EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 4, 2014, 419-428 ISSN 1307-5543 – www.ejpam.com Some Approximation Properties of Szasz-Mirakyan-Bernstein Operators Tuncay Tunç1, Ersin Şimşek 2,∗ 1 Department of Mathematics, Mersin University, Mersin, Turkey 2 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. 2010 Mathematics Subject Classifications: 41A25, 41A36, 26A48 Key Words and Phrases: Positive linear operators, Korovkin’s Theorem, Szasz-Mirakyan Operators, Bernstein Operators 1. Introduction LetN denotes the set of natural numbers and letN0 = N∪{0}. Let f be real-valued function defined on the closed interval [0,1]. The n-th Bernstein operator of f , Bn( f ) is defined as Bn � f ; x � = n ∑ k=0 pn,k (x) f � k n � , x ∈ [0,1], n ∈ N (1) where pn,k (x) = � n k � xk(1− x)n−k, 0≤ k ≤ n. The Bernstein polynomials Bn( f ) was introduced to prove the Weierstrass approximation the- orem by S. N. Bernstein [2] in 1912. 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, 10, 12, 14] and references therein. ∗Corresponding author. Email addresses: ttunc77@hotmail.com (T. Tunç), simsek.ersin@gmail.com (E. Şimçek) http://www.ejpam.com 419 c© 2014 EJPAM All rights reserved. T. Tunç, E. Şimşek / Eur. J. Pure Appl. Math, 7 (2014), 419-428 420 For the function f which is continuous on [0,∞), the Szasz-Mirakyan operators which are introduced by G. M. Mirakyan [8] in 1941 and then, are investigated by J. Favard [4] and O. Szasz [15], are defined as Sn( f ; x) = ∞ ∑ m=0 qn,m(x) f � m n � , x ∈ [0,∞), n ∈ N where qn,m(x) = e−nx (nx)m m! , m ∈ N0. (2) 2. Construction of the Generating Operators Let I is a fixed interval (bounded or not) in R and $m be a sequence of density functions on the interval I , that is, the functions $m have the following properties: i. $m non-negative for all x ∈ I and m ∈ N0 ii. ∑∞ m=0$m(x) = 1 for all x ∈ I Let (Ln) be a sequence of positive linear operators defined on the set of the continuous func- tions on the interval I , say C(I). Now we define the generating operators Gn on C(I). For every n ∈ N, x ∈ I and f ∈ C(I) Gn( f ; x) = ∞ ∑ m=0 $m(nx)Lϕn,m ( f ; x) m ∈ N0, (3) where $m are density functions on I and ϕn,m := ϕ(n, m) = αnβm where (αn) is a non- decreasing and (βm) is a strictly increasing natural sequence. It is easy to check that the operators Gn are positive and linear on C(I). Taking I = [0,1], βm = m+ 1, $m = qn,m and Lϕn,m = Bϕn,m where Bϕ and qn,m defined in (1) and (2) respectively, we can rewrite (3) as En � f ; x � = ∞ ∑ m=1 e−nx(nx)m−1 (m− 1)! mαn ∑ k=0 � mαn k � xk(1− x)mαn−k f � k mαn � . (4) The operators En defined in (4) is called the Szasz-Mirakyan-Bernstein (SMB) operators. In this study, we investigate some approximation properties of these operators and find Voronovskya- type 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 prop- erties concerning space of functions and moduli of smoothness of first and second order. For more information see [1] or [11]. T. Tunç, E. Şimşek / Eur. J. Pure Appl. Math, 7 (2014), 419-428 421 1. Let C[0, 1] be the space of real-valued continuous function on [0, 1] equipped with the uniform norm: ‖ f ‖ :=max{| f (x)| : x ∈ [0,1]} and C r[0,1], r ∈ N0, be the set all r-times continuously differentiable functions f ∈ C[0,1]. 2. For the real-valued function f defined on [0,1] and δ ≥ 0, the modulus of continuity ω( f ,δ) and the second modulus of smoothness ω2( f ,δ) of f are defined by ω( f ,δ) := sup |x−y|≤δ {| f (x)− f (y)|}, ω2( f ,δ) := sup 0≤h≤δ sup 0≤x≤1−2h {| f (x + 2h)− 2 f (x + h) + f (x)|}, respectively. It is known that, for a function f ∈ C[0, 1], we have limδ→0ω( f ,δ) = 0 and, for any δ > 0, � � f (t)− f (x) � �≤ω( f ,δ) � |t − x | δ + 1 � (5) 3. As usual, a function f ∈ LipMµ, (M > 0 and 0< µ≤ 1), if the inequality � � f (t)− f (x) � �≤ M |t − x |µ (6) holds for all t, x ∈ [0,1] 4. Let ei denote the test functions defined by ei(t) = t i , t ∈ R, i = 0,1, 2, . . .. Theorem 1 (Korovkin [6]). Let Ln : C[a, b]→ C[a, b] be a sequence of positive linear operators. If lim n→∞ Ln(ei; x) = ei(x), i = 0,1, 2, uniformly on [a, b], then lim n→∞ Ln( f ; x) = f (x). uniformly on [a, b], for every continuous function f defined on [a, b]. 4. Approximation Properties of En In this section we give some classical approximation properties of the operators En. By simple calculations, we get the following lemmas. Lemma 1. For x ∈ [0, 1] and n ∈ N, we have En(e0; x) =1; En(e1; x) =x; En(e2; x) =x2 + (1− x)(1− e−nx) nαn ; T. Tunç, E. Şimşek / Eur. J. Pure Appl. Math, 7 (2014), 419-428 422 En(e3; x) =x3 + 3x (1− x) � 1− e−nx � nαn + (1− x) (1− 2x) nα2 n ∞ ∑ m=1 e−nx(nx)m m.m! ; En(e4; x) =x4 + 6x2 (1− x) � 1− e−nx � nαn + x (1− x) (7− 11x) nα2 n ∞ ∑ m=1 e−nx(nx)m m.m! + (1− x) � 6x2 − 6x + 1 � nα3 n ∞ ∑ m=1 e−nx(nx)m m2m! . Lemma 2. For x ∈ [0,1] and n ∈ N, the following holds: En(e1 − x; x) =0, En((e1 − x)2; x) = (1− x)(1− e−nx) nαn , En((e1 − x)3; x) = (1− x)(1− 2x) nα2 n ∞ ∑ m=1 e−nx(nx)m m.m! , En((e1 − x)4; x) = 3x(1− x)2 nα2 n ∞ ∑ m=1 e−nx(nx)m m.m! + (1− x)(6x2 − 6x + 1) nα3 n ∞ ∑ m=1 e−nx(nx)m m2m! . Lemma 3. For all j ∈ N0, we have ∞ ∑ m=1 e−x xm m jm! ≤ � j + 1 � ! x j , x ∈ (0,∞) . Lemma 4. For all n ∈ N, we have En � � e1 − x �4 ; x � ≤ cn (x) � 1 nαn �2 , x ∈ (0,1] where limn→∞ cn(x) = 6. Proof. For x ∈ (0,1]. By Lemma 2 and Lemma 3, it results that En � � e1 − x �4 ; x � ≤ 3x(1− x)2 nα2 n 2! nx + (1− x) � 6x2 − 6x + 1 � nα3 n 3! n2 x2 ≤ � 1 nαn �2� 6+ 6x2 − 6x + 1 x2nαn � = � 1 nαn �2 cn (x) Theorem 2. If f ∈ C[0, 1], then the sequence of positive linear operators � En converges uni- formly to f on [0, 1]. T. Tunç, E. Şimşek / Eur. J. Pure Appl. Math, 7 (2014), 419-428 423 Proof. From Lemma 1, we get En � ei � [0,1] ⇒ ei i = 0, 1,2 n→∞. Then, using Korovkin’s theorem, we can conclude that En � f � [0,1] ⇒ f , n→∞. Where, the symbol [0,1] ⇒ shows the uniform convergence on [0,1]. 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, 13]. Theorem 3. If f ∈ C2[0, 1], then lim n→∞ n.αn � En � f ; x � − f (x) � = 1 2 (1− x) f ′′ (x) for every fixed x ∈ [0,1]. Proof. We use the Taylor formula for a fixed point x0 ∈ [0, 1]. For all t ∈ [0, 1], we have f (t) = f � x0 � + f ′ � x0 � � t − x0 � + 1 2 f ′′ � x0 � � t − x0 �2 + g � t; x0 � � t − x0 �2 where g(t; x0) is the Peano form of the remainder, g(.; x0) ∈ C2[0,1] and lim t→x0 g(t; x0) = 0. Because En(e0; x) = 1, then En � f ; x0 � − f � x0 � = f ′ � x0 � En �� e1 − x0 � ; x0 � + 1 2 f ′′ � x0 � En � � e1 − x0 �2 ; x0 � + En � g � · , x0 � · � e1 − x0 �2 ; x0 � By Cauchy-Schwartz’s inequality, we have nαnEn � g � · , x0 � � e1 − x0 �2 ; x0 � ≤ � n2α2 nEn � � e1 − x0 �4 ; x0 ��1/2 · � En � g2 � · , x0 � ; x0 ��1/2 The function ϕ(t; x0) = g2(t; x0), t ≥ 0, satisfies the conditions of Teorem 2; therefore lim n→∞ En(g 2(t; x0); x0) = 0 T. Tunç, E. Şimşek / Eur. J. Pure Appl. Math, 7 (2014), 419-428 424 Moreover, by Lemma 4, we have nαnEn � g � ·, x0 � � e1 − x0 �2 ; x0 � ≤ � n2α2 n � cn � x0 � � 1 nαn �2��1/2 · � En �� g2 � ·, x0 �� ; x0 ��1/2 It results that limn→∞ nαnEn � g � ·, x0 � � e1 − x0 �2 ; x0 � = 0. By the above results and by Lemma 2, we obtain lim n→∞ n.αn � En � f ; x0 � − f � x0 �� = 1 2 � 1− x0 � f ′′ � x0 � . 6. Rates of Convergence In this section we shall give error estimates, the for f ∈ C[0,1] and f ∈ C1[0,1]. Theorem 4. If f ∈ C[0,1], then En � f � − f ≤ 2ω � f ; 1 p nαn � . (7) Proof. Let f ∈ C[0,1]. By linearity and positivity of the operators En we get, for all n ∈ N and x ∈ [0,1], that � �En � f ; x � − f (x) � �≤ En �� � f − f (x) � � ; x � . (8) Now using (5) in inequality (8) we have, for any δ > 0, that � �En � f ; x � − f (x) � �≤ � 1+ 1 δ En �� �e1 − x � � ; x � � ω � f ;δ � (9) Applying the Cauchy-Schwartz inequality for positive linear operators it follows from (9) that � �En � f ; x � − f (x) � �≤ � 1+ 1 δ r En � � e1 − x �2 ; x � � ω � f ;δ � Using Lemma 1 in the last inequality, we can write � �En � f ; x � − f (x) � �≤  1+ 1 δ � (1− x) � 1− e−nx � nαn �1/2  ω � f ;δ � ≤ 1+ 1 δ � 1 nαn �1/2 ! ω � f ;δ � . Choosing δn = � 1 nαn �1/2 , we have the inequality � �En � f ; x � − f (x) � �≤ 2ω � f ; 1 p nαn � . T. Tunç, E. Şimşek / Eur. J. Pure Appl. Math, 7 (2014), 419-428 425 Theorem 5. For all f ∈ LipMµ and x ∈ [0, 1], we have En � f � − f ≤ M � 1 nαn �µ/2 . Proof. Applying En to the inequality (6), we have � �En � f ; x � − f (x) � �≤ En �� � f − f (x) � � ; x � ≤ M En � � �e1 − x � � µ ; x � If we consider the Hölder inequality with p = 2 µ , q = 2 2−µ and by Lemma 2 for the last inequality, we get � �En � f ; x � − f (x) � �≤M � En � � e1 − x �2 ; x ��µ/2 = M � (1− x) � 1− e−nx � nαn �µ/2 ≤M � 1 nαn �µ/2 . Theorem 6. If f ∈ C1 [0, 1], then En � f � − f ≤ 2 p nαn ω � f ′; 1 p nαn � . Proof. By the mean value theorem, there exists ξ ∈ (t; x): f (t)− f (x) = (t − x) f ′ (ξ) . As the operators En are linear and positive and on the fact that Lemma 2 it follows immediately the equality En � f ; x � − f (x) = En �� e1 − x � f ′ � ξt,x � ; x � = En �� e1 − x � � f ′ � ξt,x � − f ′ (x) � ; x � where ξt,x ∈ (min {t, x} ,max {t, x}). Using the property of modulus of continuity, we get � � f ′ � ξx (t) � − f ′ (x) � �≤ω � f ′; � �ξx (t)− x � � � ≤ω � f ′;δ � � 1+ 1 δ � �ξx (t)− x � � � ≤ω � f ′;δ � � 1+ 1 δ |t − x | � . Consequently, � �En � f ; x � − f (x) � �≤ω � f ′;δ � En � � �e1 − x � � � 1+ 1 δ � �e1 − x � � � ; x � =ω � f ′;δ � En � � �e1 − x � �+ 1 δ � e1 − x �2 ; x � T. Tunç, E. Şimşek / Eur. J. Pure Appl. Math, 7 (2014), 419-428 426 =ω � f ′;δ � � En �� �e1 − x � � ; x � + 1 δ En � � e1 − x �2 ; x � � Using the Cauchy-Schwartz inequality and Lemma 2 for the last inequality, we have � �En � f ; x � − f (x) � �≤ω � f ′;δ � �r En � � e1 − x �2 ; x � + 1 δ En � � e1 − x �2 ; x � � =ω � f ′;δ � � √ √(1− x) (1− e−nx) nαn + 1 δ (1− x) � 1− e−nx � nαn � ≤ω � f ′;δ � � 1 p nαn + 1 δ 1 nαn � Choosing δn = � 1 nαn �1/2 , we have the inequality |En � f ; x � − f (x)| ≤ 2 p nαn ω � f ′; 1 p nαn � . Theorem 7. If f ∈ C2[0, 1], then for all n ∈ N the following inequality holds: |En( f ; x)− f (x)| ≤ f ′′ 2nαn . (10) Proof. Using the Taylor formula, we write f (t) = f (x) + f ′ (x) (t − x) + R f ,x (t) (11) where R f ,x (t) = t ∫ x (t − v) f ′′(v) dv. By the mean value theorem that there exist ξt,x ∈ (min {x , t} ,max {x , t}), which satisfies R f ,x (t) = f ′′ � ξt,x � 2 (t − x)2. we can rewrite (11) as f (t) = f (x) + f ′ (x) (t − x) + f ′′ � ξt,x � 2 (t − x)2 (12) Applying En to the formula (12), by Lemma 2, we have � �En � f ; x � − f (x) � �≤En � � � � � f ′′ � ξt,x � 2 � � � � � (e1 − x)2; x ! ≤ f ′′ 2 En � (e1 − x)2; x � T. Tunç, E. Şimşek / Eur. J. Pure Appl. Math, 7 (2014), 419-428 427 = f ′′ 2 (1− x) � 1− e−nx � nαn ≤ f ′′ 2nαn . Theorem 8. If f ∈ C[0,1], then for all n ∈ N the following inequality holds: ‖En( f )− f ‖ ≤ 3ω2 � f ; 1 p nαn � . Proof. Let x ∈ [0,1]. For 0< h≤ 1 2 min {x , 1− x} we define gh (x) = 1 h2 h/2 ∫ −h/2 h/2 ∫ −h/2 � 2 f � x + t1 + t2 � − f � x + 2t1 + 2t2 � d t1d t2. Consequently � �g ′′ (x) � �= � � � f (x + 2h)− 2 f (x + h) + f (x) + � f (x − 2h)− 2 f (x − h) + f (x) � � ≤ 2 δ2 ω2 � f ;δ � also � � � ∫ a −a h (t) d t � � �≤ 2a sup u∈[−a,a] |h (u)|. Therefore � � f (x)− gh (x) � �= � � � � � � � 1 h2 h/2 ∫ −h/2 h/2 ∫ −h/2 � f � x + 2t1 + 2t2 � − 2 f � x + t1 + t2 � + f (x) d t1d t2 � � � � � � � = 1 h2 h/2 ∫ −h/2 h/2 ∫ −h/2 � � f � x + 2t1 + 2t2 � − 2 f � x + t1 + t2 � + f (x) � �d t1d t2 ≤ω2 � f ;δ � . Using these inequalities and (10) we have En � f � − f ≤ En � f − gh � + En � gh � − gh + f − gh ≤ En f − gh + En � gh � − gh + f − gh =2 f − gh + En � gh � − gh ≤ 2ω2 � f ;δ � + ‖g ′′h ‖ 2nαn ≤2ω2 � f ;δ � + 1 2nαn 2 δ2 ω2 � f ;δ � =ω2 � f ;δ � � 2+ 1 nαnδ2 � Choosing δn = � 1 nαn �1/2 we get the desired estimate. REFERENCES 428 References [1] F Altomare and M Campiti. Korovkin-Type Approximation Theory and Its Applications. Gruyter Studies in Mathematics, Berlin, 1994. [2] S N Bernstein. Demonstration du theoreme de Weierstrass Fondee sur le Calcul de Prob- abilites. Communications of the Kharkov Mathematical Society, 13(2):1–2, 1912. [3] A Ciupa. A Voronovskaya- Type Theorem for a Positive Linear Operators. International Journal of Mathematics and Mathematical Sciences, 2006(ID 42368):1–7, 2006. [4] J Favard. Sur les multiplicateurs d’interpolation. Journal de Mathématiques Pures et Appliquées, 23(9):219–247, 1944. [5] A Aral-V Gupta and R P Agarwal. Applications of q-Calculus in Operator Theory. Springer, Berlin, 2013. [6] P P Korovkin. Linear Operators and Approximation Theory. Hindustan Publishing Corpo- ration, Delhi, 1960. [7] G G Lorentz. Bernstein Polynomials. Chelsea Publishing Company, New York, 1986. [8] G M Mirakyan. Approximation of continuous functions with the aid of polynomials of the form e−nx ∑mn k=0 Ck,n xk. Comptes rendus de l’Académie des sciences de l’URSS, 31(2):201– 205, 1941. [9] G Nowak and V Gupta. The Rate of Pointwise Approximation of Positive Linear Operators Based on q-Integer. Ukrainian Mathematical Journal, 63(3):350–360, 2011. [10] S Ostrovska. On the limit q-Bernstein operators. Mathematica Balkanica, 18(1-2):165– 172, 2004. [11] R Paltanea. Approximation Theory using Positive Linear Operators. Birkhauser, Boston, 2004. [12] G M Phillips. Bernstein polynomials based on the q-integers. Annals of numerical Math- ematics, 4(1-4), 1997. [13] L Rempulska and M Skorupka. The Voronovskaya Theorem for some Operators of the Szasz-Mirakyan Type. Le Matematiche, L(2):251–261, 1995. [14] D D Stancu. Approximation of functions by a new class of linear polynomial operator. Revue Roumaine de Mathématiques Pures et Appliquées, 13(8):1173–1194, 1968. [15] O Szasz. Generalizations of S. Bernstein’s polynomials to the infinite interval. Journal of Research of the National Bureau of Standards, 45(3):239–245, 1950.