The Approximation Research Of Bernstein Type Operators

The thought and method of approximation permeate in almost all the disciplines, including discipline in the natural science and humanities. Bernstein approximation is one kind of classics and abundant research subjects in the theory of approximations, which mainly makes use of good properties and graceful structures of Bernstein polynomials to discuss a great deal of relations between Bernstein operators and the functions approached by it. This thesis has studied the approximation properties of Bernstein operators and its modified operators in detail.The first part in this article has expounded the formulation about theory of approximations problems, and pointed out the research purpose and main contents of this thesis. The second part has summarized from the one-dimension and multivariate respects the abundant approximation properties of Bernstein polynomials, mainly including the estimation of approximation degree, derivative approximation, linear combination approximation and weighted approximation. The third part has proposed the model of modified Bernstein operators, and defined a new function type on which has discussed the convergence of derivative operators of modified Bernstein operators and modified Bernstein operator on simplex and achieved the smallest terms which guarantee the convergence. In order to describe a higher degree of smoothness, the fourth part has investigated the extended results of pointwise approximation of Bernstein operators with the help of linear combinations of Bernstein polynomials and properties of modulus of smoothness.This subject has certain value to understand the development history of Bernstein approximation and probe into its extensive properties and application.