Abstract
The aim of this work is to consider bicomplex Bernstein polynomials attached to analytic functions on a compact C2-disk and to present some approximation properties extending known approximation results for the complex Bernstein polynomials. Furthermore, we obtain and present quantitative estimate inequalities and the Voronovskaja-type result for analytic functions by bicomplex Bernstein polynomials.
Highlights
The approximation theory might be considered as a numerical approach to mathematical analysis problems with wide theoretical methods
One of the most fundamental polynomials, the celebrated one of this theory, is the so-called Bernstein polynomials, which can be used to uniformly approximate any continuous function f ∈ C [0, 1] defined on the interval [0, 1]
Our goal is to introduce Bernstein polynomials for bicomplex numbers inspired by Equation (1)
Summary
The approximation theory might be considered as a numerical approach to mathematical analysis problems with wide theoretical methods (see [1,2]). The basic idea behind the approximation theory is to obtain a representation of any arbitrary function in terms of simple fundamental functions (usually polynomials) with the known properties. One of the most fundamental polynomials, the celebrated one of this theory, is the so-called Bernstein polynomials, which can be used to uniformly approximate any continuous function f ∈ C [0, 1] defined on the interval [0, 1] (see [3]). The first asymptotic formulation of the pointwise approximation of continuous functions whose second derivatives exist at a point x on [0, 1] was presented by Voronovskaja (see [4]). Determination of the convergence rate of the sequences of linear positive operators converging to the function f is a key point in the approximation theory. The following limit, which is known as Voronovskaja formula: lim m( Lm ( f ; x ) − f ( x )), m→∞
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