A Bernstein-Stancu type operator which preserves e 2

Transcription

1 A. Şt. Uiv. Ovidius Costaţa Vol. 7), 009, 45 5 A Berstei-Stacu type operator which preserves e Igrid OANCEA Abstract I this paper we costruct a Berstei-Stacu type operator followig a J.P.Kig model. Itroductio Most of liear ad positive operators o Ca,b preserve e 0 ad e : L e 0 )x) = e 0 x) L e )x) = e x) for each = 0,,,... ad x a,b. J.P. Kig defied i 3 a iterestig class of operators which preserve e. Let s x)) N be a sequece of cotiuous fuctios o 0, so that 0 s x). For ay f C0, ad x 0, let V : C0, C0, be defied by V f) x) = k=0 ) s k k x) s x)) k f ) k. ) For s x) = x, N operators V become Berstei operators. The values of the operators V o test fuctios e j = x j, j = 0,, are give by V e 0 ) x) = Key Words: Positive liear operator; Berstei operator; Stacu operator. Mathematics Subject Classificatio: 4A36. Received: July, 008 Accepted: March,

2 46 Igrid OANCEA V e ) x) = s x) V e ) x) = s x) + s x). Usig Bohma-Korovki theorem, 4) it follows immediately that lim V f = f uiformly o 0, if ad oly if lim s x) = x uiformly o 0,. I order to preserve e, the s sequece has to be as it follows: { s x) = x Mai results s x) = ) + x + 4 ), =,3,... D.D. Stacu see 5, 6) defied for two positive umbers 0 β idepedet of ad for ay fuctio f C0, the operator, P,β) f)x) = p,k x)f k=0 ) k +. ) + β The Berstei Stacu operator uses the equidistat kots a 0 = +β, a = ) ) x 0 +h,..., a = x 0 +h where h = +β ad because P,β) f 0) = f +β ) ) ad f ) = f, iterpolates fuctio f i x = 0 if = 0 ad P,β) + +β i x = if = β. Values o test fuctio are give by: P,β) e 0 ) x) = 3) ) P,β) e x) = x + βx + β ) P,β) e x) = x x x) + βx)x + βx + ) + + β) 5) so we ca state that for ay f C0, the sequece coverges uiformly to fx) o 0,. We defie ow the operators V,β) : C0, C0, by V,β) f ) x) = k=0 ) r k k x) r x)) k f P,β) ) f)x) 4) N ) k +, 6) + β

3 A Berstei-Stacu type operator 47 for ay fuctio f C0, ad x 0,. It s obvious that for r x) = x, N the Berstei-Stacu operators are obtaied, ad for a = β = 0 the V operators defied by ) are obtaied. The V,β) operators are liear ad positive. Usig the relatios 3)-5) the followig theorem ca be easily proved: Theorem. The operators V,β) have the followig properties:. ) V,β) e 0 x) = 7) V,β) e )x) = ) V,β) e x) = + β r x) + + β 8) + β) )r x) + + )r x) + ) 9). For ay fuctio f C0, şi x 0, we have uiformly o 0, if ad oly if uiformly o 0,. or lim V,β) f = f lim r x) = x Next we impose the coditio V,β) e = e, that is If we deote + β) )r x) + + )r x) + ) = x )r x) + + )r x) + x + β) = 0. the the discrimiat is give by a = ) b = + ) c = + β) x = + ) 4 ) + β) x ) = = ) + 4 ) + β) x 0

4 48 Igrid OANCEA for ay x 0,. For the solutios of the equatio are r x)), = + ) ± + ) 4 ) + β) x ). ) We choose rx) = + ) + + ) 4 ) + β) x ), > ) 0) ad rx) = x. ) Lemma. For ay x r x). +β, + +β Proof. Because r x) = b+ b 4ac a Sice a > 0 we get which leads to 0 b + b 4ac a the followig iequality holds 0 the iequality 0 r x) becomes 0 b + b 4ac a 0 b b 4ac a + b b b 4ac 4a + 4ab + b 0 ac a + ab. It results that we have to fid x 0, such that { c 0 a + b + c 0. Replacig a,b,c we obtai c 0 if + β) x 0, that is x +β,, ad a + b + c 0 becomes therefore x ) + + ) + + β) x 0 0, + +β x ) +, + β which evetually gives us that x +β, + +β.

5 A Berstei-Stacu type operator 49 If we deote I = +β, + +β from the iequalities + β + + β + + β β + it follows that I I +, N; moreover for the iterval I becomes 0,. Oe ca otice that lim r x) = x, so we have the followig Theorem.3 The operators V,β) give by 6 with the sequece rx)) N defied by 0, have the followig properties:. they are liear ) ad positive o C0,. V,β) e x) = e x), N for ay x 3. lim f = f for ay f C0,, x V,β) +β, + +β +β, + +β If L is a liear ad positive operator o Ca,b, the for ay cotiuous fuctio f Ca,b ad x a,b we have the evaluatio see, pg. 30) Lf)x) fx) fx) Le 0 )x) + Le 0 )x) + Lϕ ) x)x) ωf,) ) Le0 )x)lϕ fx) Le 0 )x) + Le 0 )x) + x)x) ωf,) ) ) x I, ) > 0, where ϕ x = e xe 0 If the operator L satisfies the coditios Le 0 = e 0 şi Le = e the the evaluatio ) ca be writte as: ) Lϕ Lf)x) fx) + x )x) ωf, ) ad sice Lϕ x)x) = L e xe 0 ),x ) = Le ) x) xle ) x) + x Le 0 ) x) = = x xle ) x) = xx Le ) x)), 3) we ca also write that ) xx Le ) x)) Lf)x) fx) + ωf, ).

6 50 Igrid OANCEA for ay f Ca,b ad x a,b. Sice the operator L is positive ad ϕ x 0 we get that Lϕ x 0 which is equivalet with xx Le ) x)). It follows that for ay x a,b,a 0 the iequality Le ) x) x. holds true. Takig a,b = I ad L = V,β) as a particular case we obtai: Lemma.4 For ay x I if r x) = rx) we have ) V,β) e x) x. We got that for ay x I we have e x) şi V,β) V,β) e ) x) = e 0 )x) = e 0 x), V,β) e )x) x; therefore the followig evaluatio stads: V,β) f)x) fx) + x x V,β) e )x) ) ωf,). The order of approximatio is at least as good as i case of approximatio by Berstei-Stacu polyomials for those x I for which the followig iequality is true V,β) ϕ x)x) P,β) ϕ x)x). 4) For > β the secod order momet of Stacu operator is give by P,β) ϕ x)x) = P,β) e xe 0 ) x x) + βx ) )x) = + β). 5) Takig ito accout the expressios of the momets for the two operators from relatios 3) ad 5), we ca rewrite the iequality 4) as: x x + β r x) + ) x x) + βx ) + β + β). 6) We preset the graphics of the two members of iequality for some particular cases:

7 A Berstei-Stacu type operator x 0 7 P V = ; = 0; β = 00 4 x 0 7 P V = ; = 00; β =.000

8 5 Igrid OANCEA x 0 6 P V Refereces = ; = 900; β =.000 H. Bohma, O approximatio of cotious ad aalytic fuctios, Ark. Mat. 95), R.A. DeVore, The approximatio of cotiuous fuctios by positive liear operators, Lecture Notes i Mathematics 93, Spriger-Verlag, New York, J.P. Kig, Positive liear operators which preserve x, Acta Math. Hugar. 993)003), P.P. Korovki, O covergece of liear positive operators i the space of cotiuous fuctios, Dokl. Akad. Nauk. SSSR NS), ), D.D. Stacu, Folosirea iterpolării liiare petru costruirea uei clase de polioame Berstei, Studii şi Cercetări Matematice, 3 8), 976), D.D. Stacu, Asupra uei geeralizări a polioamelor lui Berstei, Studia Uiversitatis Babeş-Bolyai, 4 ) 969), Valahia Uiversity of Târgovişte, Departmet of Mathematics, Bd. Uirii No 8, 3008, Târgovişte, Romaia igrid.oacea@gmail.com