King operators which preserve $x^j$

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Constructive Mathematical Analysis
Volume:6 Issue:2
King operators which preserve $x^j$

King operators which preserve $x^j$

Pages : 90-101

View : 52 | Download : 74

Publication Date : 2023-06-15

Article Type : Research Paper

Abstract :We prove the unique existence of the functions $r_n$ $insert ignore into journalissuearticles values(n=1,2,\\ldots );$ on $[0,1]$ such that the corresponding sequence of King operators approximates each continuous function on $[0,1]$ and preserves the functions $e_0insert ignore into journalissuearticles values(x);=1$ and $e_jinsert ignore into journalissuearticles values(x);=x^j$, 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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