Abstract
In this paper, we construct Bernstein type operators that reproduce exponential functions on simplex with one moved curved side. The operator interpolates the function at the corner points of the simplex. Used function sequence with parameters α and β not only are gained more modeling flexibility to operator but also satisfied to preserve some exponential functions. We examine the convergence properties of the new approximation processes. Later, we also state its shape preserving properties by considering classical convexity. Finally, a Voronovskaya-type theorem is given and our results are supported by graphics.
Highlights
Over the last 60 years, the study of linear approximation has been revealed powerful and important tools in approximation theory, mainly due to their possible applications in mathematics and in other ...elds such as statistics, engineering and computer science
The operator interpolates the function at the corner points of the simplex
We examine the convergence properties of the new approximation processes
Summary
Over the last 60 years, the study of linear approximation has been revealed powerful and important tools in approximation theory, mainly due to their possible applications in mathematics and in other ...elds such as statistics, engineering and computer science. In recent years, there is an increasing interest in modifying linear operators so that the new versions reproduce certain exponential functions. In [3], the authors proposed the modi...cation of Bernstein operators to reproduce some exponential functions and perform better compared to the classical Bernstein operators, under su¢ cient conditions. We give de...nition of a new family of generalized Bernstein operators and their certain elementary properties. Certain shape preserving properties including generalized convexity for bivariate functions are obtained. In the last two section, we have an inequality showing that the new operator is closer to function f and present examples of graphics supporting the results
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