Higher order approximation of functions by modified Goodman-Sharma operators

Year 2024, , 180 - 195, 15.12.2024

Rumen Uluchev , Ivan Gadjev , Parvan Parvanov

https://doi.org/10.33205/cma.1563047

Abstract

Here we study the approximation properties of a modified Goodman-Sharma operator recently considered by Acu and Agrawal in [1]. This operator is linear but not positive. It has the advantage of a higher order of approximation of functions compared with the Goodman-Sharma operator. We prove direct and strong converse theorems in terms of a related K-functional.

Keywords

Bernstein-Durrmeyer operator, Goodman-Sharma operator, Direct theorem, Strong converse theorem, K-functional

Project Number

BG-RRP-2.004-0008

Thanks

This study is financed by the European Union-NextGenerationEU, through the National Recovery and Resilience Plan of the Republic of Bulgaria, project No. BG-RRP-2.004-0008.

References

• A. M. Acu, P. Agrawal: Better approximation of functions by genuine Bernstein-Durrmeyer type operators, Carpathian J. Math., 35 (2) (2019), 125–136.
• A. M. Acu, I. Rasa: New estimates for the differences of positive linear operators, Numer. Algorithms, 73 (3) (2016), 775–789.
• H. Berens, Y. Xu: On Bernstein-Durrmeyer polynomials with Jacobi weights, In Approximation Theory and Functional Analysis, (Edited by C. K. Chui), pp. 25–46, Acad. Press, Boston (1991).
• L. Beutel, H. Gonska and D. Kacsó: Variation-diminishing splines revised, In Proceedings of International Symposium on Numerical Analysis and Approximation Theory, (Edited by R. Trâmbi¸ta¸s), pp. 54–75, Presa Universitar˘a Clujean˘a, Cluj-Napoka (2002).
• W. Chen: On the modified Bernstein-Durrmeyer operator, Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou (China), (1987).
• Z. Ditzian, K. G. Ivanov: Strong converse inequalities, J. Anal. Math., 61 (1993), 61–111.
• H. Gonska, R. P˘alt˘anea: Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions, Czechoslovak Math. J., 60 (135) (2010), 783–799.
• H. Gonska, R. P˘alt˘anea: Quantitative convergence theorems for a class of Bernstein-Durrmeyer operators preserving linear functions, Ukrainian Math. J., 62 (2010), 913–922.
• T. N. T. Goodman, A. Sharma: A modified Bernstein-Schoenberg operator, In Constructive Theory of Functions, Varna 1987, (Edited by Bl. Sendov et al.), pp. 166–173, Publ. House Bulg. Acad. of Sci., Sofia, (1988).
• T. N. T. Goodman, A. Sharma: A Bernstein-type operator on the simplex, Math. Balkanica (New Series), 5 (2) (1991), 129–145.
• K. G. Ivanov, P. E. Parvanov: Weighted approximation by the Goodman-Sharma operators, East J. Approx., 15 (4) (2009), 473–486.
• G. G. Lorentz: Bernstein Polynomials, Mathematical Expositions 8, University of Toronto Press, (1953).
• R. P˘alt˘anea: A class of Durrmeyer type operators preserving linear functions, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity, 5 (2007), 109–117.
• P. E. Parvanov, B. D. Popov: The limit case of Bernstein’s operators with Jacobi weights, Math. Balkanica (N.S.), 8 (2–3) (1994), 165–177.