Topics in Nonlinear Approximation

近似理论的中心任务之一是确定被近似的对象（函数、曲面等）的近似和光滑性质之间的联系，提出的研究计划的主要目标是进一步调查（类）函数（曲面，体等）的光滑性之间的确切联系。以及它们的近似值 *一般来说，非线性逼近方法比线性逼近方法表现得好得多，并且非线性流形的逼近阶比线性空间的元素的逼近阶好得多，这是研究非线性逼近技术的主要原因。拟议的研究计划福尔斯分为以下主要类别：*（A）多项式和样条逼近：逼近空间的表征，光滑度的测量，加权和未加权逼近，同时逼近和应用;*（B）约束逼近和插值：约束逼近空间的特征（有和没有权重），逼近阶，柯尔莫哥洛夫，约束函数类的线性和伪维宽度，多变量约束逼近，惠特尼型插值结果k-单调函数和应用，近似k-单调函数，放松约束;*（C）自适应算法：数据适应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。每一个类别都由几个子类别组成。例如，提出者计划研究的一些具体问题是：*（i）新的光滑模及其性质，（ii）通过根据新的光滑模测量的光滑度表征各种类的逼近阶，（iii）加权多项式逼近的加权光滑度测量和杰克逊和惠特尼型定理，（iv）加权多项式逼近，（v）（分段线性）多项式的多元约束逼近，（vi）（加权）Lp范数的形状保持逼近。

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.