The Weighted Approximation Of Some Linear Operators

This dissertation consists of two parts. In Part one, the weighted approximation by the linear operators in classical spaces and approximation in Orlicz spaces are studied; In Part two, the approximation of multivariate linear operators is discussed.In chapter 1, we point out a mistake of ZHAO DeJun in paper [1], the direct and inverse theorems of a kind of Bernstein-type operators are established:Theorem 1.1 If, ThenTheorem 1 .2 If , Then there exists ,In the second chapter, using Wickeren's method, the relations between the derivatives of the Bernstein operators and the smoothness of the approximated functions with weights are studied; and an equivalent theorem of the relations between the derivatives of the Bernstein operators and the smoothness of function is obtained:Theorem2. 1.1 If Theorem2.1.2If , thenis equivalent toSimultaneously, the direct theorem of the weighted approximation is established. Theorem2.1.3 If and , thenthenFinally, the inverse inequality of weighted approximation and the Steckin-Marchaud type inequality are established:Theorem2.2.1 If , thenTheorem2.2.2 If thenIn Chapter 3, the approximation with Jacobi weights by the Baskakov operators is discussed and a pointwise and global theorem is obtained:Theorem3. 1.1 If , and, thenTheorem3. 1.2 If, thenis equivalent toIn addition, the relations between the derivatives of the Baskakov operators and the smoothness of the approximated functions with weights are studied; we haveTheorem3.2.1 If and, thenTheorem3.2.2 If and there exists a positive constant v which satisfies the equality ofand , thenis equivalent toThe approximation with Jacob! weights by the linear combinations of the Baskakov operators is also studied.Theorem3.3.1 If , and , thenTheorem3.3.2 If , thenis equivalent toAt the same time, the characterization of higher order derivatives of the Baskakov operators with Jacobi approximation is studied, we haveTheorems. 3. 3 If , then Theorem3.3.4 If , and there exists a positive constant v which satisfies the equality of and , thenis equivalent toFinally, the inverse inequality of weighted approximation of a kind of Baskakov-type operators and the Steckin-Marchaud type inequality are established:Theorem3.4.1 If , then there exists , we haveTheorem3 .4.2 If, thenTheoremTheorems. 4.4 If ,thenIn the fourth chapter, the approximation by the Quasi-Kantorovic operators i n Orlicz spaces is discussed, and the following theorem is obtained:Theorem4.1If , for example satisfies the conditionof 2 , thenTheorem4.2 If , thenIn the fifth chapter, an asymptotic expansion formula of Sikkema operators on the simplex is established:Theorems If , then In chapter 6 and chapter 7, using the multivariate K-functional and the modulus of smoothness, the approximation with weights in classical spaces by the multivariate Bernstein operators and the mul--tivariate Durrmeyer operators are studied, the relations between the derivatives these operators and the smoothness of the approximated functions with weights are established; the direct and the inverse approximated theorems of the multivariate Durrmeyer operators with Jacob! weights are obtained:Theorem6 If , thenTheorem 7. 1 If thenand there exists 0 < δ < 1,Theorem7.2 If , then...