Multivariate Approximation

逼近理论研究通过使用更简单和更容易计算的函数来逼近一般的、可能复杂的函数的能力。解决了一元函数逼近理论中的许多问题。然而，这些方法并不总是容易地推广到对多变量函数的逼近。我提出的研究的主要目的是改进现有的多元函数逼近的方法，并创造新的方法。我将集中讨论与光滑性度量(我们如何容易地测量或“预测”逼近的精度)、多项式的不等式(证明各种理论结果的有用工具)和保形逼近(如果目标函数或对象具有某种几何形状，例如凸性，如何构造也具有这种形状的逼近)有关的问题。在所提出的研究中，凸性的概念出现在不同的语境中，许多论点具有非常几何的味道。培养高素质人才的机会将不计其数。

Approximation theory studies the ability to approximate general, possibly complicated, functions by using simpler and more easily calculated functions. Many problems in approximation theory of functions of one variable have been solved. However, the methods do not always easily generalize to the approximation of functions of several variables. The main goal of my proposed research is to improve existing and create new methods for the approximation of functions of several variables. I will focus on questions related to measures of smoothness (how can we easily measure or "predict" the accuracy of approximation), inequalities for polynomials (useful tools for proving various theoretical results) and shape preserving approximation (if the target function or object has a certain geometric shape such as convexity, how to construct an approximation which also possesses such a shape). In the proposed research, the concept of convexity appears in different contexts and many arguments have very geometric flavour. There will be numerous opportunities for training of highly qualified personnel.