Annals of Functional Analysis

Modified α-Bernstein operators with better approximation properties

Arun Kajla and Tuncer Acar

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In the present note, following a new approach recently described by Khosravian-Arab, Dehghan, and Eslahchi, we construct a new kind of α-Bernstein operator and study a uniform convergence estimate for these operators. We also prove some direct results involving the asymptotic theorems. Finally, we illustrate the convergence of the operators to a certain function with the help of Maple software.

Article information

Ann. Funct. Anal., Volume 10, Number 4 (2019), 570-582.

Received: 8 January 2019
Accepted: 10 March 2019
First available in Project Euclid: 30 October 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10}
Secondary: 41A25: Rate of convergence, degree of approximation 41A36: Approximation by positive operators 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)

approximation by polynomials Bernstein polynomials Voronovskaya-type theorem


Kajla, Arun; Acar, Tuncer. Modified $\alpha $ -Bernstein operators with better approximation properties. Ann. Funct. Anal. 10 (2019), no. 4, 570--582. doi:10.1215/20088752-2019-0015.

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  • [1] T. Acar, A. Aral, and I. Raşa, The new forms of Voronovskaya’s theorem in weighted spaces, Positivity 20 (2016), no. 1, 25–40.
  • [2] T. Acar and A. Kajla, Degree of approximation for bivariate generalized Bernstein type operators, Results Math. 73 (2018), no. 2, art. ID 79.
  • [3] A. M. Acu, P. N. Agrawal, and T. Neer, Approximation properties of the modified Stancu operators, Numer. Funct. Anal. Optim. 38 (2017), no. 3, 279–292.
  • [4] A. M. Acu, S. Hodiş, and I. Raşa, A survey on estimates for the differences of positive linear operators, Constr. Math. Anal., published online 7 November 2018.
  • [5] A. M. Acu and I. Raşa, New estimates for the differences of positive linear operators, Numer. Algorithms 73 (2016), no. 3, 775–789.
  • [6] A. Aral, V. Gupta, and R. P. Agarwal, Applications of $q$-Calculus in Operator Theory, Springer, New York, 2013.
  • [7] D. Bǎrbosu, On the remainder term of some bivariate approximation formulas based on linear and positive operators, Constr. Math. Anal., published online 7 November 2018.
  • [8] X. Chen, J. Tan, Z. Liu, and J. Xie, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl. 450 (2017), no. 1, 244–261
  • [9] A. D. Gadjiev and A. M. Ghorbanalizadeh, Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables, Appl. Math. Comput. 216 (2010), no. 3, 890–901.
  • [10] I. Gavrea and M. Ivan, An answer to a conjecture on Bernstein operators, J. Math. Anal. Appl. 390 (2012), no. 1, 86–92.
  • [11] H. Gonska, On the degree of approximation in Voronovskaja’s theorem, Stud. Univ. Babeş-Bolyai Math. 52 (2007), no. 3, 103–115.
  • [12] H. Gonska and I. Raşa, Asymptotic behaviour of differentiated Bernstein polynomials, Mat. Vesnik 61 (2009), no. 1, 53–60.
  • [13] V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, Cham, 2014.
  • [14] V. Gupta and G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Dev. Math. 50, Springer, Cham, 2017.
  • [15] V. Gupta, G. Tachev, and A. M. Acu, Modified Kantorovich operators with better approximation properties, Numer. Algorithms 81 (2019), no. 1, 125–149.
  • [16] A. Kajla and T. Acar, Blending type approximation by generalized Bernstein-Durrmeyer type operators, Miskolc Math. Notes 19 (2018), no. 1, 319–336.
  • [17] H. Khosravian-Arab, M. Dehghan, and M. R. Eslahchi, A new approach to improve the order of approximation of the Bernstein operators: Theory and applications, Numer. Algorithms 77 (2018), no. 1, 111–150.
  • [18] L. Lupas and A. Lupas, Polynomials of binomial type and approximation operators, Stud. Univ. Babeş-Bolyai Math. 32 (1987), no. 4, 61–69.
  • [19] S. A. Mohiuddine, T. Acar, and A. Alotaibi, Construction of a new family of Bernstein-Kantorovich operators, Math. Methods Appl. Sci. 40 (2017), no. 18, 7749–7759.
  • [20] M. Mursaleen, K. J. Ansari, and A. Khan, On $(p,q)$-analogue of Bernstein operators, Appl. Math. Comput. 266 (2015), 874–882. Erratum, Appl. Math. Comput. 278 (2016), 70–71.
  • [21] G. Nowak, Approximation properties for generalized $q$-Bernstein polynomials, J. Math. Anal. Appl. 350 (2009), no. 1, 50–55.
  • [22] A. A. Opriş, Approximation by modified Kantorovich-Stancu operators, J. Inequal. Appl. 2018, no. 346.
  • [23] R. Păltănea, Approximation Theory Using Positive Linear Operators, Birkhäuser Boston, Boston, 2004.
  • [24] D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 13 (1968), 1173–1194.
  • [25] D. D. Stancu, The remainder in the approximation by a generalized Bernstein operator: A representation by a convex combination of second-order divided differences, Calcolo 35 (1998), no. 1, 53–62.
  • [26] G. Tachev, The complete asymptotic expansion for Bernstein operators, J. Math. Anal. Appl. 385 (2012), 1179–1183.