Multivariate approximation

逼近理论研究通过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。单变量函数逼近论中的许多问题已经得到解决。然而，这些方法并不总是容易推广到多个变量的函数的近似。我提出的研究的主要目标是开发现有的方法并创建新的方法来逼近多个变量的函数。我将重点关注与平滑度测量（我们如何轻松测量或“预测”近似的准确性）和形状保持近似（如果目标函数具有特定形状，如何构造也具有这种形状的近似）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).