Topics in Nonlinear Approximation

非线性近似主题

• 批准号: RGPIN-2020-05678
• 负责人: Kopotun, Kirill
• 金额: $ 1.31万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=717916
• 关键词: Topics; Nonlinear; Approximation

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.

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，函数(类)的光滑性与它们的逼近阶之间的联系。程序的主要类别是：(A)多项式和样条逼近：新的光滑模及其性质，用新的光滑模刻画各类逼近阶，加权多项式逼近(Jackson，Whitney和Bernstein型定理)，带权的同时多项式逼近，多项式和样条的光滑模的等价性，及其应用；(B)约束逼近和内插：约束函数类的宽度；约束逼近空间的刻画，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。 一些具体的研究课题包括： 一、新的平滑模数及其应用。 在2014-2015年发表的一系列论文中，作者与合著者一起，引入了新的光滑模，并应用它们得到了函数在Lp中用代数多项式逼近的Jackson-型估计，并证明了匹配逆定理，从而通过函数的多项式逼近程度得到了各类函数光滑性的构造性刻画。这些模正是LP(准)范数中约束逼近的直接结果所需要的。它们的类比已被成功地用于建立L中几类函数的各种等价关系，但线性规划中的约束逼近的类比问题仍是一个开放的问题，需要新的技术。随着新的模数的引入，现在就知道这些估计应该采取什么形式了。 加权约束和无约束逼近。 该研究的另一个主题是具有一定形状性质的各类函数的加权多项式逼近和加权逼近。在2014-2016年间，作者得到了所有Lp，p&gt；0，(拟)范数上的匹配正逆定理，其加权平均取决于逼近多项式的程度，对于具有零点和奇点的重权的情形也得到了类似的结果(奇点的存在使逼近过程变得更加困难)。这一领域仍有许多问题需要调查。此外，关于具有一定形状性质的各类函数的加权逼近几乎没有结果，建议继续研究这些课题。