Quantitative estimates by linear and non-linear Bernstein-Chlodowsky-Kantorovich Operator

Bernstein polynomials were introduced by Bernsteinin 1912 and possess many interesting prop erties. These properties have led to the discovery of new applications and developments. These generalizations aim to provide adequate and powerful tools for applications such as numerical analysis, computer-aided geo metric design, and differential equation solutions. With these applications, the importance of Bernstein poly nomials increased and became an important topic in approximation theory. Classical approximation theory is concerned with the representation of continuous functions by simpler functions such as polynomials and trigonometric functions. In the last century, significant attention has been given to the realization that linearity is not a necessary condition for approximation operators. The positive nonlinear operators with maximum and product were presented by Bede. The Choquet integral has a wide range of applications in finance, the study of cooperative games, statistical mechanics, and potential theory. Approximation of max-product operators and Choquet integral operators, which can generate better approximation estimates than their classical coun terparts, has been developed in recent years. In this study, firstly we introduce the Choquet integral in relation to Bernstein-Chlodowsky-Kantorovich operators and obtain quantitative estimates in uniform and pointwise approximation using these operators. Then the max-product type of these operators is denoted, and their ap proximation properties are investigated