Rate of approximation and smoothness of functions

给定一个函数（在某些类和某些域中），用更简单的对象来近似它是方便和有用的，例如用多项式（代数的、三角的或球面的）。我们将从很多方面研究逼近函数的程度和函数的平滑程度之间的关系。将研究各种平滑度量和给定度量的不同级别之间的关系。由于函数的平滑性及其与更简单的一类函数的距离是近似理论的基石，我希望部分研究将被近似理论的其他科学家，相关领域和任何对定量估计感兴趣的人使用。在过去的几年里，我成功地与几位博士后一起工作（目前与a . Prymak合作，他从2006年7月1日起获得太平洋数学研究所博士后奖学金），我打算在未来五年内继续与博士后或学生一起参与研究。我的论文显然会引起一些人的兴趣，因为它们经常被引用（Math Sci Net在过去六年中被100多位作者引用了大约300次），而且它们也经常被下载（例如，在Journal of Approximation上，我的两篇论文在几个月的时间里都在下载论文的前10名中）。

Given a function (in some class and on some domain), it is convenient and useful to approximate it by simpler objects, for example by polynomials (algebraic, trigonometric or spherical).  Many aspects of the relations between how well we approach the function and how smooth that function is will be studied.  The relations between various measures of smoothness and different levels of a given measure will be investigated.  As the smoothness of a function and its distance from a simpler class of functions are building blocks in approximation theory, I expect parts of the research will be used by other scientists in approximation theory, allied fields and whoever is interested in quantitative estimates.In the last few years I worked successfully with several post-doctoral fellows (presently with A. Prymak who was awarded a Pacific Institute of Mathematics Postdoctoral fellowship from 1 July 2006) and I intend to continue my participation in research with Postdoctoral fellows or students in the next five years.My papers are clearly of interest to some as they are often cited (Math Sci Net mentions about 300 citations by over 100 authors in the last six years) and they are also often downloaded (e.g. in the Journal of Approximation two of my papers were in the top 10 of downloaded papers for a few months).