Certain Linear Operator And The Proposed Interpolation Approximation Theorem

The study of direct and inverse theorems on the approximation of linear operators to functions in normed linear spaces is an important subject in the approximation theory. It is significant in theory and application. In this paper we use pointwise modulus of smoothness (f, t) to study approximation direct theorem and equivalent theorem for some linear operators and quasi-interpolant operators; Using pointwise modulus we discuss the strong converse inequality on K-functional; and using a modified weighted K-functional and weighted modulus of smoothness we study approximation with Jacobi weight on operator with non-zero first order moments.Firstly, using the equivalent relation between of K-functional and weighted modulus of smoothness, we discussed the linear combinations approximation for Bernstein-Kantorovich operators on direct theorem and equivalent theorem and obtained: when 0 ≤ λ ≤ 1,0 < α < 2r/(2 λ), we can get an equivalent theorem; and when α > 2r/(2 λ),it is not true. Thus we improve the approximation order to 2r by Bernstein-Kantorovich operater. So we generate the result of Bernstein-Kantorovich operater from Ditzian-Totik modulus and classic modulus to the unified pointwise modulus of smoothness.Secondly, using the relation between the weighted modified K-functional, the weighted modulus of smoothness ,the weighted main-part modulus of smoothness . we get the pointwise direct and inverse approximation theorem with Jacobi weight for S'zdsz-Kantorovich operator. Thus some results on w(x) = 0(w(x) denotes the weight function), Ditzian-Totik modulus and classic modulus are extend.Thirdly, we obtain strong converse inequality on K-functional for Szaz operator with pointwise modulus, which extend the result with Ditzian-Totik modulus.4. Finaly, we introduce the Bernstein-Kantorovich quasi interpolant and study the approximation equivalent theorem with pointwise modulus (f. t) (o≤λ≤1 ) in space L[0,1] and approximation equivalent theorem with Ditzian-Totik modulus(f, t)in space L_p[0,1](1 ≤p ≤ω).