在一些函数空间内研究有关的最佳逼近、算子逼近、插值逼近、有理逼近等逼近论的问题，给出相应的特征刻画、正定理、逆定理、Jackson型估计等结果；同时还要在一些函数空间内研究宽度理论与极值问题这一函数逼近理论中的重要课题，对一些重要函数类的各种宽度进行渐近估计和精确估计，找出相应的极子空间并建立相应的对偶定理，用泛函分析的思想和方法揭示不同提法的极值问题与宽度理论之间的联系。在此基础上还要研究一些神经网络、压缩感知等问题。数学物理中的好多近似求解问题是函数逼近理论的直接背景，而函数逼近理论研究的理论成果可以为数学物理问题的近似求解以及选择“最优的”求解方案提供理论分析依据，函数空间又是从各类实际问题中提出的数学模型。所以，这一项目是既有学术意义又有实际背景和应用前景的研究课题。

In certain function spaces we study the concern problems in the approximation theory such as best approximation, operator approximation, interpolation approximation, rational approximation etc., obtain the characteristic theorem, direct and inverse theorems as well as the Jackson-type estimate; besides, in certain function spaces we study the width theory and extreme value problem which is a significant research subject in the approximation theory of functions, give respectively the asymptotic estimates and precise estimates of various widths of some important function classes, construct the corresponding optimal subspaces and establish the corresponding duality theorems, use ideas and methods of functional analysis to reveal the relationship between extreme value problem in kinds of different formulations and width theory; what’s more, discuss neural networks on these bases. Many problems of approximate solution in Mathematics Physics are background of the approximation theory, on the other hand, research results of the approximation theory provide theoretical basis for the approximate solution and the selection of “the best” solution. Therefore, this project has not only academic significance but also practical background and application prospects.