The Average Error Of The Bernstein Etc. Interpolation Approximate Function (and Derivative) In The Wiener Space

In this paper ,on one hand ,we establish the weakly asymptotic order of the classical Bernstein interpolation sequence approximate function(or derivative)in the Wiener space(or 1-fold integrated Wiener space),on the other hand,we discuss the asymptotically order for the average error of Lagrange interpolation sequence, Hermite-Fejer interpolation sequence and Hermite interpolation sequence based on the Chebyshev nodes on the 1-fold integrated Wiener space. According to the contents this paper is divided into three chapters.In the first chapter, we give the preface.In the second chapter,we establish the weakly asymptotic order of the classical Bernstein interpolation sequence approximate function(or derivative)in the Wiener space(or 1-fold integrated Wiener space).From our results, it can be known that the weakly asymptotic order of the information-based algorithm whose recover function (or derivative) is the classical Bernstein interpolation sequence is below that of the corresponding minimal information radius whose permissible information operator is function value.Therefore,we can know that under the mean of statistics,the classical Bernstein interpolation sequence approximate function(or derivative) are not ideal computing tool to realize optimal information-based operation.In the third chapter, we obtain the average errors(asymptotic order or weakly asymptotic order) of the Lagrange interpolation sequence, the Hermite-Fejer interpolation sequence and the Hermite interpolation sequence based on the Chebyshev nodes on the 1-fold integrated Wiener space. From our results we know that the average error of the Lagrange interpolation sequence and the Hermite interpolation sequence based on the Chebyshev nodes in the 1-fold integrated Wiener space equal weakly to the average error of their corresponding optimal approximation polynomial in the 1-fold integrated Wiener space,and as a kind of information-based operation,they have simple form and their recover functions are polynomials,in the 1-fold integrated wiener space,their average error equal weakly to the corresponding minimal information radius whose permissible information operators class is function values(or Hermite data).Therefore,we know that under the mean of statistics, interpolation operators are not only ideal algorithm for realizing optimal approximation polynomials computation,but also ideal computing tool for realizing optimal information- based operation,and the property of their recover functions are good.