Some Approximation Properties of Szasz-Mirakyan-Bernstein Operators

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1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol 7, No 4, 04, ISSN wwwejpamcom Some Approximation Properties of Szasz-Mirakyan-Bernstein Operators Tuncay Tunç, Ersin Şimşek, Department of Mathematics, Mersin University, Mersin, Turkey Mersin University Graduate School of Natural and Applied Sciences Department of Mathematics, Mersin, Turkey Abstract In this study, we have constructed a new sequence of positive linear operators by using Szasz- Mirakyan and Bernstein operators on space of continuous functions on the unit compact interval We also find order of this approximation by using modulus of continuity and give the Voronovskaya-type theorem 00 Mathematics Subject Classifications: 4A5, 4A36, 6A48 Key Words and Phrases: Positive linear operators, Korovkin s Theorem, Szasz-Mirakyan Operators, Bernstein Operators Introduction Let denotes the set of natural numbers and let 0 = {0} Let f be real-valued function defined on the closed interval [0, ] The n-th Bernstein operator of f, B n (f ) is defined as where n B n f = p n,k (x) f k=0 p n,k (x) = k, x [0, ], n () n n x k ( x) n k, 0 k n k The Bernstein polynomials B n (f ) was introduced to prove the Weierstrass approximation theorem by S N Bernstein [] in 9 They have been studied intensively and their connection with different branches of analysis, such as convex and numerical analysis, total positivity and the theory of monotone operators have been investigated Basic facts on Bernstein polynomials and their generalizations can be found in [5, 7, 9, 0,, 4] and references therein Corresponding author addresses: ttunc77@hotmailcom (T Tunç), simsekersin@gmailcom (E Şimçek) 49 c 04 EJPAM All rights reserved

2 T Tunç, E Şimşek / Eur J Pure Appl Math, 7 (04), For the function f which is continuous on [0, ), the Szasz-Mirakyan operators which are introduced by G M Mirakyan [8] in 94 and then, are investigated by J Favard [4] and O Szasz [5], are defined as where S n (f ) = q n,m (x)f m=0 m, x [0, ), n n nx (nx)m q n,m (x) = e, m 0 () m! Construction of the Generating Operators Let I is a fixed interval (bounded or not) in and ϖ m be a sequence of density functions on the interval I, that is, the functions ϖ m have the following properties: i ϖ m non-negative for all x I and m 0 ii m=0 ϖ m(x) = for all x I Let (L n ) be a sequence of positive linear operators defined on the set of the continuous functions on the interval I, say C(I) Now we define the generating operators G n on C(I) For every n, x I and f C(I) G n (f ) = ϖ m (nx)l ϕn,m (f ) m 0, (3) m=0 where ϖ m are density functions on I and ϕ n,m := ϕ(n, m) = α n β m where (α n ) is a nondecreasing and (β m ) is a strictly increasing natural sequence It is easy to check that the operators G n are positive and linear on C(I) Taking I = [0, ], β m = m, ϖ m = q n,m and L ϕn,m = B ϕn,m where B ϕ and q n,m defined in () and () respectively, we can rewrite (3) as mα n mαn k E n f = x k ( x) mαn k f (4) (m )! k mα n m= k=0 The operators E n defined in (4) is called the Szasz-Mirakyan-Bernstein (SMB) operators In this study, we investigate some approximation properties of these operators and find Voronovskyatype theorem and the order of this approximation by using modulus of continuity 3 Some Notations and Auxiliary Facts In this section we will give some basic definitions, theorems and some elementary properties concerning space of functions and moduli of smoothness of first and second order For more information see [] or []

3 T Tunç, E Şimşek / Eur J Pure Appl Math, 7 (04), Let C[0, ] be the space of real-valued continuous function on [0, ] equipped with the uniform norm: f := max{ f (x) : x [0, ]} and C r [0, ], r 0, be the set all r-times continuously differentiable functions f C[0, ] For the real-valued function f defined on [0, ] and δ 0, the modulus of continuity ω(f, δ) and the second modulus of smoothness ω (f, δ) of f are defined by ω(f, δ) := sup { f (x) f (y) }, x y δ ω (f, δ) := sup sup 0 h δ 0 x h { f (x h) f (x h) f (x) }, respectively It is known that, for a function f C[0, ], we have lim δ 0 ω(f, δ) = 0 and, for any δ > 0, f (t) f (x) t x ω(f, δ) (5) δ 3 As usual, a function f Lip M µ, (M > 0 and 0 < µ ), if the inequality f (t) f (x) M t x µ (6) holds for all t, x [0, ] 4 Let e i denote the test functions defined by e i (t) = t i, t, i = 0,,, Theorem (Korovkin [6]) Let L n : C[a, b] C[a, b] be a sequence of positive linear operators If lim n L n(e i ) = e i (x), i = 0,,, uniformly on [a, b], then lim L n(f ) = f (x) n uniformly on [a, b], for every continuous function f defined on [a, b] 4 Approximation Properties of E n In this section we give some classical approximation properties of the operators E n By simple calculations, we get the following lemmas Lemma For x [0, ] and n, we have E n (e 0 ) =; E n (e ) =x; E n (e ) =x ( x)( e nx ) ;

4 T Tunç, E Şimşek / Eur J Pure Appl Math, 7 (04), E n (e 3 ) =x 3 3x ( x) e nx E n (e 4 ) =x 4 6x ( x) e nx ( x) 6x 6x nα 3 n ( x) ( x) nα n ; mm! m= x ( x) (7 x) nα n m m! m= Lemma For x [0, ] and n, the following holds: E n (e x) =0, E n ((e x) ) = ( x)( e nx ), E n ((e x) 3 ( x)( x) ) = nα n E n ((e x) 4 3x( x) ) = nα n, mm! m= m= mm! m= ( x)(6x 6x ) nα 3 n Lemma 3 For all j 0, we have e x x m j! m j m! x j, x (0, ) m= Lemma 4 For all n, we have where lim n c n (x) = 6 E n e x 4 c n (x), x (0, ] Proof For x (0, ] By Lemma and Lemma 3, it results that E n e x 4 3x( x)! ( x) 6x 6x 3! nα n nx nα 3 n n x 6 6x 6x = c x n (x) mm! m m! m= Theorem If f C[0, ], then the sequence of positive linear operators E n converges uniformly to f on [0, ]

5 T Tunç, E Şimşek / Eur J Pure Appl Math, 7 (04), Proof From Lemma, we get E n ei [0,] e i i = 0,, n Then, using Korovkin s theorem, we can conclude that E n f [0,] f, n Where, the symbol [0,] shows the uniform convergence on [0, ] 5 Voronovskaya-Type Theorem The Voronovskaya theorem for the Bernstein operators is given in [7] or [6] Also, for the sequence of positive linear operators can be found in [3, 3] Theorem 3 If f C [0, ], then for every fixed x [0, ] lim nα n En f f (x) = n ( x) f (x) Proof We use the Taylor formula for a fixed point x 0 [0, ] For all t [0, ], we have f (t) = f x 0 f x 0 t x0 f x 0 t x0 g t0 t x0 where g(t 0 ) is the Peano form of the remainder, g( 0 ) C [0, ] and Because E n (e 0 ) =, then lim g(t 0 ) = 0 t x 0 E n f 0 f x0 =f x 0 En e x 0 0 By Cauchy-Schwartz s inequality, we have f x 0 En e x 0 ; x0 E n g, x 0 e x 0 ; x0 E n g ;, x 0 e x 0 x0 n α n E 4; / n e x 0 x0 En g /, x 0 0 The function ϕ(t 0 ) = g (t 0 ), t 0, satisfies the conditions of Teorem ; therefore lim E n(g (t 0 ) 0 ) = 0 n

6 T Tunç, E Şimşek / Eur J Pure Appl Math, 7 (04), Moreover, by Lemma 4, we have E n g, ; / x 0 e x 0 x0 n α n c n x0 E, n g / x 0 0 It results that lim n E n g, ; x 0 e x 0 x0 = 0 By the above results and by Lemma, we obtain lim nα n En f 0 f x0 = x0 f x 0 n 6 Rates of Convergence In this section we shall give error estimates, the for f C[0, ] and f C [0, ] Theorem 4 If f C[0, ], then En f f ω f ; (7) nαn Proof Let f C[0, ] By linearity and positivity of the operators E n we get, for all n and x [0, ], that En f f (x) En f f (x) (8) Now using (5) in inequality (8) we have, for any δ > 0, that En f f (x) δ E e n x ω f ; δ (9) Applying the Cauchy-Schwartz inequality for positive linear operators it follows from (9) that En f f (x) E n e x ω f ; δ δ Using Lemma in the last inequality, we can write En f f (x) ( x) e nx / ω f ; δ / ω f ; δ δ δ Choosing δ n = /, we have the inequality En f f (x) ω f ; nαn

7 T Tunç, E Şimşek / Eur J Pure Appl Math, 7 (04), Theorem 5 For all f Lip M µ and x [0, ], we have µ/ En f f M Proof Applying E n to the inequality (6), we have En f f (x) En f f (x) M En e x µ If we consider the Hölder inequality with p = µ, q = we get µ and by Lemma for the last inequality, En f f (x) M E n e x µ/ ( x) e nx = M M µ/ µ/ Theorem 6 If f C [0, ], then En f f ω nαn Proof By the mean value theorem, there exists ξ (t): f ; f (t) f (x) = (t x) f (ξ) nαn As the operators E n are linear and positive and on the fact that Lemma it follows immediately the equality E n f f (x) = En e x f ξ t,x = En e x f ξ t,x f (x) where ξ t,x (min {t, x}, max {t, x}) Using the property of modulus of continuity, we get Consequently, f ξ x (t) f (x) ω f ; ξx (t) x ω f ; δ ξx (t) x δ ω f ; δ δ t x En f f (x) ω f ; δ E n e x e x δ =ω f ; δ e E n x e x δ

8 T Tunç, E Şimşek / Eur J Pure Appl Math, 7 (04), =ω f ; δ E e n x δ E n e x Using the Cauchy-Schwartz inequality and Lemma for the last inequality, we have En f f (x) ω f ; δ E n e x δ E n e x =ω f ; δ ( x) ( e nx ) ( x) e nx δ ω f ; δ nαn δ Choosing δ n = /, we have the inequality E n f f (x) ω nαn f ; nαn Theorem 7 If f C [0, ], then for all n the following inequality holds: f E n (f ) f (x) (0) Proof Using the Taylor formula, we write f (t) = f (x) f (x) (t x) R f,x (t) () where R f,x (t) = t (t v)f (v) d v x By the mean value theorem that there exist ξ t,x (min {x, t}, max {x, t}), which satisfies R f,x (t) = f ξ t,x (t x) we can rewrite () as f (t) = f (x) f (x) (t x) f ξ t,x (t x) () Applying E n to the formula (), by Lemma, we have f ξ En f f (x) t,x f En (e x) E n (e x)

9 T Tunç, E Şimşek / Eur J Pure Appl Math, 7 (04), f ( x) e nx f = Theorem 8 If f C[0, ], then for all n the following inequality holds: E n (f ) f 3ω f ; nαn Proof Let x [0, ] For 0 < h min {x, x} we define g h (x) = h h/ h/ f x t t f x t t d t d t h/ h/ Consequently g (x) = f (x h) f (x h) f (x) f (x h) f (x h) f (x) also δ ω f ; δ a a h (t) d t a f (x) gh (x) = h = h sup u [ a,a] h/ h/ h/ h/ h (u) Therefore h/ h/ h/ h/ ω f ; δ f x t t f x t t f (x) d t d t f x t t f x t t f (x) d t d t Using these inequalities and (0) we have En f f En f gh En gh gh f gh En f gh En gh gh f gh = f gh En gh gh ω f ; δ g h ω f ; δ δ ω Choosing δ n = / we get the desired estimate f ; δ = ω f ; δ δ

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