Note o operators of Szász Miraja type, J. Approx. Theory 47 (1986) [4] V.V. Petrov, Limit Theorem of Sums of Idepedet Radom Variables (i Russio), Mosow Naua, Mosow, [5] A.N. Shiryayev, Probability, Spriger, New Yor/Beli, [6] X.H. Su, O the covergece of the modified Szász Miraja operator, Approx. Theory Appl. 10 (1) (1994) [7] G.Z. Zhou, S.P. Zhou, A remar o a modified Szász Miraja operator, Colloq. Math. 79 (1999) (33)