函数逼近论是现代数学的一个重要分支。以往主要考虑的是定义在环面、方体、球面、球体等加权函数空间中函数逼近问题，这些问题的理论大多已经发展的相当完善。但是加权的一般紧黎曼流形上函数逼近问题还有很多有待解决。本项目中，我们考虑加权紧黎曼流形上的函数逼近问题，包括正逆定理、Kolmogorov n-宽度和线性n-宽度的估计。

The theory of approximation is a significant branch in modern mathematics. Previously the problems in the theory of weighted approximation were about approximation in the function space on the torus, cube, sphere, ball and so on. The research in all the special compact Riemann manifolds has almost been done. But there are many problems left about approximation on general weighted compact Riemann manifold. In the proposal, we consider the problems of approximation on the weighted compact Riemann manifold. It includes direct and inverse theorems and the estimates of Kolmogorov n-width and linear n-width.