Stancu and Kantorovich-Type Generalizations of a Bernstein Operator: Approximating Locally Integrable Functions

This study investigates the approximation properties of two different types of parameter-dependent generalizations of Stancu type operators. In the first step, it is determined that this operator defined on the interval [-1,1] is an operator of Korovkin type satisfying the theorem and its important properties are analyzed. Then, a new class of operators of Kantorovich type is defined using this operator and the study on the approximation properties of these operators is elaborated. Another important part of the study is to investigate the convergence properties of both classes of operators in Lp spaces. In this context, the effect of the operators on functions and their convergence properties are evaluated and the advantages of the newly defined operators over the classical approximations are demonstrated. In addition, graphs of the approximation of these operators are presented and the effects of the operators on the functions are visually analyzed. By presenting the theoretical analysis and visual results of both operators, the study provides important information about their convergence.