On Several Positive Linear Operators

The operator approximation is one of the most important branches of the approximation theory. In recent decades, reseaches on the operator approximation have achieved rapid progress, ranging from continuous function spaces to measurable function spaces. What is more, it has widespread impact on other mathematics branches.The present thesis deals with Bernstein-Sikkema operator and Cesàro averaging operator. Bernstein-Sikkema operator was introduced first by Sikkema in [1], 1975. It is a kind of generalization of the classical Bernstein operator. The Cesàro averaging operator is an important positive linear operator which has been widely studied. Many authors have been focused on these operators, and they have obtained a large number of remarkable properties. This thesis based on these results shows some new items, which enrich the theories of Bernstein-Sikkema operator and Cesàro averaging operator. Following is the structure of this thesis.In the first section, the background of Bernstein-Sikkema operator and Cesàro averaging operator is given, together with the necessary declarations of the definitions and marks.In the second section, we investigate the simultaneous approximation of the Bernstein-Sikkema operator, and establish the direct and equivalent theorems by using the Ditzian-Totik modulus of smoothnessω_φλ² (0≤λ≤1).In the third, we establish a new equivalence theorem for the approximation by the Cesàro averaging operator in the continuous space and the L_2Ï€^p space, which gives the optimal approximation rate.