Lagrange Interpolation Operators Approaching Protons

In this paper, we determine the priority of the Lagrange interpolation based on the Chebyshev nods. Firstly, we consider the rate of mean convergence of derivatives by Lagrange interpolation operators based on the Chebyshev nodes. Some estimates of error of the derivatives approximation in terms of the error of best approximation by polynomials are derived. Our results are sharp. Secondly, for weighted approximation in Lp-norm, we determine the strongly asymptotic orders for the average errors of function approximation by the Hermite interpolation sequence which based on the Chebyshev nodes on the1-fold integrated Wiener space. In the sense of Information-Based Complexity, if permissible information functionals consist of standard information, then the average errors of derivative approximation by above mentioned interpolation sequence are weakly equivalent to the corresponding sequences of minimal average radii of nonadaptive information.