Multivariate approximation

逼近理论研究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标是发展现有的方法，并创造新的方法来逼近多元函数。我将集中讨论与光滑性度量(如何轻松地测量或“预测”逼近的精度)和保形逼近(如果目标函数具有某种形状，如何构造也具有这种形状的逼近)相关的问题。

Approximation theory studies the ability to approximate general, possibly complicated, functions by 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 develop 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) and shape preserving approximation (if the target function has a certain shape, how to construct an approximation which also possesses such a shape).