Topics in Nonlinear Approximation

逼近理论的中心任务之一是确定被逼近的对象(函数、曲面等)的逼近与光滑性之间的联系，而所提出的研究计划的主要目标是进一步研究函数(曲面、物体等)(类)的光滑性之间的确切联系。以及它们的近似阶。 一般而言，非线性逼近方法的性能远远好于线性逼近方法，并且非线性流形的逼近阶远好于线性空间元素的逼近阶，这是研究非线性逼近技术的主要原因。 拟议的研究方案主要分为以下几类： (A)多项式和样条线逼近：逼近空间的特征、光滑度、加权和未加权逼近、同时逼近和应用； (B)约束逼近和内插：约束逼近空间的特征(有权和无权)，逼近阶，Kolmogorov，约束函数类的线性和伪维宽度，多元约束逼近，k-单调函数的Whitney型插值结果及其应用，k-单调函数的逼近，松弛约束； (C)自适应算法：数据适应的各向同性和各向异性三角剖分、收敛、误差估计、“最佳”网格生成。 这些类别中的每一个都由几个子类别组成。例如，提议人计划调查的一些具体问题包括： (I)新的光滑模及其性质，(Ii)用新的光滑模度量的光滑性刻画各类逼近阶，(Iii)带权光滑度量和加权多项式逼近的Jackson和Whitney型定理，(Iv)带权多项式的同时逼近，(V)用(分段线性)多项式的多元约束逼近，(Vi)(加权)Lp范数下的保形逼近。

One of the central tasks in Approximation Theory is to determine a connection between approximation and smoothness properties of what's been approximated (functions, surfaces, etc.), and the main goal of the proposed research program is further investigation of the exact connection between smoothness of (classes of) functions (surfaces, bodies, etc.) and their approximation orders. In general, nonlinear approximation methods perform much better than linear ones, and the order of approximation  from nonlinear manifolds is much better than the order of approximation by the elements of linear spaces, which is the main reason for investigating nonlinear approximation techniques. The proposed research program falls into the following main categories: (A) Polynomial and Spline Approximation: characterization of approximation spaces, measures of smoothness, weighted and unweighted approximation, simultaneous approximation, and applications; (B) Constrained Approximation and Interpolation: characterization of constrained approximation spaces (with and without weights), approximation orders, Kolmogorov, linear and pseudo-dimensional widths of constrained function classes, multivariate constrained approximation, Whitney type interpolatory results for k-monotone functions and applications, approximation by k-monotone functions, relaxing constraints; (C) Adaptive Algorithms: data adapted isotropic and anisotropic triangulations, convergence, error estimates, "optimal" mesh generation. Each of these categories consists of several subcategories. For example, some of the specific problems that the proposer is planning to investigate are: (i) New moduli of smoothness and their properties, (ii) Characterization of approximation orders of various classes via smoothness measured in terms of the new moduli of smoothness, (iii) Measure of smoothness with weights and Jackson and Whitney type theorems for weighted polynomial approximation, (iv) Simultaneous polynomial approximation with weights, (v) Multivariate constrained approximation by (piecewise linear) polynomials, (vi) Shape preserving approximation in (weighted) Lp norms.