On the approximation of real continuous functions by series of solutions of a single system of partial differential equations
Volume 104 / 2006
Colloquium Mathematicum 104 (2006), 57-84
MSC: 34A05, 26E10, 35C05.
DOI: 10.4064/cm104-1-4
Abstract
We prove the existence of an effectively computable integer polynomial $P(x,t_0,\dots ,t_5) $ having the following property. Every continuous function $f: {\mathbb R}^s \to {\mathbb R} $ can be approximated with arbitrary accuracy by an infinite sum $$ \sum_{r=1}^{\infty} H_r(x_1,\dots ,x_s) \in C^{\infty}({\mathbb R}^s) $$ of analytic functions $H_r $, each solving the same system of universal partial differential equations, namely $$ P\bigg( x_{\sigma}; H_r , \frac{\partial H_r}{\partial x_{\sigma}} , \dots , \frac{{\partial}^5 H_r}{\partial x_{\sigma}^5} \bigg) = 0 \quad\ (\sigma =1, \dots ,s) . $$
