Topics in Nonlinear Approximation

逼近理论的中心任务之一是确定逼近（函数，曲面等）的逼近性和平滑性之间的联系，所提出的研究计划的主要目标是进一步研究（类）函数（曲面，体等）的平滑性与其逼近阶之间的确切联系。一般来说，非线性近似方法的性能要比线性近似方法好得多，而且非线性流形近似的阶数要比线性空间元素近似的阶数好得多，这是研究非线性近似技术的主要原因。******提出的研究计划分为以下主要类别：***(A)多项式和样条逼近：逼近空间的表征，平滑度的度量，加权和非加权逼近，同时逼近和应用；***(B)约束逼近和插值：约束逼近空间（有权和无权）的表征、逼近阶数、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-单调函数逼近、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，“最优”网格生成。******这些类别中的每一个都由几个子类别组成。例如，提议者计划研究的一些具体问题是：***(i)光滑的新模及其性质，（ii）通过根据光滑的新模测量的光滑来表征各种类别的近似阶数，（iii）加权多项式近似的加权平滑度量和Jackson和Whitney型定理，（iv）带权重的同时多项式近似，（iv）带权重的多项式近似。(v)（分段线性）多项式的多元约束近似，（vi）（加权）Lp范数的保形近似。

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.