King operators which preserve $x^j$

Year 2023, Volume: 6 Issue: 2, 90 - 101, 15.06.2023

Zoltán Fınta

https://doi.org/10.33205/cma.1259505

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

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