THE REPRESENTATION OF THE REMAINDER IN CLASSICAL BERNSTEIN APPROXIMATION FORMULA

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1 Global Joural of Advaced Research o Classical ad Moder Geometries ISSN: , Vol.6, 2017, Issue 2, pp THE REPRESENTATION OF THE REMAINDER IN CLASSICAL BERNSTEIN APPROXIMATION FORMULA DAN MICLĂUŞ ABSTRACT. I the preset paper we establish for the first time the represetatio of the remaider i classical Berstei approximatio formula, usig divided differeces. As a cosequece, we get a upper boud estimatio for the remaider term, whe approximated fuctio fulfills some give properties. 1. INTRODUCTION AND AUXILIARY RESULTS Let N be the set of positive itegers ad N 0 N 0. A recet study cocerig the geeralizatio of Berstei operator associated to ay real-valued fuctio F : a, b R, where a, b are two real ad fiite umbers, such that a < b, is doe i the papers 2, 3, 5, 4, 8. This classical Berstei operator is give by B F; x p,k xf 1 b a k k b x k F 1.1 for ay x a, b ad ay N. If a : 0, b : 1, from 1.1 for ay x 0, 1 ad ay N, the it follows the Berstei operator 1 associated to ay real-valued fuctio f : 0, 1 R, give by B f ; x k xk 1 x k f k. 1.2 Thaks to some importat properties as approximatio, shape preservatio ad variatio dimiishig, Berstei operator is a idispesable tool i computer aided geometric desig as well as i other areas of mathematics. Now, let a x 0 < x 1 < < x b be some odes, for ay N. Defiitio The divided differece of the fuctio F with respect to the poits x 0, x 1,, x, is defied to be the coefficiet at x i the Lagrage iterpolatig polyomial L F; x ad is 2010 Mathematics Subject Classificatio. 26A51, 41A36, 41A80. Key words ad phrases. Berstei operator, divided differece, covex fuctio, Popoviciu theorem, remaider term. 119,

2 120 Da Miclăuş deoted by x 0, x 1,, x ; F, where L F; x Fx k x 0 x x 1 x x. lx x x k l x k ad lx x I most books o Numerical Aalysis, divided differece for distict odes is defied recursively x 0, x 1,, x ; F 1 x x 0 x 1,, x ; F x 0,, x 1 ; F. 1.3 The first order divided differece of the fuctio F with respect to the distict odes x 0, x 1 is defied byx 0, x 1 ; F 1 x 1 x 0 Fx 1 Fx 0, respectively the secod order divided differece of the fuctio F with respect to the distict odes x 0, x 1, x 2 is defied by Fx x 0, x 1, x 2 ; F 0 x 0 x 1 x 0 x 2 + Fx 1 x 1 x 0 x 1 x 2 + Fx 2 x 2 x 0 x 2 x A fuctio F is called covex, o-cocave, polyomial, o-covex, respectively, cocave of th order, if the followig hold x 0, x 1,, x +1 ; F > 0, 0, 0, 0, respectively < 0, for ay +2 distict odes i a, b ad N 0. Remark 1.1. i For 0 we get mootoe fuctios: icreasig, o-decreasig, costat, o-icreasig, respectively, decreasig. ii For 1 we get: covex, o-cocave, liear, o-covex, respectively, cocave fuctios i the classical sese. The covex fuctios of higher order represeted a theme of study for Popoviciu. I 1944 his cotributios i this area were collected i 10. The ext result is a well-kow extesio of Lagrage s mea value theorem to the case of divided differeces. Theorem 1. 9 If F : a, b R is cotiuous o a, b ad has a derivative of order o a, b, the there exists a ξ a, b such that x 0, x 1,, x ; F 1! F ξ. I 12, Stacu established a represetatio of the remaider i Berstei approximatio formula fx B f ; x+r f ; x, 1.5 for ay real-valued fuctio f : 0, 1 R. Theorem For ay x 0, 1\ k k 0,, ad ay N, the represetatio of the remaider associated to the Berstei operator 1.2 is give by R f ; x x1 x 1 p 1,k x x, k, k+1 ; f. 1.6 Further research cocerig the represetatio of the remaider 1.6 was doe o quadrature formulas based o liear positive operators Berstei type operators, which represeted a theme of itesive study for Stacu. His cotributios i this area were collected i the secod volume of the moograph Numerical Aalysis ad Approximatio Theory 13. The aim of this paper is to establish for the first time the represetatio of the remaider i geeralized Berstei approximatio formula for arbitrary fuctios, as a covex combiatio

3 The represetatio of the remaider i classical Berstei approximatio formula 121 of divided differeces of secod order o kow odes. As a cosequece, we get a upper boud estimatio for the remaider term whe approximated fuctio possesses bouded divided differeces of secod order. 2. MAIN RESULTS Berstei polyomials 1.2 have opeed a ew era i approximatio theory startig with the year 1912, whe Berstei preseted his famous proof of the Weierstrass approximatio theorem ad cotiuig with thousads of papers util today. The approximatio by Berstei polyomials is made o the iterval 0, 1. Thikig at practice utility of Berstei operators, it is easy to remark that approximatio of fuctios defied o the iterval a, b, where a, b beig real ad fiite umbers, exceptig a : 0, b : 1 appears more ofte tha approximatio of fuctios defied o the iterval 0, 1. It is also true the fact that ay iterval a, b ca be trasformed ito 0, 1 usig the applicatio lx b a, for all x a, b ad the apply the Berstei operators. I this case occurs the problem: Is it worth to trasform the iterval a, b ito 0, 1 i ay situatio? Now, havig a arbitrary real-valued fuctio F : a, b R arise the questio: Ca we approximate fuctios defied o the itervala, b by Berstei operators, without trasformig itervala, b ito0, 1? The aswer is affirmative ad is give by classical Berstei operators 1.1. Lemma For ay x a, b ad ay N we preset the computatio of the test fuctios for geeralized Berstei operator 1.1. Be 0 ; x 1, Be 1 ; x x, Be 2 ; x x 2 + b x. Remark 2.1. The above lemma show us that preseted operator 1.1 is a Berstei type operator, amely classical Berstei operator. For ay F Ca, b, ay x a, b ad ay N, the followig Fx B F; x+r F; x 2.1 is called classical Berstei approximatio formula, where R is the remaider operator associated to the classical Berstei operator B. The formula 2.1 is a idispesable tool i approximatio of ay arbitrary fuctio defied o the iterval a, b, with the coditio to kow the form of remaider term. I 4, a upper boud estimatio for the remaider term R associated to the Berstei operator 1.1 was used. Lemma Suppose that F C 2 a, b. For ay x a, b, the remaider term of the relatio 2.1 verifies R F; x b x M 2b a 2 2 F, 2.2 where M 2 F max ξ a,b F ξ. Remark 2.2. A first form of the upper boud estimatio 2.2 appeared i 2, ad the used also i the paper 4. It is essetial to metio the fact that a rigorous proof of this upper boud estimatio it was ot doe i either of the recalled papers. Moreover, the result cocerig upper boud estimatio seems to be wrog.

4 122 Da Miclăuş Beig motivated by the Remark 2.2, i the followig we establish the represetatio of the remaider i classical Berstei approximatio formula for arbitrary fuctios, as well as a upper boud estimatio whe approximated fuctio possesses bouded divided differeces of secod order. Theorem 3. The represetatio of the remaider term associated to the classical Berstei operator 1.1, is give by R F; x b x for ay N ad x a, b\ 1 p 1,k x x, k 0,,., a+ k+1b a ; F, 2.3 Proof. I order to establish the represetatio of the remaider term, we otice that the classical Berstei operator 1.1 preserves costats, such that R F; x Fx BF; x Fx p,k xf p,k Fx x F Usig the defiitio of divided differece of first order, it follows R F; x p,k x x 1 b x b a 1 b x k k k+1 b a 1 p 1,k x b x b a p,k x k kb x x, x, 1 p 1,k x b x ; F k1 1 ; F x, 1 p 1,k x. x, x, ; F 1 b x +1 k k 1 k b a 1 b x k k k+1 b a ; F x, x, a+ k+1b a ; F, a+ k+1b a ; F. x, x, a+ k+1b a ; F ; F I what follows, we wat to preset a alterative way for provig the Theorem 3, which implies the trasformatio of the iterval a, b ito 0, 1 ad vice versa. A alterative way for provig Theorem 3. Proof. The classical Berstei approximatio formula 2.1 provides R F; x Fx B F; x ; F

5 The represetatio of the remaider i classical Berstei approximatio formula 123 ad the, usig the equality Fx : f b a, for all x a, b, it follows R F; x f b a B f ; b a R f ; b a. 2.4 Now, takig the relatios 1.6 ad 1.4 ito accout, we get b x 1 R f ; b a b x b a 2 1 p 1,k x 1 b a 2 p 1,k b a b a, k, k+1 ; f f b a kb a k+1b a 2 b a 2 Usig the equality fx : Fa+xb a, for all x 0, 1, we get b x 1 p 1,k x b x R F; x Fx kb a k+1b a 2 1 for ay N ad x a, b\ p 1,k x x, F f k kb a 2 b a kb a 2 b a 2, a+ k+1b a ; F + + F, f k+1 k+1b a. 2 b a a+ k+1b a k+1b a 2 b a 2, k 0,,. We establish the correct form of upper boud estimatio for the remaider i classical Berstei approximatio formula 2.1. Corollary 2.1. Suppose that F C 2 a, b ad the divided differeces of the secod order of F are all bouded o a, b, the we get for ay N, where M 2 F : max ξ x a,b F ξ x. R F; x b x 2 M 2 F, 2.5 Proof. Takig ito accout the represetatio 2.3 of the remaider term i classical Berstei approximatio formula, respectively Theorem 1, it follows R 1 F; x b x F p 1,k x x,, a+ k+1b a ; b x 2 1 p 1,k x F ξ x b x 2 b x 2 M 2 F, for ay N. 1 M 2 F p 1,k x Remark 2.3. Usig the estimatio 2.5 ad the iequality betwee arithmetic ad geometric mea, also we ca establish a upper boud estimatio for the remaider i classical Berstei approximatio formula, give by R F; x b a2 8 M 2 F. 2.6

6 124 Da Miclăuş Remark 2.4. Havig the represetatio of the remaider term 2.3, respectively the upper boud estimatio 2.5 proved ad well defied, it could be itegrate the classical Berstei approximatio formula 2.1, i order to get the correct form of classical Berstei quadrature formula i 4. If a, b are two real ad fiite umbers, such that a < b, the we ca approximate ay fuctio F defied o the itervala, b by classical Berstei operator 1.1 ad we ca evaluate the error i classical Berstei approximatio formula. We give a example i this sese. Example 2.1. Takig a : 0, b : +1, the for ay real-valued fuctio F : 0, +1 R it follows the associated Berstei operator 3, 11, give by, 2.7 B 0, +1 F; x +1 k xk +1 x k F k +1 for ay x 0, +1 ad ay N. The remaider term associated to the above Berstei operator admits the followig represetatio +1 R 0, F; x x1 x x +1 for ay N ad x 0, we get the upper boud estimatio +1 F; x x R0, 1 +1 \ k +1 k 0,, x +1 2 p 0, +1 1,k x k x, +1, k+1 +1 ; F, 2.8. Supposig that F C 2 0, +1, the M 2 F M 2 F, for ay N. 2.9 Ackowledgemet The results preseted i this paper were obtaied with the support of the Techical Uiversity of Cluj-Napoca through the research Cotract o. 2011/ , Iteral Competitio CICDI REFERENCES 1 Berstei, S. N., Démostratio du théorème de Weierstrass fodée sur le calcul de probabilités, Commu. Soc. Math. Kharkow , No. 2, Bărbosu, D. ad Miclăuş, D., O the composite Berstei type quadrature formula, Rev. Aal. Numér. Théor. Approx , No. 1, Bărbosu, D. ad Deo, N., Some Berstei-Katorovich operators, Automat. Comput. Appl. Math , No. 1, Bărbosu, D. ad Ardelea, G., The Berstei quadrature formula revised, Carpathia J. Math , No. 3, Dascioglu, A.A. ad Isler, N., Berstei collocatio method for solvig oliear differetial equatios, Math. Comput. Appl , No. 3, Goska, H. ad Raşa, I. O the sequece of composite Berstei operators i Frech, Rev. Numér. Théor. Approx , No. 2, Iva, M., Elemets of Iterpolatio Theory, Mediamira Sciece Publisher, Cluj-Napoca, Miclăuş, D., The geeralizatio of Berstei operator o ay fiite iterval, Georgia Mathematical Joural, accepted 9 Popoviciu, T., Sur quelques propriété des foctios d ue ou de deux variables réelles, Mathematica Cluj , Popoviciu, T., Les Foctios Covexes, Actualités Scietifiques et Idustrielles, Fasc. 992, Paris, 1944, Publiés sous la directio de Paul Motel

7 The represetatio of the remaider i classical Berstei approximatio formula Siddiqui, M. A., Agrawal, R. R. ad Gupta, N., O a class of modified ew Berstei operators, Advaced Stud. Cotemp. Math , No. 1, Stacu, D. D., The remaider of certai liear approximatio formulas i two variables, SIAM J. Numer. Aal., Ser. B , Stacu, D. D., Coma, G. ad Blaga, P., Numerical aalysis ad approximatio theory i Romaia, Presa Uiv. Cluj. II, Cluj-Napoca, 2002 TECHNICAL UNIVERSITY OF CLUJ-NAPOCA NORTH UNIVERSITY CENTER AT BAIA MARE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE VICTORIEI 76, BAIA MARE, ROMANIA address: damiclausrz@yahoo.com