Weighted Simultaneous Approximation By The Linear Combinations Of Gamma Operators

The major subjects in the approximation of operators are to study the convergent property of a sequence of linear operators, the quantization of its convergence rate and the saturation results in the theory of approximations. In this paper, we will introduce the weighted modulus of smoothness and the weighted K-functional to discuss the direct and the converse theorems, the local saturation results on the simultaneous approximation on an infinite interval with Jacobi-weight by the integral operator, the linear combinations of Gamma operators, for functions which is defined in the space Loo(0, 皁). The strong converse inequalities are proved for the Gamma operator.Firstly, using the equivalent relation between the weighted modulus of smoothness and the weighted K-functional, we discuss the direct theorems of simultaneous approximation with Jacobi-weight for functions with s-order derivatives.Secondly, by introducing a modified K-functional and the relation between the weighted modulus of smoothness and the weighted main-part modulus of smoothness, we obtain the inverse theorems of the weighted simultaneous approximation.Thirdly, after getting the direct theorems, we will investigate the local saturation results.Fourthly, the strong converse inequalities are proved for the Gamma operator that state the equivalence of two terms of error in approximation to the second modulus of smoothness with step-weight function