Trigonometric aproximation and function spaces with generalized smoothness

三角逼近和广义光滑函数空间

• 批准号: 241673541
• 负责人: Professor Dr. Hans-Jürgen Schmeißer
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/241673541?language=en
• 关键词: Trigonometric; aproximation; function; spaces; generalized

During the last years in a series of joint papers with Dr. K. Runovski a unified approach to approximation of functions in periodic Lebesgue spaces for the full range of parameters p between zero and infinity has been developed. It relies on the introduction of families of polynomial linear operators and their use as constructive approximation method. Classical approximation processes are contained as special cases. The key result is a General Equivalence Theorem. It means a general theorem on the equivalence of approximation error in an appropriate (quasi) metric, polynomial.K-Functional, generated by an associated differential operator, and modulus of smoothness, generated by an associated periodic function. Hereby it turned out that the equivalence of approximation error and modulus requires an adapted modulus of smoothness. According to the asymptotic behaviour of the modulus of smoothness new scales of function spaces of Sobolev-Besov type generalizing the classical spaces are introduced. The main goal of our research is to study these new smoothness spaces and their interrelations with corresponding approximation processes.We want to determine saturation order and saturation classes of approximation processes, polynomial K-functionals and generalized moduli of smoothness. Moreover, the characterization of function spaces via approximation processes, necessary and sufficient conditions for continuous and compact embeddings of Sobolev type will be investigated. Our method is based on the use of the above mentioned general equivalence theorem. New spaces and classical spaces are compared.The long-term successful cooperation with Dr. K. Runovski has been supported several times by German Research Foundation (DFG) as well as by AVH-Foundation. The present project serves also as starting point for cooperation of our research group with specialists from the Sevastopol branch of the Moscow State University in a field connected with the study of oscillations in various mathematical and physical problems.

在过去的几年中，已经开发了针对周期性的Lebesgue空间中功能近似的统一函数的统一方法，以介于零和无穷大之间的全部参数p。它依赖于引入多项式线性运算符的家族及其用作建设性近似方法。经典近似过程包含特殊情况。关键结果是一般等效定理。这是指适当（准）度量，多项式功能，由相关的差分运算符生成的近似误差等效的一般定理，以及由相关的周期函数生成的平滑度。因此，事实证明，近似误差和模量的等效性需要适应的光滑模量。根据平滑度模量的渐近行为，引入了Sobolev-Besov类型的功能空间的新量表，从而介绍了经典空间。我们研究的主要目的是研究这些新的平滑度空间及其相互关系，我们希望确定近似过程的饱和顺序和饱和类别，多项式k功能和平滑度的广义模量。此外，将研究通过近似过程的功能空间表征，即将研究Sobolev类型的连续和紧凑嵌入的必要条件。我们的方法基于上述一般等效定理的使用。比较了新的空间和古典空间。与K. Runovski博士的长期成功合作已受到德国研究基金会（DFG）的支持以及AVH-Founchation的支持。本项目也是我们研究小组与莫斯科州立大学Sevastopol分支的专家合作的起点，该领域与研究各种数学和物理问题的振荡有关。

项目成果

Moduli of Smoothness Related to Fractional Riesz-Derivatives

与分数 Riesz 导数相关的平滑模

• DOI: 10.4171/zaa/1531
• 发表时间: 2015
• 期刊: Zeitschrift Fur Analysis Und Ihre Anwendungen
• 影响因子: 1.2
• 作者: K. Runovski;H.-J. Schmeisser
• 通讯作者: H.-J. Schmeisser

Moduli of Smoothness Related to the Laplace-Operator

与拉普拉斯算子相关的平滑模

• DOI: 10.1007/s00041-014-9373-y
• 发表时间: 2014
• 期刊: Journal of Fourier Analysis and Applications
• 影响因子: 1.2
• 作者: K. Runovski;H.-J. Schmeisser
• 通讯作者: H.-J. Schmeisser