King operators which preserve $x^j$

Abstract

We prove the unique existence of the functions $r_n$ $(n=1,2,\ldots )$ on $[0,1]$ such that the corresponding sequence of King operators approximates each continuous function on $[0,1]$ and preserves the functions $e_0(x)=1$ and $e_j(x)=x^j$, where $j\in\{ 2,3,\ldots\}$ is fixed. We establish the essential properties of $r_n$, and the rate of convergence of the new sequence of King operators will be estimated by the usual modulus of continuity. Finally, we show that the introduced operators are not polynomial and we obtain quantitative Voronovskaja type theorems for these operators.

Keywords

• Bernstein operator
• King operator
• Korovkin theorem
• modulus of continuity
• polynomial operator

References

1. T. Acar, M. C. Montano, P. Garrancho and V. Leonessa: On sequences of J. P. King-type operators, J. Funct. Spaces, 2019 (2019), Article ID 2329060, 12 pages.
2. A. M. Acu, H. Gonska and M. Heilmann: Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render, J. Numer. Anal. Approx. Theory, 50 (2001), 3–11.
3. J. M. Aldaz, O. Kounchev and H. Render: Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces, Numer. Math., 114 (2009), 1–25.
4. M. Birou: A proof of a conjecture about the asymptotic formula of a Bernstein type operator, Results Math., 72 (2017), 1129–1138.
5. D. Cárdenas-Morales, P. Garrancho and I. Ra¸sa: Asymptotic Formulae via a Korovkin-Type Result, Abstr. Appl. Anal., 2012 (2012), Article 217464, 12 pages.
6. R. A. DeVore and G. G. Lorentz: Constructive Approximation, Springer, Berlin (1993).
7. R. A. DeVore: The Approximation of Continuous Functions by Positive Linear Operators, Lecture Notes in Mathematics, 293, Springer, New York, (1972).
8. Z. Finta: Direct and converse theorems for King operators, Acta Univ. Sapientiae, 12 (1) (2020), 85–96.

9. Z. Finta: Estimates for Bernstein type operators, Math. Inequal. Appl., 15 (1) (2012), 127–135.
10. Z. Finta: Bernstein type operators having 1 and $x^j$ as fixed points, Centr.Eur. J. Math., 11 (12) (2013), 2257–2261.
11. Z. Finta: New properties of King’s operators, Positivity, 17 (1) (2013), 101–109.
12. Z. Finta: A quantitative variant of Voronovskaja’s theorem for King-type operators, Constr. Math. Anal., 2 (3) (2019), 124–129.
13. I. Gavrea and M. Ivan: Complete asymptotic expansions related to conjecture on a Voronovskaja-type theorem, J. Math. Anal. Appl., 458 (2018), 452–463.
14. J. P. King: Positive linear operators which preserve $x^2$, Acta Math. Hungar., 99 (3) (2003), 203–208.
15. W. Rudin: Principles of Mathematical Analysis, Third Edition, McGraw-Hill, New York (1976).