Abstract
In 2010, Long and Zeng introduced a new generalization of the Bernstein polynomials that depends on a parameter and called -Bernstein polynomials. After that, in 2018, Lain and Zhou studied the uniform convergence for these -polynomials and obtained a Voronovaskaja-type asymptotic formula in ordinary approximation. This paper studies the convergence theorem and gives two Voronovaskaja-type asymptotic formulas of the sequence of -Bernstein polynomials in both ordinary and simultaneous approximations. For this purpose, we discuss the possibility of finding the recurrence relations of the -th order moment for these polynomials and evaluate the values of -Bernstein for the functions , is a non-negative integer
Highlights
In 2012, [1] introduced another proof of the Weierstrass approximation theorem by using a sequence of positive linear operators, named the classical Bernstein polynomials, as
In 1962 [6], Schurer introduced a sequence based on a parameter and proved that the sequence has an approximation order depending on the parameter
We give a modification of the Voronovaskaja formula for the -Bernstein sequence in the ordinary approximation
Summary
Let be the linear space of all real functions acting on a set is linear and positive if it satisfies: i). In 2012, [1] introduced another proof of the Weierstrass approximation theorem by using a sequence of positive linear operators, named the classical Bernstein polynomials, as [] []. In 1953, Korovkin [3] introduced a simple tool to decide that, for a sequence of linear positive operators, is converges to the function [ ] by checking the sequence's values of conditions. In 1962 [6], Schurer introduced a sequence based on a parameter and proved that the sequence has an approximation order depending on the parameter. Many kinds of research were developed and studied sequences depending on parameters; here we refer to [7, 8, 9, 10, 11,12]. In 2018 [14], Lain and Zhou studied the uniform convergence and obtained a Voronovaskaja-type asymptotic formula for the sequence in ordinary approximation
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