A GENERALIZED BERNSTEIN APPROXIMATION THEOREM

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1 Ø Ñ Å Ø Ñ Ø Ð ÈÙ Ð Ø ÓÒ DOI: /v x Tatra Mt. Math. Publ , A GENERALIZED BERNSTEIN APPROXIMATION THEOREM Miloslav Duchoň ABSTRACT. The preset paper is cocered with soe geeralizatios of Berstei s approxiatio theore. Oe of the ost elegat ad eleetary proofs of the classic result, for a fuctio fx defied o the closed iterval [0,1], uses the Berstei s polyoials of f, B x B f x f x 1 x We shall cocer the -diesioal geeralizatio of the Berstei s polyoials ad the Berstei s approxiatio theore by taig a 1-diesioal siplex i cube [0,1]. This is otivated by the fact that i the field of atheatical biology aturally arouse dyaic systes deteried by quadratic appigs of stadard 1-diesioal siplex { x i 0, i 1,...,, i1 x i 1 } to self. The last coditio guaratees savig of the fudaetal siplex. The there are surveyed soe other the -diesioal geeralizatios of the Berstei s polyoials ad the Berstei s approxiatio theore. 1. Itroductio For a fuctio fx defied o the closed iterval [0,1] the expressio B x Bx f f x 1 x 1 is called the Berstei polyoial of order of the fuctio fx. B x is a polyoial i x of degree. The polyoials B x were itroduced by S. Berstei see [1] to give a especially siple proof of Weierstrass approxiatio theore. Naely, if fx is a fuctio cotiuous o [0,1], the as it ca be see li x fx 2 uiforly i [0, 1]. c 2011 Matheatical Istitute, Slova Acadey of Scieces Matheatics Subject Classificatio: 28E10, 81P10. K e y w o r d s: Berstei polyoial, Berstei approxiatio theore, geeralized siplex. Supported by grat agecy VEGA, o. 2/0212/10. 99

2 MILOSLAV DUCHOŇ A celebrated theore of Weierstrass says that ay cotiuous real-valued fuctio f defied o the closed iterval [0,1] R is the liit of a uiforly coverget sequece of polyoials. Oe of the ost elegat ad eleetary proofs of this classic result is that which uses the Berstei polyoials of f, B f,x f x 1 x x [0,1], oe for eachiteger 1. Berstei s theore states that B f f uiforly o [0,1] ad, sice each B f is a polyoial of degree at ost, we have as a cosequece Weierstrass theore. See, for exaple, [5]. The operator B defied o the space C [0,1];R with values i the vector subspace of all polyoials of degree at ost has the property that B f 0 wheever f 0. Thus Berstei s theore also establishes the fact that each positive cotiuous real-valued fuctio o [0,1] is the liit, of a uiforly coverget sequece of positive polyoials. The preset paper is cocered with soe geeralizatios of Berstei s theore. For the followig ote that the expressios p p x x 1 x 3 cotaied i 1 are the bioial or Newto probabilities well ow i the theory of probability. If 0 x 1 is the probability of a evet E, the p x isthe probability that E will occur exactly tiesiidepedet trials. May of the properties of the p, ad of their sus which we shall eed are othig but theores of the theory of probability. As a exaple, cosider Beroulli s theore of large ubers. Let ǫ > 0 ad δ > 0 be fixed ad suppose that aog the idepedet trials, is the uber of those for which the evet E occurs. The for sufficietly large, the probability P δ that differs fro x by less tha δ is greater tha 1 ǫ. By the theore of additio of probabilities, this ay be writte i the for P δ x <δ x 1 x 1 ǫ, 4 for all sufficietly large. The last su is tae for all values 0,1,..., which satisfy the coditio x < δ; otatio of this type is used i the sequel without explaatio. We ca see that 2 easily follows fro this iequality. 100

3 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM 2. The theore of Weierstrass Berstei polyoials of the fuctio fx are liear with respect to the fuctio fx, i.e., B f x a 1B f 1 x+a 2B f 2x 5 if fx a 1 f 1 x+a 2 f 2 x. Sice p x x 1 x 0 o the iterval 0 x 1 ad 0 p 1, we have B f x M 0 x 1, 6 wheever fx M o this iterval. With the help of the polyoials B x we ay prove the faous theore of Weierstrass, which asserts that for each fuctio fx, cotiuous o a closed iterval [a,b], ad for each ǫ > 0 there is a polyoial Px approxiatig fx uiforly with a error less tha ǫ, fx Px < ǫ. 7 By a liear substitutio, the iterval [a,b] ay be trasfored ito [0,1]. The theore of Weierstrass is therefore a corollary of the followig theore: Ì ÓÖ Ñ 1 [1, Berstei]º For a fuctio fx bouded o [0,1], the relatio li B x fx 8 holds at each poit of cotiuity x of f ad the relatio holds uiforly o [0,1] if fx is cotiuous o this iterval. Proof. We shall copute the value of { T x 2 p 1 2x 1+ 2 x 2} p. 9 Obviously, p 1, oreover, we have 1 1 p x x 1 x 1 x p 1x 2 x 1 x 2 1x 2 ad therefore, T 2 x 2 2x 1x+ 1x 2 x1 x

4 MILOSLAV DUCHOŇ Sice x1 x 1 4 o [0,1], we obtai the iequality 1 2 p δ 2 x p 1 x1 x 2 δ2t δ 2 1 4δ 2 11 x δ x δ Now if the fuctio f is bouded, say fu M i 0 u 1 ad x a poit of cotiuity, for a give ǫ > 0, we ca fid a δ > 0 such that x x < δ iplies fx fx < ǫ. We have { fx B x fx f } p fx f p x <δ + x δ 12 The first su is ǫ p ǫ, the secod oe is, by 11, 2M4δ 2 1. Therefore, fx B x ǫ+m 2δ ad if is sufficietly large, fx B x < 2ǫ. Fially, if fx is cotiuous i the whole iterval [0,1], the 13 holds with a δ idepedet of x, so that B x fx uiforly. This copletes the proof. 3. -diesioal geeralizatio of the Berstei theore Now we shall discuss -diesioal geeralizatio of the Berstei theore. First we give -diesioal geeralizatio of the Berstei polyoials. We recall the ultioial theore. Ì ÓÖ Ñ 2º Multioial theore. The followig equality is valid see [3]. 102 x 1 + +x a 1,...,a a 1 + +a a 1 + +a x a 1 1 xa,! a 1! a! xa 1 1 xa,

5 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM A iportat -diesioal geeralizatio is obtaied by taig a 1- -diesioal siplex p {x i 0, i 1,...,, x x 1}. If fx 1,...,x is defied o p, we ay write with regard to the ultioial theore B f x 1,...,x l i 0, + +l l1 f,..., l p l1,...,l ;x 1,...,x,...,l!,...,l!...l!.!!...l! x 1...x l, 14 x 1...x l, Here agai we have the covergece B f f at a poit of cotiuity of f. Ì ÓÖ Ñ 3º Let fx 1,...,x be a cotiuous fuctio o p. The uiforly o p. li Bf x 1,...,x fx 1,...,x Proof. The proof is quite siilar to that of Theore 1, ad is based o the followig properties of the p l1,...,l : a the su of the p s which occur i 14 is equal to 1, b if ǫ > 0,δ > 0 are give, the su of those p l1,...,l i 14 for which l i x i δ for at least oe idex i is saller tha ǫ for each sufficietly large. To prove b, we observe that by 11 the su of p s with x 1 δ is equal to l i x i δ, + +l p l1,...,l ; x 1 l 2,...,l l1 l 2,...,l x l x l, where the suatios are exteded over x 1 δ ad l2 + +l, respectively, ad this is x 1 x 1 4δ 2 1. Aother -diesioal geeralizatio is obtaied by taig a -diesioal siplex {x i 0, i 1,...,, x 1 + +x 1}. See [5]. 103

6 l i 0, + +l MILOSLAV DUCHOŇ Ì ÓÖ Ñ 4º If fx 1,...,x is defied o, we write Bx f 1,...,x l1 f,..., l p l1,...,l ;x 1,...,x,...,l,...,l p l1,...,l ;x 1,...,x, 15 x 1...x l 1 x 1 x l,!!...l! l!. Here agai we have the covergece B f f at a poit of cotiuity of f. Proof. The proof is quite siilar to that of Theore 1, ad is based o the followig properties of the p l1,...,l : a the su of the p s which occur i 15 is equal to 1, b if ǫ > 0, δ > 0 are give, the su of those p l1,...,l i 15 for which l i x i δ for at least oe idex i is saller tha ǫ for each sufficietly large. To prove b, we observe that by 11 the su of p s with x 1 δ is equal to l l x i δ, + +l p l1,...,l ; x 1 l 2,...,l l1 l 2,...,l x l x l 1 x 1 x l, where the suatios are exteded over x 1 δ ad l2 + +l, respectively, ad this is x 1 x 1 4δ 2 1. We shall ow derive a ore -diesioal geeralizatio of the Berstei theore. First we give -diesioal geeralizatio of the Berstei polyoials. See [2], [4]. Ì ÓÖ Ñ 5º Let fx 1,x 2,...,x be defied ad bouded i the -diesioal cube 0 x i 1, i 1,...,. The the Berstei polyoial defied by 104 B f 1,..., x 1,...,x l 0 l l1 f,..., l 1 x 1 1 x x l 1 x l

7 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM coverges towards fx 1,...,x at ay poit of cotiuity of this fuctio, as all i. 4. Boha-Borovi theore We shall use a odified versio of Boha-Korovi s Theore proved by Pǎltieau to prove a geeralized Berstei theore. See Pǎltieau [6]. Let X be a copact Hausdorff space cotaiig at least two poits. ÈÖÓÔÓ Ø ÓÒ 1 Boha-Koroviº Let there be give 2 fuctios f 1,...,f, a 1,...,a CX,R, with the properties: Px,y a i yf i x 0, for all x,y X 2 i1 ad Px,y 0 x y. If H is a sequece of positive liear operators o CX;R, with the property: H f i f i as for all i 1,2,...,; the H f f as for all f CX;R. See Pǎltieau [6]. We shall forulate a iterestig applicatios of Propositio 1, aely Korovi theore. Ì ÓÖ Ñ 6 Koroviº Let H be a sequece of positive liear operators o C [a,b] ad f 1,f 2,f 3 be the fuctios defied as f 1 x 1, f 2 x x, f 3 x x 2, If H f i f i, i 1,2,3, the H f f, for all f C [a,b],r. for all x [a,b]. Proof. See Pǎltieau [6]. Let a 1 y y 2,a 2 y 2y,a 3 y 1 ad let 3 Px,y a i yf i x y x 2. We ca see that the coditios of Propositio 1 are satisfied. i1 Ì ÓÖ Ñ7S. Bersteiº Let B be a sequeceof positiveliear operators o C [0,1], defied by B fx f x 1 x, for all x [0,1] 105

8 MILOSLAV DUCHOŇ The B f f for all f C [0,1],R. Proof. See Pǎltieau [6]. It is clear that B is a sequece of positive liear operators. Let f 1 x 1, for all x [0,1]. The If we deote the we have Further we have Let It follows hece Further we obtai p x B f 1 f 1. x 1 x p x x. 1 2 p x 2 x 2 xx 1. f 2 x x, f 3 x x 2, for all x [0,1]. B f 2 x x B f 2 f 2. B 3 f 3 f 3. Sice there are satisfied the coditios of Borovi theore, we have B f f, for all f C [0,1],R. Ì ÓÖ Ñ 8 Geeralized -diesioal Berstei theoreº Let X R be a copact ad let p x x 1 x ad B fx 1,...,x p 1 x 1...p x f The for ay f CX;R,B f f as, uiforly o X ,...,

9 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM Proof. See Pǎltieau [6]. To prove this we ay put X [0,1]. I fact, there exists a -diesioal cube Y X. By Tietze theore every f CX,Rcabeextededtoafuctio f CY,Rwith f f.ifb f f the B f f. Therefore it is eough to prove theore for the cube Y, sice every-diesioalcubey isliearlyhoeoorphicwiththecube X [0,1]. Cosider the followig fuctios: f 1 x 1,...,x 1, g i x 1,...,x i,...,x x i, h i x 1,...,x i,...,x x 2 i, If we use the idetities p x x, we obtai i 1,...,. 2 p x 2 x 2 xx 1, B f 1 f 1, B g i g i, B h i h i, Fro the other side if we deote px,y y i x i 2, i1 i 1,...,. we ca see that the coditios of Propositio 1 are fulfilled. Hece B f f, for all f CX,R. A earlier versio of Boha-Borovi s theore was proved by Prolla [7, Theore 4 ad Corollary 7]. However, the versio of Pǎltieau allows to prove the ultidiesioal Berstei s theore. 5. Multioial theore For coveiece of the reader we add oe of the possibilities of the proofs of the ultioial theore o the base of See [3]. If we ultiply out the expressio x 1 + +x ad collect coefficiets we get a su i which each ter has the for x a 1 1 a 1,...,a xa xa with soe coefficiet, a 1,...,a where a i are oegative itegers with a 1 +a 2 + +a. We shall prove the followig propositio. See [3, p. 18]. 107

10 MILOSLAV DUCHOŇ ÈÖÓÔÓ Ø ÓÒ 2º a 1,...,a!, where 0! 1. a 1!a 2! a! Proof. The case 2 is the bioial theore. We could ow to prove the case 3 but we shall tae ay > 2. We shall do it by iductio o. For > 2, we have Now x 1 + +x x 1 + +x 1 a x a a x 1 + +x 1 a a x a 1 1 a 1 a xa so the coefficiet of x a 1 1 xa 1 x a is a a 1 a a a 1 a 1 Now use iductio ad defiitio of a. We have thus proved Ì ÓÖ Ñ 9º Multioial theore. The followig equality is valid x 1 + +x a 1,...,a a 1 + +a a 1 + +a I particular, if x 1 + +x 1, the 108 x a 1 1 xa,! a 1! a! xa 1 1 xa, x 1 + +x 1 a 1,...,a a 1 + +a a 1 + +a x a 1 1 xa,! a 1! a! xa 1 1 xa,

11 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM REFERENCES [1] BERNSTEIN, S.: Déostratio du théorèe de Weierstrass, fodeé sur le calcul des probabilités, Cou. Soc. Math. Kharov /13, 1 2. [2] BUTZER, P. L.: O two-diesioal Berstei polyoials, Caad. J. Math , [3] CHILDS, L.: A Cocrete Itroductio to Higher Algebra, i: Udergrad. Texts Math., Spriger-Verlag, Berli, [4] HILDEBRANDT, T. H. SCHOENBERG, I. J.: O liear fuctioal operatios ad the oet proble for a fiite iterval i oe or several diesios, A. Math , [5] LORENTZ, G. G.: Berstei Polyoials, i: Matheatical Expositios, Vol. 8, Uiversity of Toroto Press, Toroto, [6] PǍLTINEANU, G.: Clase de ulţii de iterpolare î raport cu u subspaţiu de fucţii cotiue, i: Structuri de ordie î aaliza fucţioalǎ, Vol. 3, Editura Acadeiei Roǎe, Bucureşti, 1992, pp [7] PROLLA, J. B.: A geeralized Berstei approxiatio theore, Math. Proc. Cabridge Phil. Soc , Received February 7, 2011 Matheatical Istitute Slova Acadey of Scieces Štefáiova 49 SK Bratislava SLOVAKIA E-ail: iloslav.ducho@at.savba.s 109