# Approximation by Bézier variant of the Baskakov-Kantorovich operators in the case 0<α<1

## Abstract

The present paper deals with the approximation of Bezier variants of Baskakov-Kantorovich operators $$V_{n,\alpha}^{\ast}$$ in the case $$0<\alpha<1$$. Pointwise approximation properties of the operators $$V_{n,\alpha}^{\ast}$$ are studied. A convergence theorem of this type approximation for locally bounded functions is established. This convergence theorem subsumes the approximation of functions of bounded variation as a special case.

## Authors

Xiao-Ming Zeng,
Department of Mathematics, Xiamen University, Xiamen 361005, China

Vijay Gupta
Department of Mathematics, Netaji Subhas Institute of Technology,New Delhi-110078, India

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

## Keywords

Approximation; Baskakov-Kantorovich operators; Bezier variant; locally bounded functions; Lebesgue-Stieltjes integral.

## Paper coordinates

M. Zeng, V. Gupta, O. Agratini, Approximation by Bézier variant of the Baskakov- Kantorovich operators in the case, The Rocky Mountain Journal of Mathematics, 44 (2014) no. 1, pp. 317-327. https://doi.org/10.1216/RMJ-2014-44-1-317

## PDF

##### Journal

Rocky Mountain Journal of Mathematics

##### Publisher Name

Rocky Mountain Mathematics

357596

##### Online ISSN

1. U. Abel and V. Gupta, An estimate of the rate of convergence of a Bezier variant of the Baskakov-Kantorovich operators for bounded variation functions, Demo. Math. 36 (2003), 124 136.

2. R. Bojanic and F. Cheng, Rate of convergence of Bernstein polynomials of functions with derivative of bounded variation, J. Math. Anal. Appl. 141 (1989), 136 151.

3. F.L. Cao and C.M. Ding, Lp approximation by multivariate Baskakov-Kantorovich operators, J. Math. Anal. Appl. 348 (2008), 856 861.

4. F. Cheng, On the rate of convergence of Bernstein polynomials of functions of bounded variation, J. Approx. Theor. 39 (1983), 259 274.

5. , On the rate of convergence of Szasz-Mirakyan operator for functions of bounded variation, J. Approx. Theor. 40 (1984), 226 241.

6. S. Guo, Q. Qi and G. Liu, The central approximation theorems for Baskakov-Bezier operators J. Approx. Theor. 147 (2007), 112 124.

7. V. Gupta, An estimate on the convergence of Baskakov-Bezier operators, J. Math. Anal. Appl. 312 (2005), 280 288.

8. P. Pych-Taberska, Some properties of the Bezier-Kantorovich type operators, J. Approx. Theor. 123 (2003), 256 269.

9. Z. Walczak, A note on the convergence of Baskakov type operators, Appl. Math. Comp. 202 (2008), 370 375.

10. X.M. Zeng, On the rate of two Bernstein-Bezier type operators for bounded variation functions II, J. Approx. Theor. 104 (2000), 330 344.

11. X.M. Zeng and V. Gupta, Rate of convergence of Baskakov-Bezier type operators for locally bounded functions, Comp. Math. Appl. 44 (2002), 1445 1453.

12. X.M. Zeng and J.N. Zhao, Exact bounds for some basis functions of approximation operators, J. Ineq. Appl. 6 (2001), 563 575.