Topics in Nonlinear Approximation

所提出的研究计划的主要目标是进一步研究在各种设置下（类）函数的平滑性与其近似阶之间的联系。程序的主要类别有：(A)多项式和样条近似：光滑的新模及其性质，用光滑的新模测量的各种光滑的近似阶数的表征，加权多项式近似（Jackson， Whitney和bernstein型定理），带权的同时多项式近似，多项式和样条的光滑模等价及其应用；(B)约束逼近与插值：约束函数类的宽度；约束近似空间的表征，约束函数类的精确近似阶数，多元约束近似，q-单调函数近似，放松约束。具体的研究课题有：1 .光滑度新模及其应用。在2014-2015年发表的一系列论文中，与合著者一起引入了新的平滑模，并将其应用于用代数多项式逼近Lp中函数的jackson型估计，并证明了匹配逆定理，从而通过多项式逼近的程度得到了函数的各种平滑类的建设性表征。这些模正是在Lp（拟）范数中约束近似的直接结果所需要的。它们的类似物已经成功地用于建立L中若干类函数的各种等价关系\infty，但Lp中约束近似的类似问题仍然开放，需要新的技术。随着新的模量的引入，现在我们知道这些估计应该采取什么形式。2。加权约束和无约束近似。提出的另一个研究课题是加权多项式逼近和具有一定形状性质的各类函数的加权逼近。2014-2016年，在所有Lp， p>, 0,（拟）范数中，根据多项式的近似程度，用一定的加倍权值的平均值加权，得到了匹配的正逆定理，对于具有零和奇点的加倍权值，也得到了类似的结果（奇点的存在使得近似过程更加困难）。这方面还有许多问题有待研究。此外，对于具有一定形状性质的各种函数的加权逼近，目前几乎没有结果，建议对所有这些主题进行进一步的研究。

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.