光滑函数类上的几个逼近问题

项目名称： 光滑函数类上的几个逼近问题

项目编号： No.11201104

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 段立芹

作者单位： 上海师范大学

项目金额： 22万元

中文摘要： 逼近的核心问题是寻找用简单函数代替复杂函数的有效方法并研究其逼近误差。线性逼近是比较自然且易实施的逼近方法，近来，在信号和图像处理等问题的推动下，非线性m-项逼近得到了广泛的研究。Temlyakov主要研究了具有一定光滑性的Banach空间及一些函数类上的非线性m-项逼近并给出了实现最优阶的算法，贪婪算法。许多重要函数类的线性逼近特征的研究比较完整，但非线性逼近在这些函数类上的渐近行为还有待于进一步研究。 在本项目中，首先，研究广义Besov函数类在一些特殊基及不同字典下的非线性最佳m-项逼近和不同贪婪算法的收敛阶。其次，研究各向异性Sobolev函数类在非紧嵌入到连续空间时由标准信息的恢复问题并给出在一致和随机框架下恢复问题的渐近阶。最后，研究由各向异性Sobolev函数类确定的第二类Fredholm积分方程在不同框架下局部解和整体解的数值逼近并给出数值逼近误差的渐近阶。

中文关键词： 贪婪逼近；最优恢复；数值积分；随机逼近；积分方程

英文摘要： The core problem of approximation continues to be the development of efficinet methods for replacing general functions by simpler functions and study the errors of approximation. Linear approximation is relatively natural and easily implementary methods. Recently, driven by some numerical problems from signal/image processing, the direction of approximation theory have rapidly moved toward nonlinear m-term approximation. Temlyakov mainly investigated the nonlinear m-term approximation on the Banach spaces and some classes of functions with smoothness and gave the optimal algorithm in the sense of order, greedy algorithm. Linear approximation characters of many important classes of functions have been studied completely, while the asymptotic behaviour of nonliear approximation on the classes of functions is still open. The purpose of this project is as follows. First,we will investigate the asymptotic orders of the nonlinear best m-term approximation and the convergence rates of different greedy algorithms by some special bases and dictionaries on the generalized isotropic and anisotropic Besov classes. Next, we study the recovery on the anisotropic Sobolev by the standard information,i.e.,function values under the condition of non-imbedding into the space of continuous functions and give the asymptotic orders

英文关键词： greedy approximation；optimal recovery；integral integration；randomized approximation；integral equation