A GENERALIZATION OF KANTOROVICH OPERATORS AND A SHAPE-PRESERVING PROPERTY OF BERNSTEIN OPERATORS

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1 Bulletin of the Transilvania University of Braşov Vol 5(54, No Series III: Mathematics, Informatics, Physics, A GENERALIZATION OF KANTOROVICH OPERATORS AND A SHAPE-PRESERVING PROPERTY OF BERNSTEIN OPERATORS Radu PĂLTĂNEA 1 Abstract We construct a generalization of the Kantorovich operators, depending on a parameter b and we prove that if a function f C 1 [, 1] with f(, satisfies the differential inequality f + bf, then functions B n (f, n N satisfy the same inequality, where B n are the Bernstein operators. 2 Mathematics Subject Classification: 41A36, 41A35. Key words: Kantorovich type operators, Bernstein operators, shape-preserving property. 1 Introduction The Bernstein operators on the space C[, 1] are defined by: ( B n (f, x f p n, (x, f C[a, b], x [, 1], n N, (1 n where ( n p n, (x x (1 x n. The Kantorovich modification of the Bernstein operators are given by: +1 K n (f, x ( p n, (x f(tdt, f C[, 1], x [, 1], n N. (2 We note that, the Kantorovich operators K n can be obtained by the following formula K n D B I, (3 where D is the differentiation operator: D(f f, f C 1 [, 1] and I is the antiderivative operator: I(f, x f(tdt, f C[, 1], x [, 1]. More general, if L : C[, 1] Cr [, 1] is an arbitrary linear operator and r N, if we denote by D r and I r, the iterates of operators D and I, then the operator D r L I r is named the Kantorovich modification of operator L of order r. These operators play a crucial role in simultaneous approximation. Other types of generalizations or modifications of Kantorovich operators, partially included in References, are also nown. 1 Transilvania University of Braşov, Faculty of Mathematics and Informatics Iuliu Maniu 5, Braşov 591, Romania, radupaltanea@yahoo.com

2 66 Radu Păltănea 2 Definition. Main results We consider a generalization of the Kantorovich operators in the following sense. Definition 2.1. Let a parameter b. For any n N define the operator Kn b : C[, 1] C[, 1], defined by +1 Kn(f, b +1 b x : ( + b p n, (xe e bt f(tdt + p n, (x [( + b ( be b +1 b ]e e bt f(tdt, (4 for f C[, 1], x [, 1]. Remar 2.1. If we tae b in (4 we obtain the Kantorovich operators given in (2. Theorem 2.1. Operators K b n are linear and positive, for any n N and b. Proof. The linearity is clear. In order to prove the positivity it is enough to show that ( + b ( be b. Consider function ϕ(t 1 + t + (t 1e t, t R. If we denote t it is sufficient to show that ϕ(t, for t. We have ϕ (t 1 + te t. The minimum of function ϕ is reached at point t 1 and ϕ ( 1 1 e 1 >. Hence ϕ (t >, t R. Then function ϕ is increasing on R. But ϕ( and hence ϕ(t, for t. In order to give another description of operators Kn b we consider operators D b : C 1 [, 1] C[, 1] and I b : C[, 1] C 1 [, 1], given by D b (f, x f (x + bf(x, f C 1 [, 1], x [, 1], I b (f, x e bx e bt f(tdt, f C[, 1], x [, 1. Lemma 2.1. Let n N and b. We have i (D b I b (f f, for all f C[, 1], ii (I b D b (f f, for all f C 1 [, 1], such that f(. Proof. i If f C[, 1], then I b (f is the solution of the Cauchy problem y + by f, y(. Then (D b I b (f f. ii If f C 1 [, 1] and f(, then integrating by parts we obtain, for x [, 1]: (I b D b (f, x e bx e bt (f (t + bf(tdt e bx[ e bx f(x f( b f(x. e bt f(tdt + b b e bt f(tdt

3 A generalization of Kantorovich operators 67 Theorem 2.2. For any n N and b we have: K b n D b B I b. (5 Proof. Let f C[, 1] and x [, 1]. Using the convention P n, (x, for < or > n, we have: (D b B I b (f, x (B (I b (f, x + bb (I b (f, x ( ( [p n, 1 (x p n, (x]i b +b ( [p n, 1 (x + p n, (x]i b ( [( + bp n, 1 (x ( bp n, (x]i b [ ( + 1 ( p n, (x ( + bi b ( bi b From this it follows immediately (4. The results above allow us to derive a more general shape-preservation property for Bernstein operators. For this, let b. Set We have ]. D b : {f C 1 [, 1] : D b (f, f( }. (6 Theorem 2.3. For any n N, n 2 and b, we have B n (D b D b. Proof. Let f D b. We have (D b B n (f (D b B n I b (D b (f K b n 1 (D b(f. Since D b (f and K b n 1 is a positive operator it follows Kb n 1 (D b(f, i.e. (D b B n (f. Also B n (f, f(. Hence B n (f D b. Theorem 2.4. We have for all f C[, 1]. K b n(f f (7 (The symbol means the uniform convergence on the interval [, 1]. Proof. Since operators Kn b are positive it suffices to prove relation (7 for three test functions. Let us denote e (t t, t [, 1], for, 1, 2. Then denote g I b (e,, 1, 2. From the convergence properties of Bernstein operators we have B (g g and (B (g g, for, 1, 2. Hence, for the same indices we have (D b B (g D b (g. But (D b B (g Kn(e b and D b (g e. Hence Kn(e b e, for, 1, 2. Therefore we can apply the theorem of Popoviciu- Bohmann-Korovin and we obtain (7.

4 68 Radu Păltănea References [1] Adell, J. A. and Pérez-Palomares A, Second modulus preservation inequalities for generalized Bernstein-Kantorovich operators, Stancu, Approximation and optimization. Proceedings of ICAOR: international conference, Cluj-Napoca, Romania, July 29 August 1, (D.D. Stancu et al. ed., Volume I. Cluj-Napoca: Transilvania Press, , [2] Aniol, G., On the rate of pointwise convergence of the Kantorovich-type operators, Fasc. Math. 29 (1999, [3] Bărbosu, D., Kantorovich-Stancu type operators, JIPAM, 5 (24, no. 3, article 53. [4] Cao, J. On the generalized polynomials of L. V. Kantorovich and their asymptotic behaviour, Chin. Ann. Math. 2 ( [5] Gupta, V., The Bézier variant of Kantorovitch operators, Comput. Math. Appl. 47 (24, no. 2-3, [6] Kacsó, D., Simultaneous approximation by almost convex operators, Schriftenreihe des Fachbereichs Mathemati, Univ. Duisburg, Germany, SM-DV-479, (2. [7] Kantorovich, L.V., Sur certains développements suivant les polynômes de la forme de S. Bernstein, I, II, C. R. Acad. URSS (193, , [8] Li, C., Shi, N. and Huo, X., Some approximate properties for a ind of generalized Bernstein-Kantorovich operators (Chinese, J. Fujian Norm. Univ., Nat. Sci. 24 (28, no. 4, 1-4. [9] Liu, J. and Chen, G., Locally inverse theorem in the L p [, 1], (1 p for generalized Kantorovich polynomial operator, J. Math. Res. Expo. 19 (1999, no. 3, [1] López-Moreno, A.J., Martinez-Moreno y J. and Muñoz-Delgado, F.J., Asymptotic behavior of Kantorovich type operators, Monografías del Semin. Matem. García de Galdeano, 27 (23, [11] Mache, D.H. and Zhou, D.X., Characterization theorems for the approximation by a family of operators, J. Approx. Theory 84 (1996, [12] Ren, Q., Generalized and convergence of the Bernstein type operator, J. Math. Study 31 (1998, no.1, [13] Wei, W. The construction, convergence and asymptotic formula of generalized W - Bernstein-Kantorovich operator (Chinese Acta Math. Sci. 2, Suppl. (2, [14] Zene, W. and Junfang, A generalization of the Bernstein operators, (Chinese J. Baoji Coll. Arts Sci., Nat. Sci. 2 (2, no. 4,

5 Erratum Theorem 2.2 contains an error of computation. Consequently the operators given in Definition 1 are not the real Kantorovich operators attached to Bernstein operators and the differential operator D b. The correction is made in the paper: R. Păltănea, A note on generalized Bernstein-Kantorovich operators, Bull. Transilvania Univ Brasov, Ser III, 6(55, No. 2 ( 213, The author