The Approximation Error Estimates Of Two Bernstein Type Operators

The operator approximation is an important branches of the approximation theory, in which the approximation by positive operators is a very interesting research subject. A number of mathematicians have studied it and obtained many valuable research out-comes. However, the lower estimate about the approximation error of positive operator is still very few up to now.The present thesis deals with localized Bernstein-Sikkema operator and iterated Boolean sums of Bernstein operator. It is a kind of generalization of the classical Bern-stein operator and Li cuixiang and Liu yana have given the positive inverse theory of Bernstein-sikkema operator. We will denote such an M-fold Boolean sum of the Bern-stein operators. On the approximation error of the operator, Zhou xinlong has given its approximation rate and Steckin-type inequality. This thesis based on these results shows some new items, which enrich the theories of localized Bernstein-Sikkema oper-ator and iterated Boolean sums of Bernstein operator. Following is the structure of this thesis.In the first chapter, we introduce the background of localized Bernstein-Sikkema operator and iterated Boolean sums of Bernstein operator, together with the necessary declarations of the definitions and marks.In the second chapter, we investigate the pointwise approximation of the localized Bernstein-Sikkema operators by using the central limit theorem of probability and give the approximation order. In the third chapter, we establish a new equivalence theorem for the approximation error for the iterated Boolean sums of Bernstein operators by using the Ditzian-Totik modulus of smoothness and have given estimate of its lower estimate.