Deep Univariate Polynomial and Conformal Approximation

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.