An Extension of the Szász-Mirakjan Operators

Transcription

1 A. Şt. Uiv. Ovidius Costaţa Vol. 7(), 009, A Extesio o the Szász-Mirakja Operators C. MORTICI Abstract The paper is devoted to deiig a ew class o liear ad positive operators depedig o a certai uctio ϕ These operators geeralize the Szász-Mirakja operators (case i which ϕ is the expoetial uctio). Furthermore, coditios whe these operators have properties o mootoy ad covexity are give. Itroductio Oe o the mai purpose o the approximatio theory is to id how uctios ca be approximated by simpler uctios. A directio is to use the liear, positive operators ad cosequetly, a large umber o authors have established ew properties o them. We discuss here the Szász-Mirakja operators S : C ([0, )) C ([0, )), N, give by the law (S )(x) e x (x) k, C ([0, )). Key Words: Szasz-Mirakja operators; Popoviciu-Bohma-Korovki theorem; Positivity; Liearity; Covexity. Mathematics Subject Classiicatio: 4A36. Received: October, 008 Accepted: April,

2 38 C. Mortici Obviously, these operators are liear ad i 0, the S 0, so they are also positive. The Szász-Mirakja operators was deied or the irst time by Otto Szász i the paper [9], where the origial otatio was P(u,x) e xu (ux) v ( v v! u) v, u > 0. These operators was also discussed i the paper [3], rom a dieret poit o view, while i the paper [0] the covergece o P(x,u) to (x) as u was established. This act was cosidered a geeralizatio or the iterval 0 x o the well-kow properties o S. Berstei s approximatio polyomials i a iite iterval, established i 9. The Szász-Mirakja operators play a cetral role i the theory o approximatio, so they are itesively studied. For various extesios ad urther properties ad proos, see or example [], [5], [6], [7], []. Recetly, oe directio or study more geeral versios o the Szász-Mirakja operators was give i [8], where a sequece o positive real umbers (α ) 0 was cosidered to deie the operators (S α )(x) e x α ( x α ) k, C([0, )). I case α, the classical Szász-Mirakja operators are obtaied. Our idea is to cosider a aalytic uctio ϕ : R [0, ) ad to deie the operators ϕs : C ([0, )) C ([0, )), N, give by the ormula (ϕs )(x) ϕ(x) (x) k, C ([0, )). () We will call (ϕs ) the ϕ-szász-mirakja operators. I case ϕ(y) e y, the classical Szász-Mirakja operators are obtaied. The Mai Result We remid irst the ollowig basic theorem, also called Popoviciu-Bohma- Korovki theorem. This result was irst published by the Romaia mathematicia Tiberiu Popoviciu i [9] - uortuately a local joural which was ot so kow i the mathematics world o that time. Ater this, the result was oud idepedetly by the Daish mathematicia H. Bohma i [], while

3 A Extesio o the Szász-Mirakja Operators 39 the result was clear published by the Russia mathematicia P.P. Korovki i his book [4]. Deote by e 0 (x), e (x) x, e (x) x the test uctios. The result we are talkig about is the ollowig: Theorem.. Let L : C([a,b]) C([a,b]), N be a sequece o liear, positive operators such that lim (L e j )(x) e j (x), j 0,,, () uiormly o [a,b]. The or every C([a,b]), lim (L )(x) (x), uiormly o [a, b]. Now, i order to establish the approximatios properties o the ew deied operators (ϕs ) N, we give the ollowig Lemma.. The ϕ-szász-mirakja operators satisy the ollowig relatios: a) (ϕs e 0 )(x) e 0 (x) b) (ϕs e )(x) ϕ (x) ϕ(x) x c) (ϕs e )(x) ϕ (x) ϕ(x) x + ϕ (x) ϕ(x) x. Proo. From the act that the uctio ϕ is aalytic, it results that the by derivatio, ϕ (y) a) We have k y k ϕ(y), (k )! yk ad ϕ (y) (ϕs e 0 )(x) ϕ(x) k (k )! yk. (x) k ϕ(x). ϕ(x) b) We have (ϕs e )(x) ϕ(x) (x) k k

4 40 C. Mortici c) We have x ϕ(x) k (ϕs e )(x) ϕ(x) ϕ(x) k (k )! (x)k ϕ (x) ϕ(x) x. (x) k ( ) k (x) k [k(k ) + k] [ (x) ϕ(x) (k )! (x)k + x k ϕ(x) k ] (k )! (x)k [ (x) ϕ (x) + xϕ (x) ] ϕ (x) ϕ(x) x + ϕ (x) ϕ(x) x. From this lemma, it results the ollowig Theorem.3. Let ϕ : R (0, ) be such that ϕ (y) ϕ (y) lim ad lim. (3) y ϕ(y) y ϕ(y) The the ϕ-szász-mirakja operators satisy lim (ϕs )e j e j, j 0,,, uiormly o compact itervals ad accordig with the Theorem., or every C ([a,b]), it holds: lim (ϕs ), uiormly o [a,b]. Proo. From the Lemma. ad rom the hypotesis (3), it results ad lim (ϕs )e (x) lim ( ϕ (x) ϕ(x) x ) x ( ϕ lim (ϕs (x) )e (x) lim ϕ(x) x + ) ϕ (x) ϕ(x) x x. Remark that the expoetial uctio ϕ(y) e y satisies the hypotesis (3), so we have obtaied the results rom [8] i a geeral backgroud. Furthermore, ote that our extesio is cosistet, i we take ito accout that there is a large class o uctios with the property (3), or example ϕ(y) y i e y, i N,

5 A Extesio o the Szász-Mirakja Operators 4 or more geeral, the uctios o the orm ϕ(y) P(y)e y, where P is ay polyomial uctio with o-egative coeiciets. Other iteresig particular ϕ-szász-mirakja operators ca be obtaied. I geeral, by the product dieretiatio ormula, we have so ϕ (k) (y) e y k i0 k i0 ( k i ( k i ) P (i) (y), ) P (i) (0). Now, by replacig i (), we obtai the ollowig class o operators: ( k ( ) ) ( ) k k (PS )(x) P(x)e x P (i) (0) (x) k, i i0 where P is ay polyomial uctio with o-egative coeiciets. Other ideas or extesios is to cosider certai uctio P i the previous relatio, ot ecessary a polyomial uctio. 3 Hereditary Properties Accordig with the usual procedures, we will study i this sectio the hereditary properties (mootoy ad covexity) o the ϕ-szász-mirakja operators. Theorem 3.. Assume that the aalytic uctio ϕ : [a,b] (0, ) with 0, or all itegers k 0, satisies yϕ (y) sup. (4) y [a,b] ϕ(y) [ a The i is positive, the (ϕs ) is also positive ad icreasig o, b ]. Proo. For a x x b, we have ϕ(x ) (x ) k (ϕs )(x ) (ϕs )(x ) ϕ(x ) ( ) [ k (x ) k ϕ(x ) (x ) k ϕ(x ) (x ) k ] 0,

6 4 C. Mortici because the uctio y y ϕ(y) is icreasig o [a,b] ad moreover, y yk ϕ(y) with k is also icreasig. Theorem 3.. Assume that the aalytical uctio ϕ : R (0, ) with 0, or all itegers k 0 is such that y y/ϕ(y) is covex ad icreasig. The i is positive, the (ϕs ) is covex. Proo. The uctios y y k ad y y/ϕ(y) are covex ad icreasig, so their product y y k /ϕ(y) is also covex. For x, y > 0, we have: ( ) x + y (ϕs ) ϕ(x) ϕ ( x+y ) x k We obtaied that ( ) x + y (ϕs ) ( k ) ( x+y ) k ϕ ( x+y x + y ) ( )[ k x k ϕ(x) + yk ϕ(y) + ϕ(y) (ϕs )(x) + (ϕs )(y). ] y k (ϕs )(x) + (ϕs )(y) ad by cotiuity argumets, (ϕs ) is covex. Reereces [] M. Becker, Global approximatio theorems or Szász-Mirakja ad Baskakov operators i polyomial weight spaces, Idiaa Uiv. Math. J., 7(978), o., 7 4. [] H. Bohma, O approximatio o cotiuous ad o aalytic uctios, Ask. Mat. (), 3(95), 43-5.

7 A Extesio o the Szász-Mirakja Operators 43 [3] M. Kac, Ue remarque sur les polyomes de M. S. Berstei, Studia Math., 7(938), [4] P. P. Korovki, Liear Operators ad Approximatio Theory, Delhi, 960. [5] H. G. Lehho, O a Modiied Szász-Mirakja Operator, J. Approx. Theory, 4(984), [6] M. Leśiewicz, L. Rempulska, Approximatio by some operators o the Szász-Mirakja type i expoetial weight spaces, Glas. Math. Ser. III, 3(997), [7] C. Mortici, I. Oacea, A osmooth extesio or the Berstei-Stacu operators ad a applicatio, Studia Math., (003), [8] C. M. Muraru, O the sequece o Katorovich type operators, It. J. Pure Appl. Math., 45(008), o. 3, [9] T. Popoviciu, Asupra demostraţiei teoremei lui Weierstrass cu ajutorul polioamelor de iterpolare, Lucrările Sesiuii Geerale Ştiiţiice Acad. R.P.R. (950), [0] O. Szász, Geeralizatio o S. Berstei s polyomials to the iiite iterval, J. o Research o the Natioal Bureau o Stadards, 45(950), o. 3, [] Z. Walczak, O the rate o covergece or some liear operators, Hiroshima Math. J., 35(005), 5-4. Departmet o Mathematics Valahia Uiversity o Târgovişte, Bd. Uirii 8, 3008 Târgovişte ROMANIA cmortici@valahia.ro

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