On the Durrmeyer variant of q-Bernstein operators based on the shape parameter λ
Lianta Su, Reşat Aslan, Feng-Song Zheng, M. Mursaleen
Abstract
Abstract In this work, we consider several approximation properties of a Durrmeyer variant of q -Bernstein operators based on Bézier basis with the shape parameter $\lambda \in[ -1,1]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>λ</mml:mi><mml:mo>∈</mml:mo><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:math> . First, we calculate some moment estimates and show the uniform convergence of the proposed operators. Next, we investigate the degree of approximation with regard to the usual modulus of continuity, for elements of Lipschitz-type class and Peetre’s K -functional, respectively. Finally, to compare the convergence behavior and consistency of the related operators, we demonstrate some convergence and error graphs for certain $\lambda \in[ -1,1]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>λ</mml:mi><mml:mo>∈</mml:mo><mml:mo>[</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:math> and q -integers.