Topics in Nonlinear Approximation

建议的研究计划的主要目标是进一步调查的连接，在各种设置，（类）函数和它们的逼近阶之间的光滑性。该程序的主要类别是：（一）多项式和样条逼近：新的光滑模及其性质，用新的光滑模度量的光滑性刻画各类逼近阶，加权多项式逼近（杰克逊，惠特尼和伯恩斯坦型定理），同时多项式逼近权，多项式和样条光滑模的等价性，（B）约束逼近和插值：约束函数类的宽度;约束逼近空间的特征，约束函数类的精确逼近阶，多元约束逼近，q-单调函数逼近，松弛约束。具体的研究课题有：一.新的光滑模及其应用。在2014-2015年发表的一系列论文中，与合著者一起，提出者引入了新的光滑模，并应用它们获得了Lp中函数的代数多项式逼近的Jackson型估计，并证明了匹配逆定理，从而通过多项式逼近度获得了各种光滑函数类的构造性特征。这些模正是Lp（拟）范数中约束逼近的直接结果所需要的。他们的类似物已成功地用于建立各种等价关系的几类函数在L\infty，但类似的问题，约束近似在Lp仍然是开放的，需要新的技术。随着新模数的引入，现在知道这些估计应该采取什么形式。二.加权约束和无约束近似。建议的研究的另一个主题是加权多项式逼近和加权逼近的各类功能具有一定的形状属性。在2014-2016年，提出者在所有Lp，p>0，（拟）范数中获得了匹配的正逆定理，这些范数由取决于近似多项式次数的加倍权重的某些平均值加权，并且对于具有零点和奇点的加倍权重获得了类似的结果（其存在使得近似过程明显更加困难）。在这方面仍有许多问题需要调查。此外，有几乎没有结果的加权逼近的各类函数具有一定的形状属性，并建议继续研究所有这些主题。

The main goal of the proposed research program is further investigation of the connection, in various settings, between smoothness properties of (classes of) functions and their approximation orders. The main categories of the program are: (A) polynomial and spline approximation: new moduli of smoothness and their properties, characterization of approximation orders of various classes via smoothness measured in terms of the new moduli of smoothness, weighted polynomial approximation (Jackson, Whitney and Bernstein-type theorems), simultaneous polynomial approximation with weights, equivalence of moduli of smoothness of polynomials and splines, and applications; (B) constrained approximation and interpolation: widths of constrained function classes; characterization of constrained approximation spaces, exact approximation orders of constrained function classes, multivariate constrained approximation, approximation by q-monotone functions, relaxing constraints. Some of specific research topics are: I. New moduli of smoothness and applications. In a series of papers published in 2014-2015, together with co-authors, the proposer introduced new moduli of smoothness, applied them to obtain Jackson-type estimates for approximation of functions in Lp by means of algebraic polynomials, and proved matching inverse theorems as well, thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by polynomials. These moduli are precisely what's needed for direct results for constrained approximation in the Lp (quasi)norms. Their analogs have been successfully used to establish various equivalence relations for several classes of functions in L\infty, but analogous problems for constrained approximation in Lp are still open and require new techniques. With the introduction of the new modulus it is now known what form these estimates should take. II. Weighted constrained and unconstrained approximation. Another topic of the proposed research is weighted polynomial approximation and weighted approximation of various classes of functions having certain shape properties. In 2014-2016, the proposer obtained matching direct and inverse theorems in all Lp, p>0, (quasi)norms weighted by certain averages of doubling weights depending on the degree of approximating polynomials, and analogous results were obtained for doubling weights having zeros and singularities (whose presence makes the approximation process significantly more difficult). There are numerous problems in this area that still need to be investigated. Additionally, there are almost no results on weighted approximation of various classes of functions having certain shape properties, and research on all of these topics is proposed to be continued.