Multivariate Approximation

近似理论研究的是用更简单、更容易计算的函数来近似一般的、可能复杂的函数的能力。解决了一元函数逼近论中的许多问题。然而，这些方法并不总是容易推广到多元函数的逼近。我所提出的研究的主要目标是改进现有的和创建新的方法，用于多变量函数的近似。我将专注于与光滑性度量（我们如何能够轻松地测量或“预测”近似的准确性），多项式不等式（证明各种理论结果的有用工具）和保形近似（如果目标函数或对象具有某种几何形状，如凸性，如何构建也具有这种形状的近似）相关的问题。在所提出的研究中，凸性的概念出现在不同的背景下，许多论点具有很强的几何味道。将有许多机会培训高素质的人员。

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.