本文考虑三个方面的问题：单形上Bernstein-Durrmeyer算子的逼近；球面上连续模的等价性；球面、球体和单形上的加权Sobolev类的Kolmogrov宽度、线性宽度和Gelfand宽度的渐进阶的刻画。

This thesis concerns mainly with the following three topics on the approximation of functions: approximation by Bernstein-Durrmeyer operator on a simplex; equivalence between different kinds of moduli of smoothness on the unit sphere; the asymptotic orders of the Kolmogorov widths, linear widths and Gelfand widths of the weighted Sobolev classes on the unit sphere, on the unit ball and on the simplex.