The Approximation Of Lupas-Baskakov-Type Operators

This paper consists of four parts.In the first chapter, the approximation of Lupas-Baskakov-type operators in classicalspaces is studied. In section 1, the approximation of these operators in C[0, ∞) spaces isdiscussed, and both the pointwise and global equivalent theorems for approximation are obtained by using of the modulus of smoothness ω2λ (f,t); Furthermore, the relationsbetween the derivatives of these operators and the smoothness of the approximated functions are studied; By using of the modulus of smoothness w(f,t)p and the modifiedK-functional K (f, t) p, the equivalent theorems for approximation of these operators in Xp[0,∞) (1 ≤ p ≤∞) spaces are obtained in section 2; In section 3, the global approximation by these operators is further discussed in Lp[0,∞)(1≤ p≤∞) spaces and theconverse inequalities of the strong-type are obtained.In the second chapter, the approximation of the 2r-th linear combinations of Lupas-Baskakov-type operators is studied. In section 1, the equivalent theorems for the pointwise approximation are obtained by using of the modulus of smoothness of highorder w2r(f,t); In section 2, the pointwise estimates for the simultaneous approximationare further studied, and the equivalent theorems for approximation are obtained.In the third chapter, the approximation of the generalized Lupas-Baskakov integral operators is studied. In section 1, absorbing the ideas of paper [5], both the asymptotic representations and the theorems for the simultaneous approximation are obtained; Insection 2, the approximation for a class of functions Br(v) which is wider than thefunctions of bounded variation is discussed, and the estimates for the simultaneous approximation are obtained.In the fourth chapter, the approximation by others operators is studied. In section 1,the approximation by a new linear positive operators which were defined by P.N.Agrawal and K.J.Thamer in paper [4] is discussed in Lp[0,∞)(1 ≤ p ≤ ∞) spaces. With the ideals ofE.V.Wickeren in paper [6], the converse inequalities of the weak-type are obtained; In section 2, the converse inequalities of the strong-type are further discussed; In section 3, the simultaneous approximation by the modified Szasz operators with Jacobi weights is discussed, and the converse inequalities of weak-type are obtained; In section 4, the direct and converse theorems for the pointwise approximation by Baskakov operators are given, and the relations between the derivatives of the operators and the smoothness of the approximated functions are studied, which extend the results of paper [7].