The Approximation And Simultaneous Approximation Of Combinations Bernstein Operators In The Average Setting

Function approximation theory is an important branch of modernmathematics.Establishing an approximation representation of the objective functionunder certain conditions to approximate the objective function and considering thedegree of approximation and the characteristics of the objective function itself arethe research goals of fuction approximation theory.Because of many good properties such as end interpolation and simultaneousapproximation, The research on the approximation property of Bernstein operatorsand various promotion of Bernstein operators has been a hot topic recent years.There have been many results about the approximation properties of thecombinations of Bernstein operators in the worst setting. Generally the performanceof a combination of operators is better than the original operator, so how is theperformance of operators which is constructed by the linear combinations ofBernstein operators in the average setting?The paper mainly researches the approximation property of the combinations ofBernstein operators in the Wiener space. By using some existing conclusions andcomputatinal and analytical skills the approximation orders and simultaneousapproximation orders of the combinations of Bernstein operators are obtained. Theconclusion shows that the approximation orders and the simultaneous approximationorders of the combinations of Bernstein operators are consistent with those ofBernstein operators in the above measure space with the average case setting.