Approximation of functions by a new class of Gamma type operators; theory and applications

The study of the linear methods of approximation, which are given by sequences of positive and linear operators, studied extremely, in relation to different subjects of analysis, such as numerical analysis. The principal objective of this manuscript is to develop a new and more comprehensive version of Gamma type operators and presented their approximation features. For this purpose, we benefit from two sequences of functions, which are α_n(x) and β_n(x), and from the function τ(x). To indicate how the function τ play a significant role in the construction of the operator, we reconstruct the mentioned operators which preserve exactly two test functions from the set {1, τ, τ²}. Then we established Voronovskaya type theorem and order of approximation properties of the newly defined operators utilizing weighted modulus of continuity to show that their approximation properties. At the end of this note, we present a series of numerical results to show that the new operators are an approximation technique.