Asymptotic and Non-asymptotic Results in the Approximation by Bernstein Polynomials

Abstract

This paper deals with the approximation of functions by the classical Bernstein polynomials in terms of the Ditzian–Totik modulus of smoothness. Asymptotic and non-asymptotic results are respectively stated for continuous and twice continuously differentiable functions. By using a probabilistic approach, known results are either completed or strengthened.

Estimation Features by Transformed Bernstein kind Polynomials

“…Satisfy the reduction formula(Sharma, 2016;Acu and Agrawal, 2019;Acu and Tachev, 2021;Acu et al, 2023;Adell and Cárdenas-Morales, 2022 ) 𝑆(𝑣, 𝑛, 𝑥, 𝑦) = 𝑥𝑆(𝜈 − 1, 𝑛, 𝑥, 𝑦) + 𝑛𝛽𝑆(𝑣, 𝑛 − 1, 𝑥 + 𝛽, 𝑦) By repeated use of the reduction formula, we can show that 𝑆(1, 𝑛, 𝑥, 𝑦) = ∑ 𝑛 𝑘=0 ( 𝑛 𝑣 ) 𝑣! 𝛽 𝑘 (𝑥 + 𝑦 + 𝑛𝛽) 𝑛−𝑣 as 𝑥𝑆(0, 𝑛, 𝑥, 𝑦) = (𝑥 + 𝑦 + 𝑛𝛽) 𝑛…”