Title: Deep Univariate Polynomial and Conformal Approximation Show Chinese title

Title: 深度单变量多项式和共形逼近

Abstract: A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal transformations. We show that deep approximations to $|x|$ on $[-1,1]$ achieve exponential convergence with respect to the degrees of freedom. Computational experiments suggest that a composite of two and three polynomial layers can give more accurate approximations than a single polynomial with the same number of coefficients. We also study the related problem of reducing the Runge phenomenon by composing polynomials with conformal transformations. Show Chinese abstract

Abstract: 深度近似是由多个简单函数层组合定义的近似函数。 我们研究使用由低次数多项式或简单共形变换组成的层对单变量函数进行深度近似。 我们证明在$[-1,1]$上对$|x|$的深度近似随着自由度呈指数收敛。 计算实验表明，两个或三个多项式层的组合可以比具有相同系数数量的单个多项式提供更精确的近似。 我们还研究了通过将多项式与共形变换组合来减少龙格现象的相关问题。