本项目主要研究区间[-1,1]，球面S^d, 球体B^d上Sobolev空间在概率框架下的函数逼近问题，给出概率框架下Sobolev空间的函数逼近、宽度、最优求积及最优恢复的渐近阶，再通过比较概率框架与最坏框架和平均框架下函数逼近和恢复的异同点，为函数逼近理论提供重要的理论及应用价值。另一部分研究内容是加权Paley-Wiener空间中的采样与插值和一些重要的加权不等式，给出采样和插值序列的必要密度条件。该研究内容与函数空间理论、逼近论及热门的压缩感知理论都有密切的联系。

This proposal contains two parts. In the first part, we investigate the approximation of functions on the interval [-1,1]，the sphere S^d and the ball B^d in the probabilistic case setting. We will estimate the asymptotic order of the Kolmogorov and linear widths of Sobolev classes and the asymptotic orders of the probabilistic error estimations. This results play a role in the theory of approximation of functions. The second part is devoted to studying the important weighted inequalities and the sampling and interpolating sequences for the weighted Paley-Wiener space. We will give some necessary density conditions for sampling and interpolating sequences. These results are widely used in approximation theory and harmonic analysis and compressed sensing.