光滑函数类的熵数估计

This project is mainly devoted to studying the estimates of entropy numbers of different smooth classes of function. First we consider the estimates of entropy numbers of Sobolev classes with Jacobi weight on the interval [-1,1] in weighted Lebesgue space， we want to obtain the asymptotic order of corresponding enteopy numbers. Secondly，we study the entropy numbers of the space of classes with mixed smoothness in the anisotropic spaces and those of the anisotropic classes in the spaces of functions with mixed smoothness，we compute the asymptotic orders of corresponding enteopy numbers. Results will have broad applications in related field of approximation theory and play important roles in nonlinear functional analysis, compressed sensing, and the learning theory of functions.

本项目的主要内容是考虑不同光滑函数类的熵数估计。首先我们考虑区间[-1,1]上带有Jacobi权的Sobolev类在加权的Lebesgue空间尺度的熵数的估计，得到对应熵数的渐进阶。其次，研究周期区间[0,1]^d上各向异性的函数空间和具有混合光滑性的函数空间之间的熵数估计，计算得到对应的熵数的渐进阶。预期所得结果将会广泛的应用于逼近论理论相关方向，在非线性泛函分析，压缩传感、函数学习理论等领域发挥重要作用。