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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740780
• 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.

拟议研究计划的主要目标是进一步研究在各种设置下函数（类）的平滑特性与其逼近阶数之间的联系。该程序的主要类别是：（A）多项式和样条逼近：新的平滑模及其属性，通过根据新的平滑模测量的平滑度表征各种类别的逼近阶数，加权多项式逼近（杰克逊，惠特尼和伯恩斯坦型定理），带权重的联立多项式逼近，模的等价 多项式和样条的平滑度及其应用； (B) 约束逼近和插值：约束函数类的宽度；约束逼近空间的表征、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。一些具体的研究课题是： 一、新的平滑模量及其应用。在2014-2015年发表的一系列论文中，提出者与共同作者一起引入了新的平滑模，应用它们通过代数多项式获得Lp中函数逼近的Jackson型估计，并证明了匹配逆定理，从而通过函数的逼近程度获得了各种平滑类函数的建设性表征： 多项式。这些模量正是 Lp（拟）范数中约束近似的直接结果所需要的。它们的类似物已成功用于为 L\infty 中的几类函数建立各种等价关系，但 Lp 中的约束逼近的类似问题仍然悬而未决，需要新技术。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。所提出研究的另一个主题是加权多项式逼近和具有某些形状属性的各类函数的加权逼近。 2014-2016年，提议者在所有Lp、p>0、（准）范数中获得了匹配的正向定理和逆向定理，这些（准）范数根据多项式的逼近程度，通过某些加倍权重的平均值进行加权，并且对于具有零和奇点的加倍权重（其存在使得逼近过程变得更加困难）也获得了类似的结果。该领域还有很多问题需要研究。此外，几乎没有关于具有某些形状属性的各类函数的加权近似的结果，并且建议继续对所有这些主题进行研究。