---

Discontinuity, Nonlinearity, and Complexity 11(2) (2022) 253--273 | DOI:10.5890/DNC.2022.06.006

Asha Ram Gairola$^1$, Karunesh Kumar Singh$^2$, Lakshmi Narayan Mishra$^3$

$^1$ Department of Mathematics, Doon University, Dehradun-248001 (Uttarakhand), India

$^2$ Department of Applied Sciences and Humanities Institute of Engineering and Technology Lucknow-226021

(Uttar Pradesh), India

$^3$ Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University,

Vellore 632 014, Tamil Nadu, India

Download Full Text PDF

 

Abstract

We obtain local and global rate of approximation by two new variants, $D_n^{M,1}(f,x)$ and $D_n^{M,2}(f,x)$ of Bernstein Durrmeyer operators, recently introduced by Acu et al. By utilizing a suitable Ditzian-Totik modulus of smoothness, we prove that the approximation process $D_n^{M,2}(f,x)$ is quadratic convergent. An error estimate for the functions of bounded variation by the B\'{e}zier variant of the operators $D_n^{M,1}(f,x)$ is obtained.

References

1. [1]	Acu, A.M., Gupta, V., and Tachev, G. (2019), Better numerical approximation by Durrmeyer type operators, Results in Mathematics, 74(3), 1-24.

2. [2]	Khosravian-Arab, H., Dehghan, M., and Eslahchi, M.R. (2018), A new approach to improve the order of approximation of the Bernstein operators: theory and applications, Numerical Algorithms, 77(1), 111-150.

3. [3]	Derriennic, M.M. (1981), Sur l'approximation de fonctions int{e}grables sur [0, 1] par des polyn\^{o}mes de Bernstein modifies, J. Approx. Theory, 31, 325-343.

4. [4]	Ditzian, Z. and Ivanov, K. (1989), Bernstein-type operators and their derivatives, J. Approx. Theory, 56(1), 72-90.

5. [5]	Ditzian, Z. and Totik, V. (1987), Moduli of Smoothness, Springer-Verlag, New York.

6. [6]	Durrmeyer, J.L. (1967), Une formule d'inversion de la transform{e}e de Laplace: applications \`{a} la th{e}orie des moments, Th{e}se de 3e cycle, Paris.

7. [7]	Goldberg, S. and Meir, A. (1971), Minimum moduli of ordinary differential operators, Proc. London Math. Soc., 23, 1-15.

8. [8]	Gonska, H.H. (1983), On approximation of continuously differentiable functions by positive linear operators, Bull. Austral. Math. Soc., 27, 73-81.

9. [9]	Gonska, H.H. and Zhou, X.L. (1991), A global inverse theorem on simultaneous approximation by Bernstein-Durrmeyer operators, J. Approx. Theory, 67(3), 284-302.

10. [10]	Gupta, V. and Agarwal, R.P. (2014), Convergence Estimates in Approximation Theory, Springer, New York.

11. [11]	Gupta, V., L{o}pez-Moreno, A.-J., and Latorre-Palacios, J.-M. (2009), On simultaneous approximation of the Bernstein Durrmeyer operators, Appl. Math. Comput., 213(1), 112-120.

12. [12]	Gupta, V. and Srivastava, H.M. (2018), A general family of the srivastava-gupta operators preserving linear functions, Eur. J. Pure Appl. Math., 11(3), 575-579.

13. [13]	Kajla, A. and Acar, T. (2019), Modified $\alpha-$Bernstein operators with better approximation properties, Ann. Funct. Anal., 10(4), 570-582.

14. [14]	Lupa\c{s}, A. (1972), Die Folge der Beta Operatoren, Dissertation, Universit\"{a}t Stuttgart.

15. [15]	\"{O}zarslan, M.A. and Aktu\v{g}lu, H. (2013), Local approximation for certain King type operators, Filomat, 27(1), 173-181.

16. [16]	Srivastava, H.M. and Gupta, V.A. (2003), Certain family of summation-integral type operators, Math. Comput. Modelling, 37, 1307-1315.