The Problem Of Approximation In The Measure Spaces

Approximation is to use simple instead of complex, concrete instead ofabstract. Because the function is a bridge between theory with practice and is abasis that mathematical methods research practical problems. Therefore, Theresearch of approximation theory has an important theoretical and practicalsignificance in mathematics and in the engineering practice.Polynomial functions are the universal approximation tool in theapproximation theory, since they are concerned with add operation and multiplyoperation so that their operations are simple and easy to implement on the computer.Bernstein operator and all kinds of Polynomial interpolation operators have been themain research subjects in the approximation theory. In the recent years, there anumber of papers have been studying these operators and accumulated many mainresearch results in the sense of worst element, some problems need to be studied inthe sense of average.The paper have four chapters, the first chapter mainly introduces base conceptssuch as Wiener space, Bernstein operator etc. The second chapter mainly considersthe average errors of linear combinations Bernstein-Kantorovich operators in theWiener space. The third chapter mainly considers the average errors of bivariateLagrange interpolation that its Chebyshev nodes are arbitrary in the Wiener spaceand multivariate Lagrange interpolation based on the Chebyshev nodes are same inthe integrated Wiener space. The integrated average in the linear combinations B-Koperators are difficult to calculate approximation errors, after multilayer groupingsummation and some analysis techniques of integral used, the ideal results areobtained. Results indicate that the linear combinations B-K operators andnon-combination B-K operators are consistent in the Wiener space, only in a smallextent, the accuracy of the error was affected by properly adjusting thecombination coefficients.