The Approximation And Weighted Approximation For Some Positive Linear Operators

In this thesis we discuss the approximation and weighted approximation for three kinds of positive linear operators. In the second chapter, we investigate the approximation for a kind of generalized Bernstein type operators. Direct and inverse theorems of approximation by this kind of operators are discussed, and the Jackson—type integral estimation of approximation and Steckin — Marchaud inequality of weak type of uniform approximation for these operators are obtained. Two-dimensional Bernstein type operators on simplex are constructed, and a necessary and sufficient condition of uniform approximation by mem is obtained, at the same time, the degree of approximation is characterized by two-order moduli of smoothness. In the third chapter, we study the approximation and weighted approximation for a kind of generalized Baskakov operators of two-variable. Using different weight functions, we discuss the convergence on weighted space of polynomials and rate of convergence with Jacobi weight for this kind of two-dimensional product-type operators, which extends the results of one-dimensional operators. In the third part we investigate a kind of two-dimensional non-product-type operators on continuous function space and some important properties and a locally inverse theorem are obtained. In the fourth chapter we construct a kind of recursive Kantorovich type operators and study the approximation for these operators on C space and on L_p space respectively, asymptotic behavior on C space and estimation of degree of approximation on L_p(p > 1) space are obtained.