Abstract
In this paper, a kind of new type Bezier operators is introduced. The Korovkin type approximation theorem of these operators is investigated. The rates of convergence of these operators are studied by means of modulus of continuity. Then, by using the Ditzian-Totik modulus of smoothness, a direct theorem concerned with an approximation for these operators is also obtained.
Highlights
Some Bézier type operators, which are based on the Bézier basis function, have been introduced and studied (e.g., see [ – ])
In view of the Bézier basis function, which was introduced by Bézier [ ], in, Chang [ ] defined the generalized Bernstein-Bézier polynomials for any α >, and a function f defined on [, ] as follows: nk Bn,α(f ; x) = f nJnα,k(x) – Jnα,k+ (x), ( ) k=where Jn,n+ (x) =, and Jn,k(x) = n i=k Pn,i(x), k =, n, Pn,i(x)
We introduce a new type of Bézier operators as follows: n
Summary
Some Bézier type operators, which are based on the Bézier basis function, have been introduced and studied (e.g., see [ – ]). In , Ren [ ] introduced Bernstein type operators as follows: n– The moments of the operators Ln(f ; x) were obtained as follows (see [ ]). We will study the Bézier variant of the Bernstein type operators Ln(f ; x), which have been given by ( ). We introduce a new type of Bézier operators as follows: n–
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