Approximation by Bézier type of Meyer–König and Zeller operators

Abstract

In this paper, we give direct, inverse and equivalence approximation theorems for the Bézier type of Meyer-König and Zeller operator with unified Ditzian-Totik modulus ω ϕ λ ( f, t) (0 ≤ λ ≤ 1).

Bézier-Summation-Integral-Type Operators That Include Pólya–Eggenberger Distribution

“…Bézier curves [22] are widely used in computer-aided geometric design, as well as in other areas of computer science. To obtain a better understanding of positive linear operators, many efforts are devoted to studying operators involving Bézier basis functions; namely, Bernstein-type [23,24], Pǎltǎnea-type involving Appell and Gould-Hopper polynomials [25,26], Meyer-König and Zeller [27], Baskakov [28], Srivastava-Gupta [29,30], Chlodowsky [31], Kantorovich [32], Bleimann-Butzer-Hahn [33], Durrmeyer [34,35], and Bleimann-Butzer-Hahn-Kantorovich [36]. From a computational point of view, it is important to remember that linear positive operators converge with inherently slow convergence rates (see the seminal book by Korovkin [37]) due to the Voronoskaja-type saturation results, but the good news from an applicative viewpoint is that many of these operators (especially those of the Bernstein-type) admit asymptotic expansion with respect to the parameter r, when the function is smooth enough (see [38][39][40][41]).…”