APPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS

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1 Hacettepe Joural of Mathematics ad Statistics Volume 32 (2003), 1 5 APPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS E. İbili Received 27/06/2002 : Accepted 17/03/2003 Abstract The weighted approximatio of cotiuous fuctios by Berstei- Chlodowsy polyomials ad their geeralizatios are studied. Keywords: Berstei polyomials, Berstei-Chlodowsy polyomials, Liear positive operators AMS Classificatio: 41 A 36, 41 A 35, 41 A Itroductio The classical Berstei-Chlodowsy polyomials have the followig form ( ) ( ) ( (1.1) B (f, x) = f b C x 1 x ), b b =0 b where 0 x b ad b is a sequece of positive umbers such that b =, = 0. These polyomials were itroduced by Chlodowsy i 1932 as a geeralizatio of Berstei polyomials (1912) o a ubouded set. Although there have bee may studies of Berstei polyomials to the preset date (see [1], [2], [6] ad [7]), the Berstei-Chlodowsy polyomials (1.1) have ot bee ivestigated well eough. The aim of this article is to ivestigate the problem of weighted approximatios of cotiuous fuctios by Berstei-Chlodowsy polyomials (1.1) (for a geeralizatio of these polyomials see [4]). Aara Uiversity, Faculty of Sciece, Departmet of Mathematics, Tadoga, Aara, Turey.

2 2 E. İbili 2. Mai Results ad Let φ(x) be a cotiuous ad icreasig fuctio i (, ) such that φ(x) = ± x ± ρ(x) = 1 + φ 2 (x). Deote by C ρ the space of all cotiuous fuctios f, satisfyig the coditio f(x) M f ρ(x) < x <. Obviously C ρ is a liear ormed space with the orm f ρ = f(x) <x< ρ(x). A Korovi type theorem for liear positive operators L, actig from C ρ to C ρ, has bee proved i [3], where the followig results have bee established Theorem. (See [3]) There exists a sequece of positive liear operators L, actig from C ρ to C ρ, satisfyig the coditios (2.1) L(1, x) 1 ρ = 0 (2.2) L(φ, x) φ ρ = 0 (2.3) L(φ2, x) φ 2 ρ = 0 ad there exists a fuctio f C ρ for which Lf f ρ > Theorem. (See [3]) The coditios (2.1), (2.2), (2.3) imply Lf f ρ = 0 for ay fuctio f belogig to the subset Cρ 0 of C ρ for which exists fiitely. f(x) x ρ(x) Settig ρ(x) = ad applyig Theorem 2.2 to the operators { B(f, x) if 0 x b L (f, x) = f(x) if x / [0, b ] we obtai, 2.3. Propositio. The assertio (2.4) L (f, x) f(x) = 0 holds for ay fuctio f C 0 ρ with ρ(x) = x 0. (2.5) (2.6) (2.7) Note that coditios (2.1),(2.2) ad (2.3) are fulfilled sice B (1, x) = 1 B (t, x) = x B (t 2, x) = x 2 +

3 Approximatio by Berstei-Chlodowsy Polyomials 3 ad therefore B (t 2, x) x 2 = 1 x(b x) b. I view of Theorem 2.1, the assertio (2.4) does ot hold i geeral for a arbitrary fuctio f C ρ, ρ(x) =. Moreover, the polyomials (1.1) are ot able to approximate eve the aalytic fuctio x 2 o the etire iterval [0, b ] without weight, sice (2.7) gives max [B (t 2, x) x 2 ] = b2 4 which does ot coverge to zero for some sequeces (b ) as. A affirmative solutio of the problem of approximatio of the fuctio f(x) = x 2 o the ubouded iterval may be obtaied by cosiderig the polyomials of Berstei- Chlodowsy with b satisfyig the coditio b2 0 as i (1.1). That is, ( ) ( ) ( (2.8) B(f, x) = f b C x 1 x ) 0 x b. b b where 2 =0 = 0. The (2.9) B(t 2, x) = x 2 + ad therefore max [ B (t 2, x) x 2 ] = b 4, which teds to zero as. b, 0 x b, We cosider ow the problem of the approximatio of arbitrary cotiuous fuctios by the polyomials (2.8). Firstly we shall cosider a special case Lemma. For ay cotiuous fuctio f vaishig o [a, ), where a > 0 is idepedet of, B (f, x) f(x) = 0. Proof. Sice by the give coditio, f is bouded, say f(x) M, 0 x a, we ca write for arbitrary small ε > 0 the iequality ( ) f b f(x) < ε + 2M ( ) 2 δ 2 b x, where x [0, b ] ad δ = δ(ε) are idepedet of. By the properties (2.5), (2.6) ad (2.9) ( ) 2 ( b x C x Therefore =0 b ) ( 1 x b ) = B (f, x) f(x) = ε + 2M δ 2 which completes the proof. b 4,.

4 4 E. İbili 2.5. Theorem. Let f be a cotiuous fuctio o the semiaxis [0, ), for which The f(x) = f <. x B (f, x) f(x) = 0. Proof. Obviously it is sufficiet to prove this theorem i the case of f = 0. I this case, for ay ε > 0 there exists a poit x 0 such that (2.10) f(x) < ε, x x 0. Cosider the fuctio g with properties: g(x) = f(x) if 0 x x 0, g(x) is liear o x 0 x x ad g(x) = 0 if x x The f(x) g(x) x 0 x x f(x) g(x) + x x f(x) ad sice we have max g(x) = f(x 0) x 0 x x f(x) g(x) 3ε by the coditio (2.10). Now we obtai B (f, x) f(x) B( f g, x)+ + B (g, x) g(x) + + f(x) g(x) 6ε + B (g, x) g(x). where g(x) vaishes i x x b. By Lemma 2.4, we obtai the desired result A Geeralizatio We ow give a geeralizatio of Berstei-Chlodowsy polyomials, which ca be used to approximate cotiuous fuctios o more geeral weighted spaces. Let ω(x) 1 be ay cotiuous fuctio for x 0. Let also F f (t) = f(t) 1 + t2 ω(t), ad cosider the followig geeralizatio of the polyomials (1.1) (3.1) L (f, x) = ω(x) ( ( ) ( F f )C b x 1 x ), b =0 where x [0, b ] ad b has the same property as i (1.1). I the case of ω(t) = 1 + t 2 the operators (3.1) coicide with (1.1). b

5 Approximatio by Berstei-Chlodowsy Polyomials Theorem. For a cotiuous fuctio f satisfyig the coditio the equality holds. f(x) x ω(x) = K f <, Proof. Obviously ad therefore L (f, x) f(x) = 0 ω(x) L (f, x) f(x) = { ω(x) ( ) ( ) ( b F f C x 1 x ) } F f (x), b b =0 L (f, x) f(x) B (F f, x) F f (x) =. ω(x) Also, F f (x) is a cotiuous fuctio o [0, ) satisfyig F f (x) M f ( ), x 0, sice we have the iequality f(x) M f ω(x) for f. Therefore, by Propositio 2.3 we obtai the desired result. Note that similar statemets may also be obtaied for the geeralizatio of Berstei- Chlodowsy polyomials cosidered i [5]. Acowledgmet. The author is thaful to Prof. Dr. Aif D. Gadjiev, who suggested the problem. Refereces [1] Bleima, G., Butzer, P. L. ad Hah, L. A Berstei-type operator approximatig cotiuous fuctios o semi-axis, Idag. Math. 42, , [2] Che, F. ad Feg, Y. Limit of iterates for Berstei polyomials defied o a triagle, Appl. Math. Ser. B 8 (1), 45 53, [3] Gadjiev, A. D. The covergece problem for a sequece of positive liear operators o ubouded sets ad theorems aalogous to that of P. P. Korovi, Dol. Aad. Nau SSSR 218 (5), Eglish Traslatio i Soviet Math. Dol. 15 (5), [4] Gadjiev, A. D., Efediev, R. O. ad İbili, E. Geeralized Berstei-Chlodowsy polyomials, Rocy Moutai Joural of Mathematics 28 (4), [5] Gadjieva E. A. ad İbili, E. O Geeralizatio of Berstei - Chlodowsy polyomials, Hacettepe Bul. of Nat. Sci. ad Eg. 24, 31 40, [6] Kha, A. Rasul. A ote o a Berstei-Type operator of Bleima, Butzer ad Hah, Joural of Approximatio Theory 53, , [7] Toti, V. Uiform approximatio by Berstei-Type operators, Idag. Math. 46, 87 93, 1984.